Wednesday July 12, 2023 - 10:00 - 2AB45 - Ehud De Shalit (Hebrew University, Giv'at-Ram)
Difference equations over elliptic function fields
Abstract: A recent theorem of Adamczewski, Dreyfus, Hardouin and Wibmer (2021) states that two power series, which satisfy a p-difference equation and a q-difference equation with polynomial coefficients, for multiplicatively independent numbers p and q, none of which is a rational function, are algebraically independent over the field of rational functions. We shall discuss an analogous theorem for power series satisfying \phi- and \psi- difference equations with elliptic functions as coefficients, where \phi and \psi are independent isogenies. While some ideas, like the use of difference Galois theory, are common to the rational and the elliptic theories, there are several new phenomena, having to do (a) with issues of periodicity (b) with the existence of non-trivial vector bundles over elliptic curves.
Friday June 16, 2023 - 13:30 - 2AB40 - Francesco Esposito
Geometric interpretation of approximations of Nichols algebras
Abstract: Nichols algebras are graded Hopf algebras playing a prominent role in the study of pointed Hopf algebras and comprising symmetric algebras, exterior algebras and the positive part of quantized universal enveloping algebras. Kapranov and Schechtman have given a geometric interpretation of primitive bialgebras in terms of factorizable perverse sheaves. Through this equivalence, Nichols algebras correspond to IC complexes. In ongoing joint work in progress with Giovanna Carnovale and Lleonard Rubio y Degrassi, we give a geometric interpretation as well of approximations of Nichols algebras. This allows to reformulate geometrically various open problems on Nichols algebras.
Friday June 9, 2023 - 13:30 - 2AB40 - Vasily Golyshev (ICTP, IITP)
Congruences via fibered motives
Abstract: I'll show how the existence of certain Fano topologies might lead to congruences between automorphic forms. A particular type of congruences between paramodular forms and Hilbert modular forms will be discussed.
Friday May 26, 2023 - 14:30 - 2AB40 - Giulia Gugiatti (ICTP Trieste)
Towards homological mirror symmetry for the Johnson-Kollár surfaces
Abstract: Homological mirror symmetry predicts a categorical equivalence between the complex geometry (the B-side) of a Fano variety and the symplectic geometry (the A-side) of its mirror. In this talk I will discuss homological mirror symmetry for certain log del Pezzo surfaces, known as Johnson-Kollár surfaces, and their Hodge-theoretic mirrors. These surfaces fall out of the standard mirror constructions since they have empty anticanonical linear system. I will describe the derived category of coherent sheaves of the stacks associated to the surfaces, and I will discuss some preliminary results obtained on the A-side. The result on the B side is joint with Franco Rota, while the work on the A-side is in progress with Franco Rota and Matt Habermann.
Monday May 15, 2023 - Daniel Disegni (Ben-Gurion University)
Algebraic cycles and p-adic L-functions for Rankin-Selberg motives
Abstract: The conjectures of Beilinson-Bloch-Kato predict that for a (smooth, proper) variety of dimension 2N-1 over a number field, the existence of nontrivial algebraic cycles of “arithmetic middle dimension” N-1 should be detected by L-functions. Moreover, the relevant Selmer group should be generated by algebraic cycles. I will talk about a “case study” that confirms a variant in p-adic coefficients, for a certain product of unitary Shimura varieties uniformised by complex unit balls. The result comes from a formula for the p-adic height of an explicit cycle (for curves, the cycle is a Heegner point). The proof is based on a comparison of relative-trace formulas. Joint work with Wei Zhang.
Friday April 21, 2023 - Juan Esteban Rodriguez Camargo
Solid locally analytic representations and D-modules (Joint work with Joaquín Rodrigues Jacinto)
Abstract: In this talk I will report some progress in the theory of solid locally analytic representations and different geometric interpretations. I will explain its relation with the theory of D-modules of Ardakov and how some basic properties or constructions are understood geometrically.
Thursday April 20, 2023 - 14:30 - 2AB40 - Andreas Hochenegger
Relations among P-twists
Abstract: On a K3 surface, rational curves and line bundles give rise to interesting autoequivalences of its derived category, so-called spherical twists. It was shown by P. Seidel and R. Thomas, that these spherical twists are mirror-dual to Dehn twists in symplectic geometry. Moreover, they showed that given a chain of rational curves, the associated spherical twists satisfy braid relations. Generalising from K3 surfaces to hyperkähler varieties, D. Huybrechts and R. Thomas showed that the corresponding generalisation of the spherical twists are P^n-twists. In this talk, I will introduce these autoequivalences, and speak about the possible relations among them. This talk is about joint work with Andreas Krug.
Friday April 14, 2023 - 14:45- 2AB40 -Heer Zhao
Log p-divisible groups and semi-stable Galois representations
Abstract: Let K be a complete discrete valuation field of mixed characteristic (0,p), let R be the ring of integers of K, and let G be the absolute Galois group of K. A famous conjecture of Fontaine (proved by Fontaine, Laffaille, Breuil, and Kisin) states that an integral p-adic representation of G with Hodge-Tate weights in [0,1] is crystalline iff it arises from a p-divisible group over R. We generalize the above to: an integral p-adic representation of G with Hodge-Tate weights in [0,1] is semi-stable iff it arises from a log p-divisible group over R (endowed with the canonical log structure). This is based on a joint work with Alessandra Bertapelle and Shanwen Wang.
Friday March 24, 2023 - Alberto Vezzani
Motivic monodromy and p-adic cohomology theories
Abstract: In this talk, we will discuss some recent advances in the theory of motives in the context of rigid analytic geometry. In particular, we offer a new definition of the Hyodo-Kato cohomology, purely defined on the generic fiber, without making any reference to log schemes or the log-de Rham Witt complex. As a consequence, we can construct Clemens-Schmidt-style complexes in the mixed characteristic setting, confirming an expectation of Flach and Morin, and simplify our proof of the p-adic weight monodromy conjecture for smooth projective hypersurfaces. This is a joint work in progress with Federico Binda and Martin Gallauer.
Tuesday February 14, 2023 - 14:30 - 2AB40 - Robert Cass
Geometrization of the mod p Satake isomorphism
Abstract: The classical Satake isomorphism relates the spherical Hecke algebra of a reductive group G over a local field F to representations of the Langlands dual group. When F is of mixed characteristic (0,p) and the Hecke algebra has characteristic prime to p, the Satake isomorphism has been geometrized by Zhu, Yu, Fargues-Scholze, and Richarz-Scholbach using techniques from p-adic geometry. In this talk, we consider the case where the Hecke algebra has characteristic p. I will speak on my recent joint work with Y. Xu, where we geometrize the mod p Satake isomorphism of Herzig and Henniart-Vignéras using mod p étale sheaves on Witt vector affine flag varieties. Our methods involve the constant term functors inspired from the geometric Langlands program, especially the geometry of certain generalized Mirković-Vilonen cycles. The situation is quite different from l-adic sheaves (l \neq p) because only three of the six functors preserve constructible sheaves.
Friday February 10, 2023 - 14:30 - 1BC50 - Thibaud van den Hove
The integral motivic Satake equivalence
Abstract: For a reductive group G over a field k, geometric Satake gives an equivalence between the category of equivariant perverse sheaves on the affine Grassmannian of G and the category of representations of the Langlands dual group of G. Depending on the field k, one can use different cohomology theories, such as Betti cohomology, étale cohomology, (arithmetic) D-modules, ... On the other hand, the representation category of the Langlands dual group remains the same, depending only on the coefficients of the cohomology theory. In this talk, I will explain how to construct a version of the Satake equivalence using a universal cohomology theory, i.e., motivically, generalizing most previously known instances of geometric Satake. This is joint work with Robert Cass and Jakob Scholbach.
Thursday February 2, 2023 - 14:30 - 2AB40 - Alexandr Buryak (Higher School of Economics - Moscow)
Counting meromorphic differentials on the Riemann sphere: explicit formulas and a relation with mathematical physics
Abstract: Counting maps between Riemann surfaces is a classical problem in algebraic geometry and combinatorics, and it was studied for more than one hundred years. In recent years, a related problem of counting meromorphic differentials on Riemann surfaces attracted considerable interest. I will talk about a complete solution of this problem for the Riemann sphere that we obtained recently with Paolo Rossi and will show beautiful explicit formulas that we found.
Thursday December 15, 2022 - 12:30 - 1A150 - Giovanni Rosso (Concordia, Montreal)
Hirzebruch--Zagier cycles in p-adic families and adjoint L-values
Abstract: Let E/F be a quadratic extension of totally real fields. The embedding of the Hilbert modular variety of F inside the Hilbert modular variety of E defines a cycle, called Hirzebruch--Zagier cycle. Thanks to work of Hida and Getz--Goresky, it is known that the integral of a Hilbert modular form g for E over this cycle detects if g is the base change of a Hilbert modular form for f, and in this case the value of the integral is related to the adjoint L-function of f. In this talk we shall present joint work with Antonio Cauchi and Marc-Hubert Nicole, where we show that the Hirzebruch--Zagier cycles vary in families when one considers deeper and deeper levels at p. We shall present applications to \Lambda-adic Hilbert modular forms and adjoint p-adic L-functions.
Monday December 5, 2022 - 15:00 - via Zoom - Simon Pepin Lehalleur (Radboud University, Nimega)
Motivic mirror symmetry for Higgs bundles
Abstract: The moduli spaces of semistable Higgs bundles over a smooth projective curve for two Langlands dual groups are conjecturally related by a form of mirror symmetry, the semi-classical limit of the Geometric Langlands correspondence. In the case of \(SL_n\) and \(PGL_n\), Hausel and Thaddeus conjectured that this mirror symmetry should be reflected in a specific identity of (twisted orbifold) Hodge numbers, which was then proven by Groechenig, Wyss and Ziegler. I will explain that this identity lifts to an isomorphism of (Voevodsky) motives with rational coefficients. The method is based on another proof of the Hausel-Thaddeus conjecture due to Maulik and Shen, combined with extensions of our previous result that motives of moduli of \(GL_n\)-Higgs bundles are direct summands of motives of abelian varieties. I will start with a brief introduction to motives and to moduli spaces of bundles on curves. Joint work with Victoria Hoskins (Nijmegen).
Thursday November 24, 2022 - 14:00 - room 1BC45 - Ju-Feng Wu (University of Warwick)
Overconvergent Eichler-Shimura morphisms for Siegel modular forms
Abstract: A classical theorem of Eichler and Shimura tells us that the Betti cohomology of modular curves can be understood by spaces of modular forms. This theorem admits an arithmetic avatar provided by Faltings. In the early 2010s, Andreatta--Iovita--Stevens announced a partial generalisation of this theorem to p-adic families by constructing the so-called `overconvergent Eichler-Shimura morphisms'. In this talk, based on joint work with Hansheng Diao and Giovanni Rosso, I will explain how to use perfectoid method to construct the overconvergent Eichler--Shimura morphisms for Siegel modular forms. Such a strategy is inspired by the work of Chojecki--Hansen--Johansson in the case of automorphic forms over compact Shimura curves.
Thursday November 24, 2022 - 15:15 - room 1BC45 - Michel Brion (Grenoble)
Actions of finite group schemes on curves
Abstract: Every action of a finite group scheme G on a variety admits a projective equivariant model, but not necessarily a normal one. As a remedy, we introduce and explore the notion of G-normalization. In particular, every curve equipped with a G-action has a unique projective G-normal model, characterized by the invertibility of ideal sheaves of all orbits. Also, G-normal curves occur naturally in some questions on surfaces in positive characteristics.
Wednesday November 3, 2022 - 15:00 - room 1BC50 - Antoine Chambert-Loir (Paris Cité)
Real differential forms and currents on Berkovich spaces
Abstract: I will describe a theory of real differential forms and currents on nonarchimedean spaces in the sense of Berkovich. The construction involves a bigraded version of differential forms invented by Aaron Lagerberg in the context of tropical geometry, nonarchimedean tropicalizations à la Bieri-Groves, and ultimately takes place on paralinear subspaces of Berkovich spaces. We have analogues of the Poincaré-Lelong equation and of the Bedford-Taylor calculus of products of closed positive currents. Formal geometry gives rise to interesting examples, as well as the particular case of abelian varieties. (Joint work with Antoine Ducros).
Monday September 26, 2022 - 11:30 - 2AB45 - Hironori Shiga (Chiba Univ.)
K3 Hypergeometric Modular Form
Abstract: In 1977 at the conference on several complex variables in Cortona (Organized by Vesentini and Bombieri) the speaker has proposed to initiate the study of “the K3 modular function”. Now, 45 years have passed. In my talk I will give just an outline of the development of this theme during almost half a century, mainly focused on the results obtained among my colleagues. As an important anecdote, in 2003 Prof B. Chiarellotto stayed in Chiba, during the stay he gave a conference talk in Hayama (Near Kamakura) on the Lamé equation. In the after talk discussion he suggested the possibility of the hypergeometric modular function coming from the Lauricella equation. I considered this suggestion for many years, and it flourished as several explicit examples of the Shimura canonical model modular functions for his higher dimensional complex multiplication theory. There are various results connected with the theme. We must look at the individual articles to see the detailed arguments. For example: [1] A.Nagano and H. Shiga, Geometric Interpretation of Hermitian Modular Forms via Burkhardt Invariants, Transformation Groups (2022). [2] A. Nagano and H. Shiga, On Kummer-like surfaces attached to singularity and modular forms, Mathematische Nachrichten, to appear, arXiv:2012.11954 2022. [3] Hironori Shiga, K3 Hypergeometric modular forms, Sugaku (2022), Review Paper in Japanese (to appear).
Wednesday September 21, 2022 - 15:30 - room 701 - Giordano Cotti (Lisbona)
Quantum differential equations, isomonodromic deformations, and derived categories
Abstract: The quantum differential equation (qDE) is a rich object attached to a smooth projective variety X. It is an ordinary differential equation in the complex domain which encodes information of the enumerative geometry of X, more precisely its Gromov-Witten theory. Furthermore, the asymptotic and monodromy of its solutions conjecturally rules also the topology and complex geometry of X. These differential equations were introduced in the middle of the creative impetus for mathematically rigorous foundations of Topological Field Theories, Supersymmetric Quantum Field Theories and related Mirror Symmetry phenomena. Special mention has to be given to the relation between qDE's and Dubrovin-Frobenius manifolds, the latter being identifiable with the space of isomonodromic deformation parameters of the former. The study of qDE’s represents a challenging active area in both contemporary geometry and mathematical physics: it is continuously inspiring the introduction of new mathematical tools, ranging from algebraic geometry, the realm of integrable systems, the analysis of ODE’s, to the theory of integral transforms and special functions. This talk will be a gentle introduction to the analytical study of qDE’s, their relationship with derived categories of coherent sheaves (in both non-equivariant and equivariant settings), some applications to number theory, and a theory of integral representations for its solutions. The talk will be a survey of the results of the speaker in this research area.
Wednesday July 6, 2022 - 14:00 - 2AB40 - David Smith (Padova)
A model-theoretic proof of Hilbert's Nullstellensatz
Abstract: Model theory, a sub-field of mathematical logic, studies relationships between formal language expressing statements and mathematical structure. With comparatively less formal rigour than other sub-topics of logic, model theory has been known to have ties to algebraic and diophantine geometry. The purpose of this talk is to give a brief introduction to model theory whilst illustrating how model-theoretic techniques could be used to prove algebro-geometric results
Thursday June 29, 2022 - 11:00 - 1BC45 - Michele Fornea (Columbia)
Plectic Jacobians and Hodge theory
Abstract: Gehrmann, Guitart, Masdeu and myself recently proposed, and gave evidence for, plectic generalizations of Stark-Heegner points. The construction is p-adic, cohomological, and unfortunately lacking a satisfying geometric interpretation. Nevertheless, we formulated precise conjectures on the algebraicity of plectic points and their significance for the arithmetic of higher rank elliptic curves. In this talk I will report on work in progress on the Archimedean side of the story where geometry has a prominent role: I will describe a collection of complex tori — called plectic Jacobians — associated with the plectic Hodge structure appearing in the middle degree cohomology of Hilbert modular varieties. Interestingly, the Oda-Yoshida conjecture can be used to prove that plectic Jacobians are modular abelian varieties defined over \bar{Q}. Moreover, the existence of exotic Abel-Jacobi morphisms (mapping zero-cycles to plectic Jacobians) further highlights the arithmetic appeal of the construction.
Thursday June 21, 2022 - 14:00 - room 2AB40 and on Zoom - Anna Barbieri (University of Padova)
On quiver categories associated to quadratic differentials
Abstract: In a paper in 2015, Bridgeland and Smith identified some moduli spaces of quadratic differentials with simple zeroes on a Riemann surface with some spaces of stability conditions on certain categories. This identification passes through associating a quiver Q and a triangulated category D(Q) to a triangulation of a marked bordered surface with boundaries defined by a quadratic differential. I will review this correspondence and discuss how the picture changes when quadratic differentials with zeroes of arbitrary order are considered. This is part of a joint work with M. Moeller and J. So.
Thursday June 9, 2022 - 12:15 - room 2AB40 - Antonio Cauchi (Concordia University, Montreal)
Quaternionic diagonal cycles and instances of the Birch and Swinnerton-Dyer conjecture for elliptic curves
Abstract: In the early nineties, Kato’s Euler system of Beilinson elements and the theory of Heegner points revolutionised the arithmetic of (modular) elliptic curves over the rationals. For instance, the former led Kato to proving instances of the Birch and Swinnerton-Dyer conjecture for twists of elliptic curves over Q by finite order characters. While the theory of Heegner points was generalised to elliptic curves E/F defined over totally real number fields, Kato’s result has not found its natural extension to twists of E/F yet. More recently, the theory of diagonal cycles, arising from the work and collective effort of Bertolini, Darmon, Rotger, Seveso, and Venerucci, has proven to be a fertile environment for proving new instances of the Birch and Swinnerton-Dyer conjecture for elliptic curves over the rationals. The aim of this talk is to discuss joint work in progress with Daniel Barrera, Santiago Molina, and Victor Rotger on the generalisation of the theory of diagonal cycles to quaternionic Shimura curves over totally real number fields F and its application to extending Kato’s result for twists of elliptic curves E/F by Hecke characters of F of finite order.
June 1, 2022 - 14:30 - room 2BC30 - Alexandr Buryak (Higher School of Economics, Moscow)
Counting meromorphic differentials with zero residues on the Riemann sphere and the KP hierarchy
Abstract: Let us fix n integers (multiplicities) and consider the set of configurations of n points on the sphere such that there exists a meromorphic differential with zero residues and with the divisor given by the linear combination of the n points with the prescribed multiplicities. Let us factorize this set by the automorphism group of the sphere. The resulting set is finite if and only if the number of poles is equal to n-2. The cardinalities of these sets can be considered as natural analogs of Hurwitz numbers. In our joint work with Paolo Rossi and Dimitri Zvonkine we proved that these numbers are exactly the coefficients in the equations of the dispersionless KP hierarchy.
Thursday June 29, 2022 - 11:00 - 1BC45 - Michele Fornea (Columbia)
Plectic Jacobians and Hodge theory
Abstract: Gehrmann, Guitart, Masdeu and myself recently proposed, and gave evidence for, plectic generalizations of Stark-Heegner points. The construction is p-adic, cohomological, and unfortunately lacking a satisfying geometric interpretation. Nevertheless, we formulated precise conjectures on the algebraicity of plectic points and their significance for the arithmetic of higher rank elliptic curves. In this talk I will report on work in progress on the Archimedean side of the story where geometry has a prominent role: I will describe a collection of complex tori — called plectic Jacobians — associated with the plectic Hodge structure appearing in the middle degree cohomology of Hilbert modular varieties. Interestingly, the Oda-Yoshida conjecture can be used to prove that plectic Jacobians are modular abelian varieties defined over \bar{Q}. Moreover, the existence of exotic Abel-Jacobi morphisms (mapping zero-cycles to plectic Jacobians) further highlights the arithmetic appeal of the construction.
Thursday June 21, 2022 - 14:00 - room 2AB40 and on Zoom - Anna Barbieri (University of Padova)
On quiver categories associated to quadratic differentials
Abstract: In a paper in 2015, Bridgeland and Smith identified some moduli spaces of quadratic differentials with simple zeroes on a Riemann surface with some spaces of stability conditions on certain categories. This identification passes through associating a quiver Q and a triangulated category D(Q) to a triangulation of a marked bordered surface with boundaries defined by a quadratic differential. I will review this correspondence and discuss how the picture changes when quadratic differentials with zeroes of arbitrary order are considered. This is part of a joint work with M. Moeller and J. So.
Thursday June 9, 2022 - 12:15 - room 2AB40 - Antonio Cauchi (Concordia University, Montreal)
Quaternionic diagonal cycles and instances of the Birch and Swinnerton-Dyer conjecture for elliptic curves
Abstract: In the early nineties, Kato’s Euler system of Beilinson elements and the theory of Heegner points revolutionised the arithmetic of (modular) elliptic curves over the rationals. For instance, the former led Kato to proving instances of the Birch and Swinnerton-Dyer conjecture for twists of elliptic curves over Q by finite order characters. While the theory of Heegner points was generalised to elliptic curves E/F defined over totally real number fields, Kato’s result has not found its natural extension to twists of E/F yet. More recently, the theory of diagonal cycles, arising from the work and collective effort of Bertolini, Darmon, Rotger, Seveso, and Venerucci, has proven to be a fertile environment for proving new instances of the Birch and Swinnerton-Dyer conjecture for elliptic curves over the rationals. The aim of this talk is to discuss joint work in progress with Daniel Barrera, Santiago Molina, and Victor Rotger on the generalisation of the theory of diagonal cycles to quaternionic Shimura curves over totally real number fields F and its application to extending Kato’s result for twists of elliptic curves E/F by Hecke characters of F of finite order.
June 1, 2022 - 14:30 - room 2BC30 - Alexandr Buryak (Higher School of Economics, Moscow)
Counting meromorphic differentials with zero residues on the Riemann sphere and the KP hierarchy
Abstract: Let us fix n integers (multiplicities) and consider the set of configurations of n points on the sphere such that there exists a meromorphic differential with zero residues and with the divisor given by the linear combination of the n points with the prescribed multiplicities. Let us factorize this set by the automorphism group of the sphere. The resulting set is finite if and only if the number of poles is equal to n-2. The cardinalities of these sets can be considered as natural analogs of Hurwitz numbers. In our joint work with Paolo Rossi and Dimitri Zvonkine we proved that these numbers are exactly the coefficients in the equations of the dispersionless KP hierarchy.
May 19, 2022 - 15:00 - room 1BC45 - Vasily Golyshev (IITP Moscow)
On Deligne and Birch--Swinnerton-Dyer periods
Abstract: We give closed hypergeometric expressions for Birch--Swinnerton-Dyer volumes of certain rank 4 weight 3 Calabi-Yau motives presumed to be of analytic rank 1, and compare them numerically to the first derivative of their $L$-functions at $s=2$.
May 12, 2022 - 11:45 - room 2AB40 - Dominik Bullach (King’s College London)
Dirichlet L-series at s = 0 and the scarcity of Euler systems
Abstract: In 1989 Coleman made a distribution-theoretic conjecture which predicts that every Euler system `for \Q' should essentially be cyclotomic in nature. In this talk I will discuss work joint with Burns, Daoud and Seo which not only allows us to prove Coleman's Conjecture but also provides an elementary interpretation of, and thereby more direct strategy to proving, the equivariant Tamagawa Number Conjecture (eTNC) for Dirichlet L-series at s = 0. As a concrete application we obtain an unconditional proof of the `minus part' of the eTNC over finite abelian CM extensions of totally real fields.
April 29, 2022 - 15:00 - room 2AB40 - Matteo Tamiozzo (Imperial College London)
Perfectoid Jacquet-Langlands correspondence and the cohomology of Hilbert modular varieties.
Abstract: The work of Tian-Xiao on the Goren-Oort stratification for quaternionic Shimura varieties provides a geometric incarnation of the Jacquet-Langlands correspondence, and leads to a geometric approach to level raising of quaternionic automorphic forms. I will describe a perfectoid version of Tian-Xiao's result, and explain how it can be used, joint with geometric properties of the Hodge-Tate period map, to prove vanishing theorems for the cohomology of quaternionic Shimura varieties with torsion coefficients. This is joint work with Ana Caraiani.
April 28, 2022 - 11:45 - room 2BC30 - Matteo Tamiozzo (Imperial College London)
Geodesics on modular surfaces and functional transcendence.
Abstract: The approach to the André-Oort conjecture suggested by Pila-Zannier relies on the study of (complex) subvarieties of Shimura varieties (and their universal cover) from the viewpoint of functional transcendence. I will first recall the main results of this theory in the simplest case of the product of two modular curves. I will then deduce analogous theorems for real subvarieties of a modular curve seen as a real algebraic surface.
April 27th, at 14.30 room SR701 - Alex Kuronya (Goethe-Universität Frankfurt)
Newton-Okounkov bodies and measures of local positivity
Abstract: Newton-Okounkov bodies are a convex-geometric way to understand the vanishing behaviour of global sections of line bundles in an asymptotic sense. While they have interesting connections with a variety of mathematical topics including representation theory, Diophantine approximation, and mathematical physics, here we will focus on their relationship with birational geometry. More specifically, we will see how they encode local positivity of line bundles and how they give rise to interesting invariants that are mostly unexplored even in the toric case.
April 22, 2022 -15:00 - online seminar room SR701 - Aleksander Horawa (University of Michigan)
Motivic action on coherent cohomology of Hilbert modular varieties
Abstract: A surprising property of the cohomology of locally symmetric spaces is that Hecke operators can act on multiple cohomological degrees with the same eigenvalues. We will discuss this phenomenon for the coherent cohomology of line bundles on modular curves and, more generally, Hilbert modular varieties. We propose an arithmetic explanation: a hidden degree-shifting action of a certain motivic cohomology group (the Stark unit group). This extends the conjectures of Venkatesh, Prasanna, and Harris to Hilbert modular varieties.
April 14, 2022 -11:45 - online seminar room 2AB40 - Óscar Rivero (Warwick University)
Anticyclotomic Euler systems and diagonal cycles.
Abstract: In this talk, I will discuss joint work with Raul Alonso and Francesc Castella where we construct an anticyclotomic Euler system for the Rankin-Selberg convolutions of two modular forms, using p-adic families of generalized Gross-Kudla-Schoen diagonal cycles. As applications of this construction, we prove new cases of the Bloch-Kato conjecture in analytic rank zero (and results towards new cases in analytic rank one), and a divisibility towards an Iwasawa main conjecture. If time permits, I will also consider the case of the symmetric square of a modular form, where the key ingredient is a factorization formula for the triple product p-adic L-function.
April 7, 2022 -11:45 - online seminar room 2AB40 - Giada Grossi (LAGA, Université Sorbonne Paris Nord)
(Anti)cyclotomic main conjectures for elliptic curves at Eisenstein primes
Abstract: I will discuss work in progress with F. Castella and C. Skinner on the anticyclotomic and cyclotomic Iwasawa main conjectures at Eisenstein primes p, generalising our earlier paper with J. Lee and the results of Greenberg and Vatsal. As a consequence, we obtain new results on the p-part of the Birch-Swinnerton-Dyer conjecture. .
April 6, 2022 - 14:30 - room 2BC30 - Enrico Fatighenti (Università La Sapienza, Roma)
Varietà di Fano di tipo K3 e loro proprietà
Abstract. Le varietà di Fano di tipo K3 sono una speciale classe di Fano studiate per il collegamento con la geometria hyperkaehler, le loro interessanti proprietà di razionalità, e molto altro ancora. In questo talk ripercorreremo alcuni risultati recenti, ottenuti in una serie di lavori in collaborazione con Bernardara, Manivel, Mongardi e Tanturri, focalizzati alla costruzione esplicita di esempi e allo studio delle loro proprietà Hodge-teoretiche.
March 31, 2022 -11:45 - online seminar - Andrea Marrama (Ecole Polytechnique)
Filtrations of Barsotti-Tate groups via Harder-Narasimhan theory
Abstract: Let p be a prime number and let R be a complete valuation ring of rank one and mixed characteristic (0,p). Given a Barsotti-Tate group H over R, its p-power-torsion parts possess a natural "Harder-Narasimhan" filtration, introduced by Fargues in analogy with the theory of vector bundles over a smooth projective curve over a field. One may wonder when these filtrations build up to a filtration of the whole Barsotti-Tate group H. I will present some sufficient conditions in this direction, especially in the case that the endomorphisms of H contain the ring of integers of a finite extension of $\mathbb Q_p$ .
March 24, 2022 -11:45 - room 2AB40 - Francesco Battistoni (Università degli Studi di Milano)
Title: Arithmetic equivalence for number fields and global function fields
Abstract: Two number fields K and L are said to be arithmetically equivalent if, for almost every prime number p, the factorizations of p in the rings of integers of K and L are analogous (in a precise sense that will be explained). A completely similar definition can be given for finite extensions of a function field F(T), where F is a finite field.
In this talk we discuss the concept of arithmetic equivalence in both contexts, focusing on the similarities and the differences between the two cases. In particular, we will show a group-theoretic analogue of the problem and we will explain the relation between arithmetic equivalence and equality of certain zeta functions (the classical Dedekind zeta function for number fields, a more complicated function for function fields). Finally, we will show how to produce examples of equivalent but not isomorphic fields in both contexts.
March 17, 2022 -11:45 - online seminar - Riccardo Pengo (École normale supérieure de Lyon)
Limits of Mahler measures and (successively) exact polynomials
Abstract: Mahler's measure is a height function of fundamental importance in Diophantine geometry, protagonist of a celebrated problem posed by Lehmer. The work of Boyd has shown that Lehmer's problem can be approached by studying Mahler measures of multivariate polynomials, and that the latter are often linked to special values of $L$-functions. In this seminar, I will talk about a generalization of the work of Boyd, obtained jointly with François Brunault, Antonin Guilloux and Mahya Mehrabdollahei, in which we find a class of sequences of polynomials whose Mahler measures converge. Furthermore, we provide an explicit upper bound for the error term, and an asymptotic expansion for a particular family of polynomials, whose terms share all the peculiar property of being "exact". If time permits, I will explain more in detail this notion of exactness, and talk about a generalization of it (the notion of "successive exactness"), studied jointly with François Brunault, which is related to a certain "weight loss" of the $L$-functions whose special values are conjecturally related to the Mahler measure of the polynomial in question.
March 10, 2022 - 11:45 - online seminar - Lennart Gehrmann (University of Duisburg-Essen)
Algebraicity of polyquadratic plectic points
Abstract: Heegner points play an important role in our understanding of the arithmetic of modular elliptic curves. These points, that arise from CM points on Shimura curves, control the Mordell-Weil group of elliptic curves of rank 1. The work of Bertolini, Darmon and their schools has shown that p-adic methods can be successfully employed to generalize the definition of Heegner points to quadratic extensions that are not necessarily CM. Numerical evidence strongly supports the belief that these so-called Stark-Heegner points completely control the Mordell-Weil group of elliptic curves of rank 1. In this talk I will report on a plectic generalizations of Stark-Heegner points. Inspired by Nekovar and Scholl's conjectures, these points are expected to control Mordell-Weil groups of higher rank elliptic curves. I will give strong evidence for this expectation in the case of polyquadratic CM fields. This is joint work with Michele Fornea.
March 2, 2022 - 14:00 - room 1BC50 - Fabien Trihan (Sophia University, Japan)
On the Tamagawa Number Conjecture in characteristic p
ABSTRACT: We will talk on the Tamagawa Number conjecture for overconvergent F-isocrystals over varieties over finite fields. This is a joint work with O. Brinon and a work in progress.
February 17, 2022 - 15:30 - room 2BC60 - Alessandro Chiodo (IMJ-PRG, Paris 6 - France)
Coomologia g-orbifold e risoluzioni crepanti
ABSTRACT: Si considera un orbifold X di tipo Gorenstein, il suo spazio grezzo S e un sua risoluzione Y crepante (discrepanza zero K_Y=f*K_S). I gruppi di coomologia di queste due desingolarizzazioni Y->S e X->S sono isomorfi nel senso della coomologia orbifold di Chen-Ruan (teorema di Yasuda). Elana Kalashnikov ed io osserviamo che questo isomorfismo si generalizza per una variante della coomologia orbifold, che chiamiamo la coomologia g-orbifold. Questo permette di spiegare una serie di teoremi di dualità mirror dimostrati precedentemente caso per caso da Artebani, Boissière, Bott, Comparin, Lyons, Priddis e Sarti.
January 20, 2022 - 14:00 - room 2AB45 - Xavier Blot (Weizmann institute, Israel)
Quantum tau functions and a Chiodo class
ABSTRACT: In 1990 Witten conjectured that a certain generating series of intersection numbers on the moduli space of curves is a so-called tau function of the KdV hierarchy. This was the first result connecting the geometry of the moduli space of curves and integrable systems. Recently, Buryak, Dubrovin, Guere and Rossi used the double ramification hierarchies and their quantization to introduce quantum tau functions. The goal of the talk is to present a conjectural description of the coefficients of the quantum tau functions in terms of intersection numbers with a particular class on the moduli space curves called a Chiodo class.
November 8, 2021 - 15:00 - room 1A150 - François Loeser (Sorbonne Université )
Monodromy and non-archimedean geometry
ABSTRACT: We will present an overview of recent results on mondoromy and Milnor fibers in relation with non-archimedean geometry and motivic integration.
October 28, 2021 - 15:00 - online - Frederik Benirschke (University of Chicago)
Equations and degeneration
Zoom meeting: https://unipd.zoom.us/j/5332063266
ABSTRACT: Linear subvarieties are a class of special subvarieties in the moduli space of differentials on Riemann surfaces, which are defined by linear equations among periods. After motivating the study of linear subvarieties, both from the viewpoint of algebraic geometry and dynamical systems, we discuss a framework to understand degenerations of families of differentials. The goal is to explain the statement “The boundary of a linear subvariety is linear
October 20, 2021 - 11:00 - room 2AB40 - Danilo Lewanski (IHES and IPhT - France)
Frobenius manifolds, moduli spaces of curves, and enumerative geometry: the topological recursion.
ABSTRACT: Topological recursion (TR) is a recently developed method, which can be thought as an algorithm recursively generating solutions to enumerative geometric problems. For instance, TR generates interesting numbers involved in Gromov-Witten theory, Mirror Symmetry, String Theory, different graphs counting on surfaces, Hurwitz theory, Weil-Petersson and Masur-Veech volumes, WKB analysis, Painlevé equations, polynomial invariants of knots, Hitchin systems, BPS states, and more. On the other hand, it provides a system of cohomology classes on the moduli spaces of curves with good properties — often a cohomological field theory, therefore in correspondence under certain conditions with the Frobenius manifolds introduced by Dubrovin. It is an exciting time for TR, many new results are being developed, and several conjectures await to be proved.
April 28, 2021 - 14:00 - David Homes (University of Leiden)
The double-double ramification cycle
ABSTRACT: A basic question in the geometry of Riemann surfaces is to decide when a given divisor of degree 0 is the divisor of a rational function (is principal). In the 19th century Abel and Jacobi gave a beautiful solution: one writes the divisor as the boundary of a 1-cycle, and the divisor is principal if and only if every holomorphic differential integrates to zero against this cycle. From a modern perspective it is natural to allow the curve and divisor to vary in a family, perhaps allowing the curve to degenerate to a singular (stable) curve so that the corresponding moduli space is compact. The double ramification cycle can then be seen as a virtual fundamental class of the locus in the moduli space of curves over which our divisor becomes principal. We will focus on two basic questions: where does the double ramification cycle naturally live, and what happens when we intersect two double ramification cycles? We will see why (logarithmically) blowing up the moduli space can make life easier. This is joint work with Rosa Schwarz, building on easlier work with Aaron Pixton and Johannes Schmitt.
Working seminar on work by Andreatta and Iovita
First meeting October 9, 2020 - 14:30. Next meetings: every Friday at 14:30
Scope of the seminars is to introduce base concepts in the theory developed in recent years by Fabrizio Andreatta and Adrian Iovita among the integral theory of authomorphic forms and to show some applications in the 1-dimensional case. Talks will be given in English. Schedule:
09/10/2020 Introduzione ai fibrati formali con sezioni marcate
16/10/2020 Fasci di sezioni: sottomoduli tagliati da un carattere e filtrazioni naturali
23/10/2020 Connessioni sulle sezioni di un fibrato formale con sezioni marcate
30/10/2020 Modello formale dello spazio di pesi p-adico e carattere universale
6/11/2020 Spazi di moduli di curve ellittiche e sottogruppo canonico
13/11/2020 Costruzione della torre di Igusa (finita)
20/11/2020 Costruzione dei fasci modulari sulla torre di Igusa, 1
27/11/2020 Costruzione dei fasci modulari sulla torre di Igusa, 2
June 10, 2020 - 14:00 - Emanuele Macrì (Université Paris-Saclay)
Superfici speciali in ipersuperfici cubiche di dimensione 4
ABSTRACT: In seguito ad articoli influenti di Harris, Hassett e Kuznetsov, ci si aspetta che l'ipersuperficie cubica complessa molto generale di dimensione 4 non sia razionale e che le cubiche razionali formino un'unione numerabile di divisori "speciali" nello spazio di moduli delle cubiche. Questi divisori speciali sono descritti mediante la teoria di Hodge. Una domanda di Hassett è se sia possibile caratterizzare questi divisori geometricamente: una cubica è in uno di questi divisori se e solamente se questa contiene una superficie "speciale". In questo seminario, basato su un lavoro in preparazione con Arend Bayer e Alex Perry, presenterò una risposta congetturale alla domanda di Hassett utilizzando spazi di moduli di complessi e categorie derivate. Questa congettura è verificata per un sottoinsieme numerabile di divisori. Infine, tempo permettendo, discuterò eventuali relazioni con la razionalità.
May 22, 2020 - 10:00 - Andres Sarrazola Alzate (Padova)
A Geometric Approach of Admissible Locally Analytic Representations
ABSTRACT: Let $\mathbb G$ be a split connected reductive group scheme over $\mathbb Z_p$. An important theorem in group theory is the localization theorem, demonstrated by A. Beilinson and J. Bernstein, and by J.L. Brylinsky and M. Kashiwara. This is a result about the D-affinity of the flag variety of the generic fiber $\mathbb G_{\mathbb Q_p} = \mathbb G\times_{Spec (\mathbb Z_p)} Spec(\mathbb Q_p)$. In mixed characteristic an important progress is found in the work of C. Huyghe and T. Schmidt. They give a partial answer by considering algebraic characters. The first part of this presentation will be dedicated to extend this correspondence (the arithmetic localization theorem) for arbitrary characters (over $\mathbb Z_p$). In the second part we will consider locally analytic representations. We will show that for an algebraic character, which is dominant and regular, the category of admissible locally analytic $\mathbb G(\mathbb Z_p)$-representations, with central character, it is equivalent to the category of coadmissible $\mathbb G(\mathbb Z_p)$-equivariant arithmetic modules over the family of formal models of the rigid analytic flag variety.
January 15, 2020 - room 701 - 14:00 - Daniele Zuddas (Università di Trieste)
Strutture complesse non Kähleriane su $R^4$
ABSTRACT: Calabi ed Eckmann dimostrarono in un famoso lavoro del '53 che lo spazio Euclideo $R^n$ ammette strutture complesse senza metriche di Kähler compatibili per ogni $n > 4$ pari. Lo scopo di questo seminario è mostrare un risultato analogo nel caso n = 4. Le tecniche usate sono molto diverse da quelle originali di Calabi-Eckmann, tuttavia le strutture complesse che si ottengono in dimensione 4 presentano certe analogie con quelle in dimensione alta. Questo lavoro è in collaborazione con Antonio J. Di Scala e Naohiko Kasuya.
December 11 2019 - room 1BC45 - 11:00 - Eleonora Romano (Warsaw)
Azioni toriche su varietà proiettive: Combinatorica VS Geometria Birazionale
Abstract: In questo seminario ci focalizzeremo su azioni di tori algebrici di dimensione uno su varietà complesse, lisce, proiettive di dimensione arbitraria. Adopereremo sia la combinatorica che deriva dall'azione del toro che strumenti propri della Geometria Birazionale. In particolare, rivisiteremo la teoria classica di aggiunzione secondo una nuova prospettiva derivante da tali azioni. Inoltre vedremo come questo nuovo approccio ci permette di ottenere dei risultati di classificazione di varietà speciali chiamate "small bandwidth". Come applicazione, inseriremo tali risultati nel contesto della classificazione di varietà di Fano contact di dimensione alta. Si tratta di un joint work con G. Occhetta, L. Sola' Conde e J. Wisniewski.
December 6, 2019 - room 1BC50 - 14:30 - Rahul Pandharipande (ETH Zürich)
Abel-Jacobi maps and double ramification cycles
Abstract: I will discuss several current developments in the study of Abel-Jacobi maps with connections to double ramification cycles, log geometry, and Gromov-Witten theory.
ovember 26, 2019 - room SR 701 - 15:15-16:15 - Sheng-Chi Shih (Univ. Lille)
On the Hilbert cuspidal eigenvariety at weight one Eisenstein points
Abstract: The irreducible components of the parallel weight Hilbert eigencurve for a totally real field F are either cuspidal or Eisenstein. The way how the two loci intersect plays a crucial role in the Iwasawa main conjecture. When the Deligne-Ribet p-adic L-function of a finite order totally odd character \phi of F has trivial zeros, any p-stabilization of the corresponding weight one Eisenstein series belongs to the Hilbert cuspidal eigencurve. In the case of elliptic modular forms, it was proved by Betina-Dimitrov-Pozzi that the cuspidal eigencurve is etale over the weight space at such points. In this talk, we will report an ongoing work with Adel Betina and Mladen Dimitrov that the Hilbert cuspidal eigencurve is etale over the weight space at such points as well when p is inert in F and the Leopoldt conjecture holds. Hence, each weight 1 Eisenstein point can be deformed into a unique cuspidal component in any direction. In addition, the cuspidal component of the eigencurve intersects with each Eisenstein component transversally at such points. The complexity of the geometry of the Hilbert cuspidal eigencurve at such points growing with the dimension of H^1(F,\phi) which equals the degree of F, a challenging question is to determine the extension classes occurring in Galois representations attached to cuspidal Hida families. A key step of our work is to construct p-ordinary irreducible Galois representations with values in certain local rings of the eigencurve and to compute the associated reducibility ideal. As applications, we give a new proof of the rank one abelian Gross-Stark conjecture relating the leading term of p-adic L-function of \phi and a non-zero algebraic L-invariant. This conjecture was first proved by Dasgupta-Darmon-Pollack under the assumption that a sum of two analytic L-invariances is non-zero. From our approach, we can show that such an assumption on the analytic L-invariances is true. Another application is that when F is a real quadratic field, Darmon-Pozzi-Vonk uses the uniqueness of the deformation of the weight 1 Eisenstein points in the anti-parallel direction to study the Gross-Stark unit.
November 26, 2019 - room SR 701 - 11:00-12:00 - Roberto Pagaria (Bologna)
Configuration spaces of surfaces
Abstract: In this talk we will introduce the ordered and unordered configuration spaces of a manifold. Our aim is to give an overview of the main classical results and then we focus on the Betti numbers in the case of surfaces. The main idea behind the aforementioned results is considering all these spaces simultaneously. The natural maps - that consist in adding or removing a point - will be described. Also inclusions between manifolds give remarkable properties. In the second part we will describe the Kriz model for rational homotopy type of configuration spaces. Using that model and some representation theory, we will compute the Betti and mixed Hodge numbers of the unordered configuration spaces of surfaces.
November 6, 2019 - room SR 701 - 15:30 - Annette Werner (Frankfurt)
Tropical Geometry of the Hodge Bundle
Abstract: After explaining some background on tropical and algebraic curves and their moduli spaces, we study the image of the tropicalization map connecting the algebraic and the tropical Hodge bundle. For every pair consisting of a stable tropical curve G plus a divisor in the canonical linear system on G, we obtain a combinatorial condition to decide whether there is a smooth curve over a non-Archimedean field whose stable reduction has G as its dual graph together with an effective canonical divisor specializing to the given one. This is joint work with Martin Möller and Martin Ulirsch.
Values of quaternionic modular forms.
Abstract: The Eichler-Shimizu-Jacquet-Langlands correspondence relates classical modular forms to modular forms on quaternion algebras. Trivial weight modular forms on definite quaternion algebras, which correspond to weight 2 elliptic modular forms, are simply functions on finite sets of ideal classes. We will explain some things one can say about the values of these functions, and describe applications to congruences of elliptic/Hilbert modular forms and nonvanishing of L-values.
October 31, 2019 - room SR701 - 13:30 - Nahid Walji
On the occurrence of Hecke eigenvalues in sectors of the complex plane.
Abstract: Let r be a non-selfdual cuspidal automorphic representation of non-solvable polyhedral type for GL(2) over a number field. We will show that r has a positive density of Hecke eigenvalues in any sector of size 150.42 degrees. The proof relies on functoriality results for automorphic representations and the ensuing analytic properties of various L-functions.
October 23, 2019 - room SR701 - 11:00 - Paolo Dolce (Udine)
Adelic geometry on arithmetic surfaces
Abstract: After briefly introducing the theory of 2-dimensional local fields and their topologies, we define the ring of adeles on an arithmetic surface completed with the fibres at infinity. We show that as topological group it is self-dual. Moreover fundamental adelic subspaces are self orthogonal with respect to a natural differential pairing, this result can be thought as a stronger version of the arithmetic 2-dimensional reciprocity laws. Joint work with W. Czerniawska.
October 17, 2019 - room 430 - 14:30 - Carolina Rivera Arredondo (Padova)
Height pairings of 1-motives
Abstract: We provide a generalization, in the context of 1-motives, of the height pairings constructed by B. Mazur and J. Tate for abelian varieties. Following their approach, we consider ρ−splittings of the Poincaré biextension of a 1- motive and establish results concerning their existence. When ρ is unramified this is guaranteed if the monodromy pairing of the 1-motive is non-degenerate. For ramified ρ, the ρ−splitting is constructed from a pair of splittings of the Hodge filtrations of the de Rham realizations of the 1-motive and its dual, generalizing previous constructions by R. Coleman and Y. Zarhin for abelian varieties. These ρ−splittings are then used to define local and global pairings.
Thursday July 11, 2019 - 2AB45 - 14:00 - Antonella Perucca (Luxembourg)
Multiplicative order of the reductions of algebraic numbers
Let $K$ be a number field, and let $\alpha$ be a non-zero algebraic number. For all but finitely primes $\mathfrak p$ of $K$ we can define the reduction of $\alpha$ modulo $\mathfrak p$ and its multiplicative order. We consider various results concerning the sequence (indexed by $\mathfrak p$) of the multiplicative order of $\alpha$ modulo $\mathfrak p$. We are interested in particular in the density of primes such that the multiplicative order is coprime to some given prime number $\ell$, or it lies in a given arithmetic progression. We also replace $\alpha$ by a finitely generated subgroup of $K^\times$. This is partly a joint work with Christophe Debry and Pietro Sgobba, and we rely on methods by Hasse and Ziegler.
Tuesday May 30, 2019 - 2AB40 - 14:00 - Alessandro Chiodo (Paris 6)
Simmetria speculare e automorfismi
ABSTRACT: Grazie a Borcea, Dolgachev, Nikulin e Voisin esiste una versione arricchita della simmetria speculare che si applica alle superfici K3. In effetti questo è un caso in cui l'enunciato classico è banale. Trattiamo questo enunciato arricchito come il punto di partenza di un enunciato di simmetria speculare in presenza di un automorfismo in dimensione qualunque. Per enunciare il teorema principale, rivisitiamo la corrispondenza di McKay che mette in relazione una singolarità e la sua risoluzione (lavoro in collaborazione con Kalashnikov e Veniani).
Friday, May 17th, 2019 - 1AD100 - 11:00 - Dimitri Zvonkine (CNRS Versailles)
Fractional quantum Hall effect and vector bundles over $M_{g,n}$ -- the first steps
ABSTRACT: Vector bundles of Laughlin states were introduced by physicists to study the fractional quantum Hall effect and their Chern classes are related to measurable physical quantities. We will explain how they are related to the vector bundle of theta-functions over the moduli space and perform the first steps in the computation of their Chern classes. Work in progress with Semyon Klevtsov.
Friday, May 17th, 2019 - 1AD100 - 9:30 - Nicola Pagani (Liverpool)
Different extensions of the double ramification cycles
ABSTRACT: Fix natural numbers g,n and integers d1, d2, ..., dn. The moduli space Mgn of n-pointed curves of genus g contains an interesting locus that parameterises pointed curves (C, p1, ..., pn) that admit a meromorphic function f such that div(f) equals \sum di pi. There is different ways of extending this cycle to the compactification of Mgn by means of stable n-pointed curves of arithmetic genus g. One way of extending this cycle is by means of the theory of relative stable maps, and another is by pulling back the Brill-Noether class w^0_0 via a (possibly rational) section to some compactified universal Jacobian. In this talk I will explain how the first can be seen as a particular case of the second (a joint work with David Holmes and Jesse Kass). If the ramification vector is of type (1, -1, 0,...0) then this gives an (unexpected to me) relation between two tautological classes.
Thursday, May 16th, 2019 - 2AB40 - 14:00 - Johannes Schmitt (ETH Zürich)
Admissible cover cycles in the moduli space of stable curves
ABSTRACT: Inside the moduli space of stable curves there are closed subsets defined by the condition that the curve C admits a finite cover of a second curve D with specified ramification behavior. I will show how these sets can be parametrized by nice smooth and proper moduli spaces. In many cases, this parametrization can be used to compute the fundamental class of such admissible cover loci in the cohomology group of the moduli space. This is joint work with Jason van Zelm.
Tuesday May 14, 2019 - 1BC45 - 14:30 - Gianluca Occhetta (Trento)
Manifolds with two projective bundles structures
ABSTRACT: Having a projective bundle structure makes a variety rather special, so having two different projective bundle structure should be quite uncommon, especially if we assume that the Picard group of the variety is two-dimensional (in this case the variety is Fano). On the other hand, such varieties appear in many different - apparently unrelated - contexts, for instance in the study of dual defective varieties, flops and horospherical varieties; therefore classification results for these varieties are very interesting. In my talk I will review old and new results about this problem.
Tuesday April 16, 2019 - room 701 - 11:30 - Daniele Turchetti (Dalhousie University, Halifax- Canada)
Triangulations of Berkovich curves and ramification
Given a smooth projective curve $C$ over a discretely valued field $K$, the semi-stable reduction theorem of Deligne-Mumford (1969) asserts that there is a finite extension $L/K$ such that $C\times_K L$ has a semi-stable model. Still, the problem to determine a minimal extension with this property remains open for a large class of curves. In this talk, I will present a strategy for studying this problem by looking at the analytification $C^{an}$ in the sense of Berkovich. After introducing the theory of Berkovich curves, I will explain how a special finite set of points of $C^{an}$, called minimal triangulation, encodes all the necessary information to retrieve $L$. When $L/K$ is tamely ramified, the minimal triangulation can be computed from the minimal regular model of $C$ with strict normal crossings. When $L/K$ is wild, this often can not be done, and I will classify and illustrate the pathologies that can appear. This is joint work with Lorenzo Fantini.
Tuesday April 2, 2019 -1BC45 - 14:00 - Roberto Pignatelli (Trento)
Rigid but not infinitesimally rigid compact complex manifolds
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Tuesday March 26, 2019 - 2BC30 - 14:00 - Michael Lönne (Bayreuth)
Monodromy groups and root lattices of elliptic fibrations
Every smooth elliptic surface contains a finite number of singular fibres, and a root lattice is associated to each type of fibre. The complement is a torus fibre bundle with monodromy in SL_2(Z). These invariants give rise to so called topological stratifications of the Miranda moduli space of Jacobian elliptic fibrations. With Klaus Hulek we looked closer at the case of K3 surfaces. We classify all irreducible closed subsets in the moduli space of Jacobian elliptic K3 which contain both a monodromy stratum and a root lattice stratum as an open subset.
Tuesday January 29, 2019 - room 1C150 - time 11:30 - Mama Foupouagnigni (Yaoundé, Cameroon)
On difference equations for orthogonal polynomials on special nonuniform lattices
In this talk, we derive an appropriate polynomial basis which is then used to: 1) provide the functional approach of the characterization theorem for classical orthogonal polynomials on nonuniform lattice; 2) providing algorithmic method to find explicit solution to holonomic divided-difference equation and 3) the derivation of some structure relations for classical orthogonal polynomials on nonuniform lattices such the determination of the coefficients of the connection and the linearisation problems involving classical orthogonal polynomials on nonuniform lattices. This is a collection of joint works with: Wolfram Koepf (University of Kassel, Germany), Salifou Mboutngam (University of Maroua, Cameroon), Maurice Kenfack-Nangho (University of Dschang, Cameroon), Daniel Duviol Tcheutia (University of Kassel, Germany) and Patrick Njionou Sadjang (University of Douala, Cameroon).
Thursday October 18, 2018 - room 2AB45 - 15:30 - Takashi Suzuki (Chuo University)
Duality for cohomology of local fields and curves with coefficients in abelian varieties
Abstract: I will explain a duality for cohomology of local fields and curves over perfect base (or residue) fields of positive characteristic with coefficients in abelian varieties. The cohomology mentioned here is equipped with a structure of a sheaf over a Grothendieck site called the "rational etale site" of the base field, and we consider a sheaf-theoretic relative duality. With this duality, we are able solve Grothendieck's duality conjecture in SGA 7 on special fibers of abelian varieties. Also, Tate-Shafarevich groups, as sheaves, are represented by unipotent algebraic groups. Cassels-Tate pairings are generalized with these geometric structures. I will then try to express my naive hope that, in this duality, abelian varieties and unipotent groups should be generalized to motives and curves should be generalized to morphisms between varieties. The big picture here is that there should be a relative duality and six operations formalism for mixed etale motives with $p$-torsion, a program that has yet to be developed.
Thursday October 18, 2018 - room 2AB45 - time 14:20 - Yukako Kezuka (Regensburg)
The p-part of Birch-Swinnerton-Dyer conjecture for the Gross family of elliptic curves
Abstract: Take $q$ to be any prime congruent to 7 modulo 8, and let $K= Q(\sqrt{-q})$. B. Gross proved the existence of an elliptic curve $A$ defined over the Hilbert class field $H$ of $K$ with complex multiplication by the ring of integers of $K$ and minimal discriminant $-q^3$. We study a large family of quadratic twists $E$ of $A$ for which the complex L-series of $E/H$ does not vanish at 1. We discuss the $p$-part of Birch-Swinnerton-Dyer conjecture for $E/H$ for any prime $p$ which splits in $K$, including $p=2$. This is a joint work in progress with J. Coates, Y.Li and Y. Tian.
Wednesday june 6, 2018 - room 2AB45 - time 14:30 - Antonio Cauchi (UCL)
Towards an Euler System for GSp(6)
Abstract: The theory of Euler systems is one of the most powerful tools available for studying the arithmetic of global Galois representations. For example, the work of Kato on the construction of an Euler system for modular forms has had applications towards cases of the Birch and Swinnerton-Dyer conjecture and the Iwasawa Main conjecture for modular forms. Recently, Lei, Loeffler and Zerbes, building on work of Bertolini, Darmon and Rotger, constructed an Euler system for Rankin-Selberg convolutions, proving new cases of the Bloch-Kato conjecture for the underlying representation. These techniques have been extended to the case of Hilbert modular forms and Siegel modular forms of genus 2. In this talk, I will explain how to construct Galois cohomology classes for Galois representations appearing in the middle degree cohomology of the Shimura variety of the similitude symplectic group GSp(6). These classes are conjectured to be constituents of an Euler system. As supporting evidence for this conjecture, I will show that these classes provide elements in the Iwasawa cohomology of these representations. This is joint work with Joaquin Rodrigues.
Wednesday May 30, 2018 - room 2AB40 - time 13:15 - Nicola Pagani (Liverpool)
Pull-backs of Brill-Noether cycles from universal Jacobians
Abstract: Let $M_{g,n}$ be the moduli space of smooth $n$-pointed curves of genus $g$. For a given vector of integers $(d_1,...,d_n)$ one can define a natural locus of pointed curves $(C, p_i)$ such that $O_C(\sum d_i p_i)$ admits a nonzero global section. We discuss how this can be extended to the moduli space of stable curves by interpreting it as the pullback of a cohomology class on (compactified) universal Jacobians. Because there are multiple ways to compactify the Jacobian, this leads to multiple classes related by wall-crossing. We explain why this gives an effective approach for the computation of the cohomology class of these cycles (in terms of tautological classes). The case where $d_1 + ... + d_n = 0$ is known in the literature as the "double ramification cycle" and it has attracted the attention of several mathematicians. This is a joint work with Jesse Kass.
Monday, May 28, 2018 - room 1BC45 - time 14:30 - Margarida Melo (Roma Tre)
On the tropical Torelli map
Abstract: The talk will survey a series of results on tropical Torelli map(s) based on joint work with Silvia Brannetti and Filippo Viviani with a view towards new developments and open questions.
Friday May 25, 2018 - room 2AB45 - time 14:30 - Paolo Rossi (Université de Bourgogne)
Intersection theory on the moduli spaces of stable curves and cohomological field theories
Abstract: In studying moduli spaces of stable curves the question of computing their cohomology (or Chow) ring is a natural but difficult one. The tautological ring is the smallest subring of cohomology which still captures, in some sense, the most important part of their topology. Recently new methods (involving cohomological field theories) were successfully employed by Pandharipande-Pixton-Zvonkine in computing tautological rings. In a joint work with Buryak and Guéré we have another approach (also involving CohFTs, but in a very different way), yielding results whose relation with PPZ is unclear.
Wednesday May 16, 2018 - room 2BC30 - time 13:15 - Slawomir Rams (Università Jagellonica, Cracovia)
Counting lines on surfaces, especially quintics
Abstract: The sharp bound on the number of lines on a smooth degree-4 surface in three-dimensional projective space has been shown only recently, whereas the maximal number of lines on smooth degree-d complex projective (hyper)surfaces for any fixed $d \geq 5$ is unknown. In my talk I will introduce certain rational functions on a smooth projective surface X in $\mathbb P^3$ which facilitate counting the lines on X. I will apply this to smooth quintics in characteristic zero to prove that they contain no more than 127 lines, and that any given line meets at most 28 others. I will construct examples which demonstrate that the latter bound is sharp (joint work with M. Schuett, LUH Hannover).
Wednesday May 2, 2018 - room Sala Riunioni VII piano - time 14:30 - Nicole Marc-Hubert (Marseille)
The Gross-Kohnen-Zagier theorem for p-adic families
Abstract: Given an elliptic curve E defined over the field of rational numbers and given an imaginary quadratic field K, one may define (using the theory of complex multiplication) a K-rational point of the elliptic curve, called a Heegner point. Heegner points are crucial tools for studying the arithmetic of elliptic curves; in particular, the celebrated theorem of Gross and Zagier relates, under suitable arithmetic assumptions, the Néron-Tate height of Heegner points and the leading term of the complex L-function of E over K. The Gross-Kohnen-Zagier theorem (GKZ), complementary to the Gross-Zagier theorem mentioned above, shows that, under suitable arithmetic assumptions, the relative positions of the Heegner points, as the imaginary quadratic field varies while the elliptic curve stays fixed, are encoded by the Fourier coefficients of a certain kind of modular form called a Jacobi form. Briefly put, Heegner points are generating series for Jacobi forms. In this talk, I will explain a variant of the GKZ theorem where we make all objects involved vary in p-adic families i.e., using crucially on Hida families of modular forms of varying p-adic weight. We like to view our result as the GL(2) instance of a p-adic Kudla program. Joint work with Matteo Longo.
Monday April 16, 2018 - room 2BC3 - time 14:30 - Matteo Penegini (Genova)
On Zariski multiplets of branch curves
Abstract: In this talk, I consider Zariski multiplets of plane singular curves obtained as branched curves of ramified covering of the plane by surfaces isogenous to a higher product with group $({\mathbb Z}/2{\mathbb Z})^k$. This is a joint work in progress with Michael Loenne.
Wednesday April 11, 2018 - room 2BC30 - time 14:00 - Andrea Fanelli (Düsseldorf)
Del Pezzo fibrations in positive characteristic
Abstract: In this talk, I will discuss some pathologies for the generic fibre of del Pezzo fibrations in characteristic p>0, motivated by the recent developments of the MMP in positive characteristic. The recent joint work with Stefan Schröer applies to deduce information on the structure of 3-dimensional Mori fibre spaces and answers an old question by János Kollár.
Wednesday March 21, 2018 - room SRVII - time 14:00 - Pietro de Poi (Udine)
Fano Congruences
Abstract: Congruences of lines in $\mathbb{P}^n$ are subvarieties of the Grassmannian of (co)dimension $n-1$. We study a special class of congruences $X$ defined by -forms. All such congruences are irreducible components of reducible linear congruences, and we shall denote their residual by $Y$. If the 3-form is general, we prove that $X$ is always a smooth Fano variety of index 3. Furthermore, the fundamental locus $F$ also is a Fano variety of index 3 if $n$ is odd and it is smooth if $n<10$. We study the Hilbert scheme of these congruences $X$, proving that the choice of the 3-form determines $X$ uniquely, except when $n=5$. The residual congruence $Y$ can be analysed in terms of the quadrics containing the linear span of $X$. In this way, we determine the singularities and the irreducible components of its fundamental locus. Joint work with Emilia Mezzetti, Daniele Faenzi and Kristian Ranestad.
< Wednesday March 7, 2018 - room 2BC30 - time 14:00 - Margherita Lelli Chiesa (L'Aquila)
Nikulin surfaces and moduli of Prym curves
Abstract: The relevance of K3 surface in the study of the moduli space of curves is well-established. Nikulin surfaces, that is, K3 surfaces endowed with a nontrivial double cover branched along eight disjoint rational curves, play a similar role at the level of the moduli space of Prym curves. I will report on a work in this direction joint with Knutsen and Verra. In particular, I will prove that a general Nikulin section of fixed genus lies exactly on one Nikulin surface with only a few exceptions occurring in low genus.
Thursday December 14, 2017 - room 2AB40 - time 13:20 - Ivan Fesenko (Nottingham)
Inter-universal Teichmueller Theory (IUT)
Monday July 3, 2017 - room 2BC60 - time 11:15 - Bernard Le Stum (Rennes)
Rigid cohomology and -adic spaces
Monday July 3, 2017 - room 2BC60 - time 10:00 - Nobuo Tsuzuki (Tohoku)
Constancy of Newton polygons of F-isocrystals on Abelian varieties and its application
Thursday June 1, 2017 - room 2BC30 - time 14:30 - Nicola Mazzari (Bordeaux)
Attorno ad un teorema di Serre e Tate
Abstract: Negli anni 70 Serre-Tate sviluppano la teoria delle deformazione di una varietà abeliana in caratteristica p>0 attraverso i loro gruppi p-divisibili. Ne descriveremo gli aspetti principali seguendo il lavori di Katz e Messing. Concluderemo enunciando alcuni sviluppi recenti ottenuti in collaborazione con A. Bertapelle.
Friday May 26, 2017 - room SRVII - time 11 - Michele Bolognesi (Montpellier)
A variation on a conjecture of Faber and Fulton
Friday May 19, 2017 - room 2AB45 - time 14:00 - Bryden Cais (Arizona)
Breuil-Kisin modules and crystalline cohomology
Abstract: The theory of Breuil-Kisin modules provides a powerful classification of stable lattices in crystalline p-adic Galois representations via certain semi-linear algebra structures over the power series ring Z_p[[u]]. On the other hand, the i-th integral p-adic etale cohomology of a smooth and proper scheme X over the ring of integers in a p-adic field provides such a stable Galois lattice, so has a Breuil--Kisin module attached to it. In this case, it is natural to ask if the associated Breuil--Kisin module can be described in terms of the cohomology of the scheme. Recent work of Bhatt, Morrow, and Scholze answers this question affirmatively, provided one is willing to extend scalars to the "big period ring" A_{inf}. I will explain how this result can be descended to give a cohomological construction of the original Breuil-Kisin module (i.e. without extending scalars) when i < p-1 and the crystalline cohomology of the special fiber of X is p-torsion-free in degrees i and i+1. This is joint work with Tong Liu.
Marc-Hubert NICOLE (Université d’Aix-Marseille)
Thursdays May 11 in SRVII, May 18 in 2BC30, May 25 2017, in 1BC45 - time 14:15 Three lectures on Drinfeld modules and Drinfeld modular forms à la Katz
Abstract: This series of lectures is my attempt to give a follow-up to the recent Padova seminar on Katz’s seminal 1972 paper titled « p-adic properties of modular schemes and modular forms », cf. the LNM 350 volume. I will essentially cover the analogue of Katz’s Chapters 1,2 and 3 suitably adapted to function fields of positive characteristic. In short, elliptic curves are replaced by Drinfeld modules; modular curves & forms by their Drinfeld analogues. In the presentation, I will stick to the setting of the rational function field F_q [T], q=p^n, p a prime number, and Drinfeld modules of rank 2 (also called elliptic modules). For details about the lectures (lecture #m should be very closely related to chapter #m of Katz’s paper), see below. In particular, new contributions will be discussed in the 2nd and 3rd lectures. Lecture 1 introduces the basic objects: Drinfeld- modules & modular curves/forms. Lecture 2 covers $\varpi$-adic Drinfeld modular forms à la Katz. As a bonus, I will explain a recent theorem of mine about the existence of n-1 generalized Hasse invariants for Drinfeld modular varieties for GL(n). Lecture 3 develops the explicity theory of the canonical subgroup and applications.
Thursday May 18, 2017 - room 2BC30 - time 11:30 - Giovanni Rosso (Cambridge)
Ordinary modular forms over some unitary Shimura varieties without ordinary locus
Abstract: If one wants to study congruence modulo p among modular forms, a key ingredients is the geometry of the ordinary locus and certain etale covers of it. Hida generalised these ideas to all PEL Shimura varieties with ordinary locus. A generalisation of the ordinary locus is is the mu-ordinary locus, which is never empty. Building on ideas and work of Goldring--Nicole-Hubert, Pilloni, and Hernandez, we shall explain a general construction of a pro-etale cover of the mu-ordinary locus (the Igusa tower) which can be used to develop Hida theory for all Shimura varieties. The talk will be very introductory and this is joint work (in progress) with Riccardo Brasca.
Thursday April 27, 2017 - room 2BC30 - time 14:30 - Daniel Caro (Caen)
$p$-basis and arithmetic $\mathcal{D}$-modules
Abstract: Let $k$ be a perfect field of characteristic $p>0$. Berthelot introduced an arithmetic theory of modules over the ring of differential operators over smooth $k$-varieties. In a common work with David Vauclair, we explain how to naturally extend these constructions for varieties having $p$-basis, e.g. smooth varieties over $k [[t]]$. This also extends Crew’s construction concerning $k [[t]]$.
Monday April 12, 2017 - room 1AD100- time 12:00 - Yves André (CNRS, Paris 6)
Recent progress on the homological conjectures in commutative algebra
Abstract: The homological conjectures form a skein of statements more or less connected to local cohomology and originating from problems in intersection theory in the 70’s. One of them is the Direct Summand Conjecture (M. Hochster 1973) which states that every (commutative noetherian) regular ring R is a direct summand, as R-module, of any finite ring extension. After a presentation of these conjectures in their historical context, we shall outline recent progresses obtained through the theory of perfectoid spaces in p-adic Hodge theory (including the Direct Summand Conjecture).
Monday April 10, 2017 - room 2BC30- time 14:15 - Yves André (CNRS, Paris 6)
Kummer extensions of rings of formal power series, and perfectoid algebras
Abstract: We consider Kummer extensions of $A := Z_p[[T_1, … , T_n]]$, obtained by adjoining the $p^i$th roots of unity and of some element $a$ of $A$, then inverting $p$ and taking the $p$-integral elements. It is very difficult to describe such extensions. They are not always flat, but Hochster’s conjecture predicts that they are pure, i. e. $A$ is a direct summand (as $A$-module) of any such extension. We shall explain how one can prove this conjecture by going to the completed colimit in $i$ and using perfectoid theory via a deformation argument.
Wednesday April 5, 2017 - room SRVII - time 16:00 - Andreas Knutsen (Bergen)
Title: Brill-Noether theory of curves on abelian surfaces
Abstract: The Brill-Noether theory of curves on K3 surfaces is well understood. Until recently, quite little has been known for curves on abelian surfaces. In the talk I will present some recent results obtained with M. Lelli-Chiesa and G. Mongardi. In particular, we show that the general curve in the linear system |L| on a general primitively polarized abelian surface (S,L) is Brill-Noether general, as in the K3 case. However, contrary to the K3 case, there are smooth curves in |L| possessing "unexpected" linear series, that is, with negative Brill-Noether number. As an application, we obtain the existence of components of special Brill-Noether loci of the expected dimension in the moduli space of curves.
Wednesday March 15, 2017 - room SRIV - time 16:00 - Hugo Torres (UNAM, Morelia, Mexico)
Title: Linear stability and the Butler conjecture.
Abstract: Let $(L,V)$ be a generated linear series on a smooth complex projective curve $C$. We study the relationship between the stability of the Lazarsfeld bundle $M_{V,L}$ and the linear stability of the linear serie $(L,V)$. We prove that the stabilities coincide in the cases: $V= H^0(L)$ and $C$ general and hyperelliptic (as conjectured by Mistretta and Stoppino). Moreover, We give precise conditions to the stability of $M_L$ when $C$ is a general curve (Work in progress with Abel Castorena).
Thursday February 9, 2017 - room 2BC60 - time 14.30 - Fumiharo Kato (Tokyo Institute of Technology )
Henselian rigid geometry
Thursday January 12, 2017 - room SRVII - time 14.30 - Christian Lehn (TU Chemnitz).
Birational geometry of singular symplectic varieties and a global Torelli theorem
Abstract: Verbitsky's Global Torelli theorem has been one of the most important advances in the theory of holomorphic symplectic manifolds in the last years. In a joint work with Ben Bakker (University of Georgia) we prove a version of the Global Torelli theorem for singular symplectic varieties and discuss applications. Symplectic varieties have interesting geometric as well as arithmetic properties, their birational geometry is particularly rich. We focus on birational contractions of symplectic varieties and generalize a number of known results for moduli spaces of sheaves to general deformations. Our results are obtained through the interplay of Hodge theory, deformation theory, and a further example of Verbitsky's technique which might carry the name "how to deduce beautiful consequences from ugly behavior of moduli spaces"
Friday December 16, 2016 - room SRVII - time 11:15 - Shinichi Kobayashi (Tohoku)
The radius of convergence of the p-adic sigma function at supersingular primes
Friday December 16, 2016 - room SRVII - time 12:15 - Nobuo Tsuzuki (Tohoku)
$F$-isocrystals on elliptic curves over a finite field
Thursday December 1, 2016 - room SRVII - time 14:30 - Ambrus Pal (Imperial)
Simplicial homotopy theory of algebraic varieties over real closed fields
Abstract: First I will introduce the homotopy type of the simplicial set of continuous definable simplexes of an algebraic variety defined over a real closed field, which I call the real homotopy type. Then I will talk about the analogue of the theorems of Artin-Mazur and Cox comparing the real homotopy type with the étale homotopy type, as well as an analogue of Sullivan's conjecture which together imply a homotopy version of Grothendieck's section conjecture. As an application I show that for example for rationally connected varieties over any real closed field the map from connected components of points to homotopy fixed points is a bijection.
Thursday, November 10, 2016 - room 2BC/30 - time 15:00 - Francesco Polizzi (Calabria)
New constructions of surfaces with $p_g=q=2$ via finite covers
Abstract: We present some new constructions of surfaces with $p_g=q=2$ and maximal Albanese dimension via finite covers of the Jacobian $J(C)$ of a smooth genus 2 curve C. If time permits, we will relate these examples to the monodromy representations of the braid group $B_2(C)$. Some of these results were obtained in collaboration with R. Pignatelli, C. Rito, X. Roulleau.
Thursday October 20, 2016 - room 2BC30 - time 14:30 - Marco Franciosi (Pisa)
Superfici stabili con K^2=1
Abstract: Il concetto di superficie stabile e’ stato introdotto da Kollar e Sheperd-Barron e lo spazio dei moduli relativo e’ una compattificazione naturale dello spazio di Gieseker dei moduli dei modelli canonici di superfici di tipo generale. In questo seminario si parlera’ di una collaborazione con Rita Pardini e Soenke Rollenske sulle superfici stabili Gorenstein con K^2=1. Verranno mostrati risultati relativi alla loro classificazione, alcuni dei quali naturalmente legati ai casi classici.
Wednesday October 19, 2016 - room SRVII - time 14:30 - Remke Kloosterman (Padova)
Alexander polynomials of curves, Mordell-Weil ranks and syzygies
Abstract: Fix a (singular) plane curve C. Recent results by Cogolludo-Agustin and Libgober, by Dimca suggest that there is an interesting interplay between
1. the Alexander polynomial of C. (A topological invariant) 2. the Mordell-Weil rank of certain isotrivial fibrations of abelian varieties with base P^2 and discriminant curve C. (An arithmetic invariant) 3. the Hilbert function of certain ideals of certain subschemes of the singular locus of C. (An algebraic invariant)
In this talk we will explain this interplay and refine it. Moreover, we will give various applications of this. In particular, if f(x,y,z) is a polynomial defining C then we show how these results can be used to find all polynomials (g,h) such that g^2+h^3=f.
Friday October 14, 2016 - room 2BC30 - time 14:30 - Christian Liedtke (TU München)
Good Reduction of K3 surfaces
Abstract: By a classical theorem of Serre and Tate, extending previous results of Néron, Ogg, and Shafarevich, an Abelian variety over the field of fractions K of a local Henselian DVR has good reduction if and only if the Galois action on its first l-adic cohomology is unramified (“no monodromy”). In this talk, we show that if the Galois action on second l-adic cohomology of a K3 surface over K is unramified, then the surface admits an ``RDP model'', and has good reduction (that is, a smooth model) after a finite and unramified extension. (Standing assumption: potential semi-stable reduction.) Moreover, we give examples where such an unramified extension is really needed. This is joint work with Yuya Matsumoto.
Thursday October 6, 2016 - room 2BC30 - 14:30 - Nero Budur (Leuven)
Cohomology jump loci.
Abstract: Firstly, we propose and illustrate a refinement of Deligne’s principle: every infinitesimal deformation problem over a field of characteristic zero with cohomology constraints is governed by a differential graded Lie algebra together with a module. Secondly, we review recent results about the global structure of cohomology jump loci of rank one local systems. Finally, we address future directions for other types of jump loci. All this is joint work with Botong Wang.
Tuesday September 27, Thursday September 29 2016; - room 1BC50 - time 11:30 - Maria Rosaria Pati (Pisa)
p-adic L-functions, following Bertolini-Darmon-Iovita-Spiess construction.
Abstract: The aim of this series of two lectures is to present the construction of the anticyclotomic p-adic L-function due to Bertolini and Darmon, following the approach developed by Bertolini-Darmon-Iovita and Spiess.
July 7 2016 - room 2AB40 - 12:00 - Francesco Baldassarri
Metodi analitici di Barsotti per gruppi p-divisibili e algebre perfettoidi.III./p>
Elena Mantovan (Caltech)
Tuesday June 21, 2016 - room 2BC30 - time 10:30 - Differential operators and families of automorphic forms on unitary groups of arbitrary signature. Thursday June 23, 2016 - room 2AB45 - time 10:30 - An approach to Hida's theory at unramified non-split primes.
Tuesday May 24, 2016 - room SRVII - 13:30 - Adrian Langer (Warsaw)
Rigid representations of quasi-projective fundamental groups
Abstract: In the talk I will survey various results concerning rigid representations of the topological fundamental group of a complex quasi-projective variety. The main part will be devoted to the conjecture due to C. Simpson that says that every rigid integral irreducible representation of the topological fundamental group of X is of geometric origin, i.e., it comes from some family of smooth projective varieties. I will also show some ideas behind my recent joint work with C. Simpson on the rank 3 representations.
Richard Crew (University of Florida)
Arithmetic Differential Operators on Adic Formal Schemes
Abstract: We show that Berthelot's theory of arithmetic differential operators relative to a smooth morphism can be generalized to the case of formally smooth morphisms satisfying a weak finiteness condition. For example if Y -> S is a smooth morphism of locally noetherian adic formal schemes and X is the completion of Y along a closed subscheme, the theory applies to X -> S. If time permits, I will give applications to overconvergent isocrystals and rigid cohomology.
Friday May 6, 2016 - room 2AB40 - 14:00-16:00 Thursday May 12, 2016 - room SRVII - 14:00-16:00 Friday May 13, 2016 - room 2AB40 - 14:00-16:00
Monday April 18, 2016 -room SRVII - 14:00 - Andrea Pulita (Grenoble)
On some recent works
Friday February 5, 2016 - room 2AB45- 11:00 - Hironori Shiga
Visualization of the Shimura complex multiplication via hypergeometric equations and Lam'e equations
The talk is based on a paper by the speaker. The paper is devoted to construction of Hilbert class fields by singular values of automorphic functions on Shimura curves associated to indefinite quaternion algebra. In the first half of the paper, authors traced Shimura’s original work with a special emphasis on arithmetic triangle groups, and connected Shimura’s theory with theory of Schwartz mapping for hypergeometric functions. In the later part, many explicit examples of singular values and Hilbert class field for higher CM fields are computed. These results add explicit and beautiful examples to “Kronecker’s Jugendtraum (Hilbert’s twelfth problem)” for higher CM fields.
Friday November 6, 2015 - room 2AB40- 11:30 - Farhad Babaee (ENS)
A tropical approach to the strongly positive Hodge conjecture
Abstract: I will briefly explain complex tropical currents, and will address their extremality, intersection theory, and approximation problems. I will discuss how in joint work with June Huh, we constructed an example of a non-approximable tropical current, which, in turn, refutes a stronger version of the Hodge conjecture.
Thursday November 5, 2015 - room 2BC30 -14:15 - Ambrus Pal (Imperial College)
Indipendence results on p-adic monodromy groups
Wednesday November 4, 2015 - room 2BC30 -14:15 - Christopher Lazda
Monodromy-weight conjecture
Friday October 30, 2015 - room 2BC60 -14:15 - Baskar Balasubramanyam
Special values of L-functions and congruences of modular forms
Abstract: Let f and g be two modular forms that are eigenvectors for all the Hecke operators. Let \lambda_f (n) and \lambda_g (n) be the respective eigenvalues for the T(n) operator. Then f and g are said to be congruent mod a prime p if \lambda_f (n) is congruent to \lambda_g (n) mod p, for all n. Such congruences have been historically important have lead to studying p-adic families of modular forms. In the early 80s, Hida proved an important relationship between congruences and the special values of certain L-functions attached to modular forms. In my talk, I will review these results and discuss some of its generalizations.
Wednesday July 1, 2015- room 2AB40 - 10:30 - Konstantin Ardakov (Oxford)
$\hat{ \cal D}$-modules on rigid analytic spaces
Wednesday June 24, 2015- room 2AB40 - 14:30 - Elisa Postinghel (Leuven)
Positivity of divisors on blown-up projective spaces
Abstract: We give cones of l-very ample divisors on the blown-up projective space P^n in a collection of points in linearly general position. For small numbers of points, less than 2n, we prove that a divisor is nef if and only if it is globally generated and ample if and only if it is very ample. We establish Fujita's conjectures in this range. Moreover we show that the strict transform of these divisors in the iterated blow-up along the linear cycles of the base locus is globally generated, for a small number of points, less than n+3. If time permits, we discuss applications of this in the direction of the F-conjecture for M_{0,n}. This is joint work with Olivia Dumitrescu
Wednesday June 24, 2015- room 2AB40 - 15:30 -Eric Katz (Waterloo, Canada)
Uniform bounds on rational points on curves of low Mordell-Weil rank
Abstract: I will discuss the recent proof with Joseph Rabinoff and David Zureick-Brown that there is a uniform bound for the number of rational points on genus g curves of Mordell-Weill rank at most g-3, extending a result of Stoll on hyperelliptic curves. Our work also gives unconditional bounds on the number of rational torsion points and bounds on the number of geometric torsion points on curves with very degenerate reduction type. I will outline the Chabauty-Coleman method for bounding the number of rational points on a curve of low Mordell-Weil rank and discuss the challenges to making the bound uniform. These challenges involve p-adic integration and Newton polygon estimates, and are answered by employing techniques in Berkovich spaces,tropical geometry, and the Baker-Norine theory of linear systems on graphs.
June 23, 26, 2015 - room 2BC60 - Eric Katz (Waterloo)
Minicourse: Introduction to Tropical Geometry
Abstract: Tropical geometry is a method of transforming problems in algebraic geometry into problems in combinatorics. The procedure of tropicalization takes as input an algebraic subvariety of an algebraIc torus and produces a combinatorial objected, a polyhedral fan. In the first two hours, we discuss tropicalization and explain the relationship between tropicalization and intersection theory. In the second two hours, we use results in tropical geometry to understand the question of which homology classes in a toric variety can be realized by homology classes of a subvariety, culminating in the recent disproof of Demailly's strongly positive Hodge conjecture by Farhad Babaee and June Huh.
Thursday June 4, 2015- room 2AB40 - 14:15 - Chris Lazda (Imperial College HIMR)
Rigid cohomology over Laurent series fields
Abstract: If F is a characterstic p local field (that is, a Laurent series field), then every ℓ-adic representation (ℓ≠p) of its absolute Galois group is potentially semistable: this is Grothendieck's ℓ-adic local monodromy theorem, and it can be viewed as a cohomological incarnation of potentially semistable reduction. If we consider the the p-adic setting, however, then the objects produced by rigid cohomology (i.e. (φ,∇)-modules over the Amice ring) are not the objects to which the correpsonding local monodromy theorem applies (that is i.e. (φ,∇)-modules over the Robba ring). I will explain some work in constructing a refinement of rigid cohomology that bridges the gap between these two kinds of objects, and which therefore 'reconnects' the monodromy theorem to the geometry of varieties over F. I will also discuss how a construction of Marmora leads to a formulation versions of weight monodromy and ℓ-independence which include the case ℓ=p, and hopefully explain why everything works in the case of smooth curves. This is joint work with Ambrus Pál.
Polytope algebra, tropical algebra and toric cycles
Abstract:The structure of algebraic geometry is as elegant as complicated. Given a variety X, it is often possible to associate it some polyhedral objects that reflect its properties (e.g. tropical geometry, Newton-Okounkov bodies). For instance, any toric variety X with maximal torus T=(C^*)^n contained in X is associated with a polyhedral complex \Sigma (a fan) in R^n, whose cones are in bijection with T-orbits in X. Any codimension 1 subvariety of X is also associated with a polytope in R^n, representing its global sections. For an ample divisor, the dual fan of its polytope is exactly Sigma. What happens for toric cycles in higher codimension? It is possible to give an algebraic structure both to polytopes (Pi) and to fans (T), and the obtained polytope algebra Pi turns out to be isomorphic to the tropical algebra T by an exponential morphism. Moreover, they are also isomorphic to the algebra of toric cycles and to the one of Minkowski weights. This seminar, after recalling some introductory notions, will focus on this identifications, which give different points of view on the positivity problem for algebraic cycles and agree with the known results on divisors.
Thursday May 21, 2015- room 1BC50 - 14:00 - Thomas Bitoun (Moscow , National Research University)
On the theory of b-function in positive characteristic
Abstract: We exhibit a construction in noncommutative nonnoetherian algebra that should be understood as a positive characteristic analogue of the Bernstein-Sato polynomial or b-function. Recall that the b-function is a polynomial in one variable attached to an analytic function f. It is well-known to be related to the singularities of f and is useful in continuing a certain type of zeta functions, associated with f. We will briefly recall the complex theory and then emphasize the arithmetic aspects of our construction.
Tuesday April 28, 2015- room SRVII - 12:00 - Alessandro Chiodo (Parigi - Jussieu)
Modelli di Néron di gruppi Picard via gruppi di Picard
Abstract: Il modello di Néron N(Pic^0C_K) fornisce, su un anello a valutazione discreta R, un prolungamento universale di Pic^0C_K, il gruppo di Picard formato dalle classi di isomorfismo dei fibrati di grado zero su una curva liscia C_K su K=Frac(R). Al fine di descrivere N(Pic^0C_K), è naturale sfruttare il funtore di Picard relativo Pic^0C_R formato dai fibrati di grado totale 0 su tutte le fibre di una riduzione (semi)stable CR. Il gruppo Pic^0C_R non è separato in generale, ma il modello di Néron N(Pic^0C_K) è uguale a Pic^0C_R modulo la chiusura della sezione nulla di Pic^0C_K (Raynaud, 1970). In alcuni casi molto speciali, si può ottenere N(Pic^0C_K) direttamente, senza passare per il quoziente, ma semplicemente identificando le componente contenente l'elemento neutro in ogni fibra di Pic^0C_R. Tale gruppo è il sotto-gruppo dei fibrati che hanno grado zero su ogni componente irriducibile delle fibre di Pic^0C_R. In generale il quoziente non possiede una simile interpretazione, ma questo seminario illustra che, quando adottiamo come riduzione di C_K una curva torta nel senso di Abramovich e Vistoli, il modello di Néron rappresenta un funtore di Picard separato costituito da fibrati di grado zero su tutte le componenti irriducibili delle fibre.
Friday April 17, 2015- room 1C150 - 10:30 - Adrian Iovita
The Spectral Halo
Abstract: Together with F. Andreatta and V. Pilloni we have proved a conjecture of R. Coleman who predicted the existence of a Banach space of (mod p) overconvergent modular forms together with a compact operator U_p, such that its characteristic series is the reduction (mod p) of the characteristic series of U_p acting on overconvergent modular forms in characteristic 0.
Thursday, April 2, 2015- room 1AD100 - 14:30 -Zhengyu Mao (Rutgers Univ. Newark)
Fourier coefficients of cusp forms
Abstract: An important result in number theory is the relation between Fourier coefficients of half integral weight forms and the L-values of integral weight forms. We will discuss this result and its generalization: the relation between Fourier coefficients of cusp forms on any reductive group and certain L-values.
Monday, March 9 2015- room 1BC45 - 14:30 - Federico Bambozzi (Regensburg)
Dagger geometry as Banach algebraic geometry
See the preprint (with Oren Ben-Bassat) http://arxiv.org/pdf/1502.01401.pdf Abstract of the paper: In this article, we apply the approach of relative algebraic geometry towards analytic geometry to the category of bornological and Ind-Banach spaces (non-Archimedean or not). We are able to recast the theory of Grosse-Kl¨onne dagger affinoid domains with their weak G-topology in this new language. We prove an abstract recognition principle for the generators of their standard topology (the morphisms appearing in the covers). We end with a sketch of an emerging theory of dagger affinoid spaces over the integers, or any Banach ring, where we can see the Archimedean and non-Archimedean worlds coming together.
Thursday, March 12, 2015, - room 1BC45 - 14:30 - Lucia Di Vizio (CNRS)
Geometria dei gruppi alle differenze e teoria di Galois parametrica delle equazioni differenziali
Abstract
Thursday, March 12, 2015, - room 1BC45 - 15:30 - Michele Bolognesi (Rennes)
Pfaffian Cubic Fourfolds and non-trivial Brauer Classes
Abstract. In this talk I will show a general class of smooth rational cubic fourfolds X containing a plane whose associated quadric surface bundle does not have a rational section. Equivalently, the Brauer class B of the even Clifford algebra over the discriminant cover (a K3 surface S of degree 2) associated to the quadric bundle, is nontrivial. These fourfolds provide nontrivial examples verifying Kuznetsov's conjecture on the rationality of cubic fourfolds containing a plane. Indeed, using homological projective duality for grassmannians, one obtains another K3 surface S' of degree 14 and a nontrivial twisted derived equivalence A_X = Db(S;B ) = Db(S'), where A_X is Kuznetsov's residual category associated to the cubic hypersurface X.
February 2014, - room 2BC30 February 10, 10:00-12:00, February 12, 9:00-11:00, February 13, 9:00-11:00, February 16, 11:00-13:00, February 17, 11:00-13:00, February 18, 9:00-11:00, February 19, 14:00-16:00, February 23, 14:00-16:00, February 24, 14:00-16:00, February 26, 9:00-11:00. - Benjamin Schraen, Stefano Morra
Topics on p-adic Langlands (Corso Dottorato)
Tuesday, October 7, 2014, - room 2BC/30 - 14:30 - Enrica Floris (Imperial College London)
Invarianza dei plurigeneri per foliazioni su superfici
Abstract Sia X una superficie algebrica complessa liscia. Una foliazione in curve su X, che indicheremo con F, è il dato di un sottofibrato TF di rango 1 del fibrato tangente di X tale che il quoziente sia localmente libero di rango uno al di fuori di un sottoinsieme finito. I punti di tale sottoinsieme sono i punti singolari di F e il duale a TF è detto canonico della foliazione F. Negli ultimi anni è stato molto utile allo studio delle foliazioni l’uso di metodi birazionali, che mettono in relazione le proprietà geometriche della foliazione con le proprietà del suo fibrato canonico. Uno degli invarianti più importanti che descrivono le proprietà di un fibrato in rette L è la dimensione di Kodaira κ(L), che misura la crescita delle sezioni di L. È possibile definire una nozione di equivalenza birazionale tra foliazioni. La dimensione di Kodaira di una foliazione κ(F) è definita come la dimensione di Kodaira del suo fibrato canonico ed è un invariante birazionale. Nei loro fondamentali lavori, Brunella e McQuillan forniscono una classificazione delle foliazioni su superfici sul modello della classificazione di Enriques-Kodaira: le foliazioni con dimensione di Kodaira κ(F) ∈ {−∞, 0, 1} sono classificate a meno di equivalenza birazionale. Il passo successivo alla classificazione delle varietà è l’analisi di come queste si comportano in famiglia. Brunella dimostra che, per una famiglia di foliazioni in curve su superfici, con qualche ipotesi tecnica sulle singolarità, la dimensione di Kodaira non dipende da t. Per analogia con il teorema di Invarianza dei plurigeneri di Siu, è naturale domandarsi se la dimensione delle sezioni globali delle potenze del canonico della foliazione dipenda o meno da t. In questo seminario discuteremo in che misura un teorema un teorema di invarianza dei plurigeneri è vero e sotto quali ipotesi sulla famiglia di foliazioni.
Tuesday, September 23, 2014, - room 2AB/45 - 14:30 - Frank Neumann (University of Leicester)
Weil Conjectures and Moduli of Vector Bundles
Abstract In 1949, Weil conjectured deep connections between the topology, geometry and arithmetic of projective algebraic varieties over a field in characteristic p, including an analogue of the celebrated Riemann Hypothesis. These conjectures led to the development of etale cohomology in algebraic geometry as an analog of singular cohomology in algebraic topology by Grothendieck and his school and culminated in the proof of the Weil conjectures by Deligne in the 70s. After giving a gentle introduction into the classical Weil conjectures for projective algebraic varieties and discussing what moduli problems and moduli stacks are, I will outline how an analog of these Weil conjectures for the moduli stack of vector bundles over a projective algebraic curve can be formulated and proved. This basically comes down to counting how many vector bundles (up to isomorphisms) there are over a given algebraic curve in characteristic p.
Thursday, July 10, 2014, - room 2AB45 - 14:30-16:00 - Fabio Saggin (Master student, Padova)
Cohomology theories and formal groups
Friday, July 4, 2014, - room 2BC60 - 10:45 - Adebisi Agboola (Santa Barbara)
On special values of p-adic L-functions
Abstract I shall discuss the relationship between the arithmetic of a large class of special values of the Katz two-variable p-adic L-function and the stucture of certain non-standard Selmer groups. This can be used to obtain precise information about certain special values whose behaviour is not known to be governed by the arithmetic of an underlying algebraic cycle.
Thursday, May 29, 2014, - room 2BC60 - 12:00 - Christopher Lazda (Imperial College)
A homotopy exact sequence and unipotent fundamental groups over function fields
Abstract If X/F is a smooth and proper variety over a global function field of characteristic p, then for all l different from p the co-ordinate ring of the l-adic unipotent fundamental group is a Galois representation, which is unramified at all places of good reduction. In this talk, I will ask the question of what the correct p-adic analogue of this is, by spreading out over a smooth model for C and proving a version of the homotopy exact sequence associated to a fibration. There is also a version for path torsors, which enables me to define an function field analogue of the global period map used by Minhyong Kim to study rational points.
Thursday, May 22, 2014, - room 1BC50 - 14:30 - Sarah Zerbes (University College London)
Euler systems and the conjecture of Birch and Swinnerton-Dyer
Abstract One special case of the Birch--Swinnerton-Dyer conjecture is the statement that if E is an elliptic curve over a number field, and the L-function of E does not vanish at s = 1, then E has only finitely many rational points and its Tate-Shafarevich group is finite. This is known to be true for elliptic curves over Q by a theorem of Kolyvagin. Kolyvagin's proof relies on an object called an 'Euler system' -- a system of elements of Galois cohomology groups -- in order to control the Tate-Shafarevich group. It has long been conjectured that Euler systems should exist in other contexts, and these should have similarly rich arithmetical applications; but only a very small number of examples have so far been found. In this talk I'll describe the construction of a new Euler system attached to pairs of elliptic curves -- or more generally pairs of modular forms -- and some of its arithmetical applications. This is joint work with Antonio Lei and David Loeffler, and it has recently been generalised by Guido Kings, David Loeffler and myself.
Thursday, May 29, 2014, - room 1BC50 - 14:00 - Sofia Tirabassi (Utah)
Varieties with $\chi$ equal to 1
Abstract We study varieties of maximal Albanese dimension and Euler characteristic one. When their Albanese image is nonsingular in codimension 1 and not fibered by tori we find that these are always birational to products of theta divisors in principally polarized abelian varieties. We are able to give a complete classification when the irregularity of the variety is 2dim X-1, extending in higher dimension results of Hacon--Pardini concerning surfaces. This is a joint work with Z. Jiang and M. Lahoz.
Wednesday February 19, 2014, - room 1BC45 - 16:00 - Vladimir Matveev
Nonintegrability of the Zipoy-Vorhees metric
Abstract I will consider natural Hamiltonian systems with two degrees of freedom and speak about the existence and nonexistence of integrals that are polynomial in momenta. This is a classical topic: I will start with a historical overview and explain the classical and modern motivation. In the mathematical part of my talk, I will mostly discuss the following questions: given a metric, how to prove the (non)existence of an integral of a given degree, and how to find it explicitly? I will explain the classical and modern methods to study this question. As an application, I will present new systems admitting an integral of degree 3 in momenta and a solution of a problem explicitly stated by J. Brink (partially joint with H. Dullin, V. Shevchishin, B. Kruglikov). The talk will be understandable for mathematicians from other fields and will emphasize how the algebraic geometry methods could be used in mathematical physics and differential geometry.
Thursday January 9, 2014, - room 2BC30 - 14:30-15:30 - Thomas Hudson (Daejeon, Corea)
Schubert classes in the algebraic cobordism of generalized flag bundles
Abstract Let V be a vector bundle over a smooth scheme X. One of the first steps needed to establish a Schubert calculus for the full flag bundle FL V consists in identifying a basis for CH^*(FL V) as a module over CH^*(X). For this purpose one usually considers Schubert classes, i.e., fundamental classes of Schubert varieties, which can be described by means of double Schubert polynomials. Analogous constructions are also available for flags of bundles which are isotropic with respect to a given quadratic (or symplectic) form on V. I will illustrate one possible way of defining Schubert classes in the context of a general oriented cohomology theory and more specifically in algebraic cobordism.
Friday December 20, 2013, - room 1BC45 - 11:30-12:30 - Tommaso Centeleghe (Heidelberg)
Sullo spezzamento dei primi nei campi di torsione di curve ellittiche
Abstract Sia E una curva ellittica sul campo Q dei numeri razionali, e sia N un intero positivo. In questo seminario consideriamo l'estensione Q(E[N])/Q ottenuta a partire da Q aggiungendo le coordinate dei punti di N-torsione di E. Il risultato principale descrive come l'invariante j di E e gli Hilbert Class Polynomials possano essere utilizzati per descrivere lo spezzamento dei primi in Q(E[N])/Q. Un ruolo importante in questo problema e' giocato dagli anelli di endomorfismi delle riduzioni di E modulo p.
Thursday December 5, 2013, - room 2BC30 - 14:30-15:30 - David Schmitz (Marburg)
Big cone decompositions and volume of chambers
Abstract I will recall some decompositions of the big cone of a smooth projective variety in terms of the mapping behavior of linear series and study their geometric properties. Furthermore, I will introduce a metric description of these decompositions and present effective results in the case of the Zariski chamber decomposition on surfaces introduced by Bauer, Küronya, Szemberg.
November 7, 2013, - room 2BC30 - 11:00-12:00 - Atsushi Shiho (Tokyo)
On the differential Artin conductor of overconvergent isocrystals
November 14, 2013, - room 2BC30 - 9:30-11:00 - Andrea D'Agnolo (Padova)
Riemann-Hilbert correspondence for holonomic D-modules
Abstract The classical Riemann-Hilbert correspondence establishes an equivalence between the triangulated category of regular holonomic D-modules and that of constructible sheaves. We prove a Riemann-Hilbert correspondence for holonomic D-modules which are not necessarily regular. The construction of our target category is based on the theory of ind-sheaves by Kashiwara-Schapira and influenced by Tamarkin's work. Among the main ingredients of our proof is the description of the structure of flat meromorphic connections due to Mochizuki and Kedlaya. This is joint work with Masaki Kashiwara.
Thursday November 14, 2013, - room 2BC30 - 14:30-15:30 - Stefano Urbinati (Padova)
Positivity for normal varieties
Abstract A key ingredient in birational geometry is intersection theory, that gives the notion of positivity. Unfortunately intersection theory requires strong assumptions that sometimes it is necessary to avoid. In this talk we will give three different possibilities to partially overcome intersection theory for defining positivity. The first, given by Boucksom, de Fernex and Favre using b-divisors. The second through the pullback for Weil divisors introduced by de Fernex and Hacon. And the third through Okounkov bodies, a work in progress of Kuronya and Lozovanu.
Thursday October 24, room SRVII - 14:00- 15:30 - Niko Naumann (Regensburg)
Introduction to perfectoid spaces
Friday October 18, 2013, - room 2BC30 - 15:30-16:30 - Pierre Colmez (CNRS)
The Montreal functor and its inverse
Wednesday October 16, 2013, - room 2BC30 - 15:30-16:30 - Pierre Colmez (CNRS)
Locally analytic representations of GL_2 and (φ,Γ)-modules
Thursday October 10, 2013, - room SRVII - 14:30-15:30, - Giovanni Moreno (Opava, Cechia)
Su alcune questioni di geometria algebrica "nascoste" nella teoria geometrica delle PDE non lineari
(In collaborazione con G. Manno) Abstract Avendo origini distanti l’una dall’altra, ed essendosi sviluppate in contesti differenti, le strade della Geometria Algebrica e della teoria geometrica delle PDE non lineari si sono incrociate assai raramente, e sono pochi ed isolati, sebbene molto profondi, i lavori che si occupano delle interazioni fra queste due discipline. Si possono citare, al proposito, gli ormai classici testi di V. Lychagin et al. [7], o di R. Bryant et al. [3]. In questo contesto, G. Manno et al. hanno eneralizzato la descrizione geometrica delle equazioni di Monge–Ampére al caso di n variabili indipendenti [1], il relatore ha individuato una distribuzione canonica sulla cosiddetta Grassmanniana “Lagrangianoide” [2], ed i due sono correntemente impegnati nella generalizzazione agli ordini superiori dei risultati contenuti in [1]. In questo seminario verranno esposte alcune questioni incontrate dai relatori nel corso degli studi summenzionati, le quali possono essere completamente avulse dal contesto delle PDE non-lineari e, come tali, formulate in termini di Geometria Algebrica elementare, con l’obiettivo di sensibilizzare gli esperti di questa disciplina ed ottenere più informazioni al proposito. Verranno menzionate le seguenti nozioni: sequenza universale associata ad una varietà Grassmanniana, Grassmanniana “Lagrangianoide” (ossia in cui la dimensione dei sottospazi non è massimale) e sue distribuzioni naturali [4, 2], ipersuperfici quadriche in spazi simplettici associate al simbolo di una PDE quasi–lineare [1, 6], varietà di bandiere di sottospazi isotropi rispetto alla forma di curvatura della distribuzione di Cartan. References [1] Dmitri V. Alekseevsky, Ricardo Alonso-Blanco, Gianni Manno, and Fabrizio Pugliese. Contact geometry of multidimensional Monge-Ampère equations: characteristics, intermediate integrals and solutions. Ann. Inst. Fourier (Grenoble), 62(2):497–524, 2012. [2] M. Bächtold and G. Moreno. On the polar distribution for singularities equations of nonlinear pdes. page http://arxiv.org/abs/1208.5880. [3] R. L. Bryant, S. S. Chern, R. B. Gardner, H. L. Goldschmidt, and P. A. Griffiths. Exterior differential systems, volume 18 of Mathematical Sciences Research Institute Publications. Springer-Verlag, New York, 1991. [4] G infinity (http://mathoverflow.net/users/22606/g infinity). A natural tower of bundles over grassmannian manifolds. MathOverflow. URL:http://mathoverflow.net/q/138243 (version: 2013-07-31). [5] G infinity (http://mathoverflow.net/users/22606/g infinity). Submanifolds in the grassmannian of ndimensional subspaces determined by a submanifold in the grassmannian of l-dimensional subspaces. MathOverflow. URL:http://mathoverflow.net/q/138544 (version: 2013-08-06). [6] G infinity (http://mathoverflow.net/users/22606/g infinity). When a hyperplane of symmetric forms is determined by a quadric hypersurface? MathOverflow. URL:http://mathoverflow.net/q/141113 (version: 2013-09-03). [7] Alexei Kushner, Valentin Lychagin, and Vladimir Rubtsov. Contact geometry and non-linear differential equations, volume 101 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 2007.
October 2013, - room 2BC30 Wednesday 2, Friday 4, Wednesday 9, Friday 11 - 14:30-16:30 room 2BC30 Wednesday 16, and Friday 18 - room 2BC30 - 14:30-15:30 Wednesday 23, room 2BC30 - 13:30-15:00 Friday 25, room 2BC30 - 14:30-15:30 (Wednesday 30, room 2BC30-14:30-16:30 TBC) - Niko Naumann (Regensburg)
Perfectoid spaces and their applications (Corso Dottorato)
Abstract We give an introduction to the deformation theory of formal groups and Drinfeld structures leading up to recent structure results of Scholze and Weinstein. Time and interest of the audience permitting, we may then explain applications of this to Rapoport-Zink spaces and/or classification of p-divisible groups.
Thursday, June 27, 2013 - room 2AB40 - 11:30 - Giuseppe Ancona (Paris 13)
Algebraic cycles of abelian schemes
Abstract Let X be a variety and consider the cycle class map, which maps the Chow groups of X to the cohomology groups of X. Hodge or Tate conjectures predict what should be the image of such a map, and Bloch-Beilinson-Murre conjecture predicts some structures on the kernel. We will investigate these conjectures when X is an abelian scheme, and especially when it is the universal abelian scheme over a PEL Shimura variety.
Friday, June 28, 2013 - room 1AD100 - 11:30 - Sophie Marques (Padova-Bordeaux)
Tameness for actions of affine group schemes and quotient stacks
Monday, June 3, 2013 - room 2BC30 - 11:30 - Stefano Morra (Toronto)
The action of compact subgroups on some p-modular automorphic representations of GL_2(Q_p)
Abstract The weight part in the generalized Serre's conjectures gives us deep information on the local factor at p of isotypical components in certain spaces of p-modular quaternionic forms, providing evidence for local-global compatibility phenomena in a hypothetical p-modular Langlands correspondence. Nevertheless the theory of p-modular representations of p-adic groups turns out to be extremely delicate and there seem to be many parameters in the local factor at p which are invisible to its socle with respect to a maximal compact subgroup. The aim of this talk is to establish some tools which let us deeply investigate the local automorphic representations of GL_2(Q_p) appearing the p-modular Langlands correspondence. This is done by purely local means, describing such objects in terms of representation theory of appropriate compact subgroups and letting one detect some of their arithmetic invariants.
Thursday, May 23, May 30, June 6, 2013, - room 2AB45 - 9:00-12:00, Friday, June 7, 2013, -room 2BC30 - 11:00-14:00 - Aaron Silberstein (Harvard)
Introduction to birational anabelian geometry (Corso Dottorato)
Friday, May 31, 2013 - room SRVII - 14:00 - Shinichi Kobayashi (Tohoku)
The p-adic height pairing on abelian varieties at non-ordinary primes
Abstract In 80's, P. Schneider constructed the $p$-adic height pairing on abelian varieties at ordinary primes by using the universal norm group. In this talk, I explain a generalization of Schneider's construction to the non-ordinary case by using certain norm systems, and explain an application for the p-adic Gross-Zagier formula.
Friday, May 31, 2013 - room SRVII- 11:00 - Wieslawa Niziol (Utah)
Syntomic cohomology and regulators
Abstract Recently Beilinson and Bhatt have developed a new approach to comparison theorems of p-adic Hodge Theory. I will explain how their constructions can be used to define a syntomic cohomology for varieties over p-adic local fields.
Friday,May 31, 2013 - room SRVII - 9:30 - Pierre Colmez (CNRS)
Representations analytiques et (phi,Gamma)-modules
Friday, May 24, 2013 - room 2AB45 - 14:30 - Vincent Pilloni (ENS Lyon)
p-Adic Weil II and classicity of overconvergent modular forms
Wednesday, May 22, 2013 - room SRVII - 14:30 Friday, May 24, 2013 - room SRVII - 10:00 - Jean-Baptiste Teyssier (Ecole Polytechnique)
An analogue for meromorphic connections of Abbes-Saito construction
Thursday, May 9, 2013 - room 1BC45 – 10:45 - Carlos de Vera (UPC Barcelona)
Shimura curves: moduli interpretation and rational points
Abstract Shimura curves have great arithmetic significance, for they are moduli spaces of abelian surfaces with quaternionic multiplication (orfake elliptic curves). The study of diophantine properties of these curves is therefore of fundamental importance in number theory.After recalling the moduli interpretation of Shimura curves, we will explain two strategies for studying the existence of global points on both Shimura curves and their Atkin-Lehner quotients. The first of them requires introducing the notion of Galois representations over fields of moduli, inspired on the work of Ellenberg and Skinner on the modularity of Q-curves. And the second one is a descent strategy for which we need to exploit the Cerednik-Drinfel'd theory of p-adic uniformization of Shimura curves. Part of the talk will be based on a work in collaboration with V.Rotger.
Friday, March 15, 2013- room 2AB40- 10:30 Friday, March 22, 2013 - room 2AB40 - 13:45 Friday, April 5, 2013- room SRVII- 11:30 - room 2AB45- 14:00 - R. Scheider (Regensburg)
The de Rham realization of the Polylogarithm
Abstract Polylogarithms in their various manifestations provide one of the most powerful tools for the study of special values of L-functions. The sheaf-theoretic and motivic formalism of the elliptic polylogarithm was introduced by Beilinson and Levin in the fundamental paper [Be-Le]: there, the polylogarithm appears as Hodge- or l-adic sheaf on a family of elliptic curves minus its zero section, and its R-Hodge realization is explicitly described. The de Rham realization of the polylogarithm was studied in [Ba-Ko-Ts] for the case of a single elliptic curve over a subfield of C, where it can be represented by very complicated formulas whose conceptual interpretation so far is unknown. In the upcoming talks I first want to introduce the basic definition of the logarithm sheaves and the de Rham polylogarithm class for an abelian scheme (resp. elliptic curve) over a base scheme. We will then see that the logarithm sheaves (very formal objects in nature) can be described in purely geometric terms using the Poincare'-Bundle and universal extension of the dual abelian scheme; if time permits we will also interpret this result in the elegant language of the Fourier-Mukai-Transform for flat vector bundles developed in [La]. The first logarithm of an abelian scheme is of motivic origin, and I will outline how to represent it as the de Rham realization of a certain one-motive using the results of [An-Ber]. At a final stage, we proceed to an explicit description of the polylogarithm in the case of the universal family of elliptic curves with level N-structure, using theta-functions defined on the universal covering. The specialization of the polylogarithm along torsion sections down to the modular curve can then explicitly be determined and reveals to be directly expressable in terms of Siegel modular units and Eisenstein series, as defined in [Ka], Ch. I. References [An-Ber] F. Andreatta, A. Bertapelle, Universal extension crystals of 1-motives and applications, Journal of Pure and Applied Algebra, Vol. 215 (2011), 1919-1944. [Ba-Ko-Ts] K. Bannai, S. Kobayashi and T. Tsuji, On the de Rham and p-adic realizations of the Elliptic Polylogarithm for CM elliptic curves, Ann. scient. de l’ENS, serie 4, Vol. 43-2 (2010), 185-234. [Be-Le] A. Beilinson and A. Levin, The Elliptic Polylogarithm, in: U. Jannsen, S. Kleiman, J.-P. Serre (eds.), Motives, Proceedings of Symposia in Pure Mathematics, Vol. 55, Part 2, American Mathematical Society, Providence RI, 1994, p.123-192. [Ka] K. Kato, p-adic Hodge theory and values of zeta functions of modular forms, in: P. Berthelot, J.-M. Fontaine, L. Illusie, K. Kato, M. Rapoport (eds.), Cohomologies p-adiques et applications arithmétiques (III), Asterisque 295 (2004), 117-290. [La] G. Laumon, Transformation de Fourier generalisee, preprint, arXiv:alg-geom/9603004v1, 5 Mar 1996.
Seminar Postponed - - Marco Boggi (Universidad de los Andes - Bogotà)
Profinite Teichmüller theory and arithmetic of curves over number fields
Abstract For $2g-2+n>0$, the Teichmüller modular group $\Gamma_{g,n}$ of a compact Riemann surface of genus $g$ with $n$ points removed $S_{g,n}$ is the group of homotopy classes of diffeomorphisms of $S_{g,n}$ which preserve the orientation of $S_{g,n}$ and a given order of its punctures. This group is naturally identified with the topological fundamental group of the moduli stack ${\mathcal M}_{g,n}$ of $n$-pointed, genus $g$ smooth complex curves. Its profinite completion $\hat{\Gamma}_{g,n}$ then identifies with the algebraic fundamental group of the moduli stack ${\mathcal M}_{g,n}$ over ${\mathbb Q}$. Profinite Teichmüller theory studies the properties of the profinite group $\hat{\Gamma}_{g,n}$ following the analogies with its discrete counterpart. The motivation is that a great deal of arithmetic but also topological properties of hyperbolic curves and their moduli spaces are embedded in this group. Let $\Pi_{g,n}$ be the fundamental group of $S_{g,n}$, with a given base point, and $\hat{\Pi}_{g,n}$ its profinite completion. There is then a natural representation $\hat{\Gamma}_{g,n}\rightarrow\mathrm{Out}(\hat{\Pi}_{g,n})$. Whether this representation is or not faithful is known as the congruence subgroup problem for the Teichmüller group. An affirmative answer would be essential for understanding the profinite Teichmüller group $\hat{\Gamma}_{g,n}$. However, this question turns out to be extremely difficult and has been solved positively only in genus $\leq 2$. The procongruence Teichmüller group $\check{\Gamma}_{g,n}$ is defined to be the image of the profinite Teichmüller group $\hat{\Gamma}_{g,n}$ inside the profinite group $\mathrm{Out}(\hat{\Pi}_{g,n})$. The idea is to study first this group which, for obvious reasons, is much more treatable. The combinatorial group-theory of the procongruence completion $\check{\Gamma}_{g,n}$ starts fixing a natural set of topological generators for this group. This is the set of profinite Dehn twists of $\check{\Gamma}_{g,n}$, defined to be the closure, inside this group, of the set of Dehn twists of $\Gamma_{g,n}$. The first main technical result is a parametrization of the set of profinite Dehn twists of $\check{\Gamma}_{g,n}$ and the subsequent description of their centralizers. This is the basis for the Grothendieck-Teichmüller Lego with procongruence Teichmüller groups as building blocks. As an application, we then show that some Galois representations associated to hyperbolic curves over number fields and their moduli spaces are faithful.
Thursday, November 22, 2012- room SR VII- 14:00 - Gloria Della Noce (Pavia)
On the Picard number of singular Fano Varieties
Abstract Fano varieties constitute a special class of varieties which plays a fundamental role in some branches of algebraic geometry. In the last decades, some classification results were achieved in particular cases. In the remaining cases, it is clear the importance of studying their geometrical invariants; one the most important is the Picard number. In this talk I will discuss some risults on the Picard number of singular Fano varieties obtained generalizing a construction of C. Casagrande for the smooth case.
Wednesday, June 6, 2012 - room 1C/150 - 14:30 - Hou-Yi Chen (Taiwan)
Reconstruction of a variety from $O[[h]]$-modules.
Abstract We prove that varieties are uniquely determined by the derived category of $O[[h]]$-modules with coherent cohomology which is the same as O-modules proved by A. Bondal and D. Orlov. We also generalize a theorem of Orlov. See arXiv: 1202.2940
Tuesday, June 5, 2012- room 1BC/45 - 11:00 - Florian Pop (Philadelphia)
The Oort Conjecture on Lifting Covers of Curves
Tuesday, May 29, 2012 - Cancelled, new date TBA - Vincent Pilloni (ENS Lyon)
Classicity of certain weight 1 p-adic modular forms and applications
Abstract We will give a classicity criterion for certain weight 1, p-adicmodular forms. This has applications to the Artin conjecture.
Friday, May 25, 2012- room SRVII - 10:30 - Benoît Stroh (Paris 13)
Overconvergence and classicity for Hilbert modular forms
Abstract We will explain how to generalize to the case of Hilbert varieties a well-known theorem due to Coleman which states that overconvergent modular forms of small slope are classical. This theorem is useful for the construction of eigenvarieties of Andreatta, Iovita and Pilloni. This talk will be linked to the talk of Pilloni next week.
Tuesday, May 22, 2012 - room SRVII - 11:00 - Hershy Kisilevsky (Concordia University, Montreal)
Mordell-Weil groups of elliptic curves under field extension
Abstract Let E be an elliptic curve defined over the rational field $Q$ with L-function $L(E,s)$.We are interested in studying $E(K)$ as $K$ varies over finite extensions of $Q$. Analytically this questions translates (under the Birch and Swinnerton-Dyer conjecture)into when $L(E,1, \chi)=0$ for Artin characters $\chi$. For $[K:Q]=2$, there is an extensive literature on this question. We present our results and conjectures when $[K:Q]>2$.
Friday, May 11, 2012- room 2BC/30 - 11:30 - Luca Migliorini (Bologna)
Ngo's support theorem
Tuesday, May 8, 2012 - room 1BC/45 - 11:00 - George Shabat (Moscow State University)
Dessin d'enfants and algebraic geometry
Abstract A dessin d'enfant is such a graph on a compact oriented surface that its complement is homeomorphic to a disjoint union of 2-cells. It turns out (understood by Grothendieck in 1970's) that the category of dessins d'enfants, being appropriately defined, is equivalent to the category of BELYI PAIRS, i.e. the category of smooth complete curves over the field of algebraic numbers together with a covering of the projective line ramified only over a three points. Therefore the combinatorial topology of graphs on surfaces is somewhat equivalent to a part of arithmetic geometry; the precise statements will be given in the first part of the talk and some examples will be presented. In the second part the deeper relations of the dessins d'enfants theory with algebraic geometry will be discussed, mainly related to the Penner-Kontzevich dessins-labelled stratification of moduli spaces of curves with marked points.
Wednesday, May 2, 2012 - room 2AB/45 - 16:30 - Gianluca Pacienza (Strasbourg)
Un versione logaritmica del "bend-and-break lemma" di Miyaoka-Mori
Abstract Il "bend-and-break lemma" di Miyaoka-Mori e' un risultato che, a partire da una famiglia di curve passanti per un punto fissato di una varietà proiettiva, permette di ottenere una curva razionale di grado limitato passante per questo stesso punto. Le due conseguenze piu' importanti di tale risultato sono il criterio numerico di unirigatezza di Miyaoka-Mori e il celebre teorema del cono. Allo scopo di studiare le variet? quasi-proiettive sarebbe importante estendere il "bend-and-break lemma" alle coppie (X,D), dove X ? una variet? proiettiva e D un divisore a incroci normali. Il solo risultato di rilievo in questa direzione ? stato ottenuto in dimensione 2 da Keel e McKernan. Nel seminario presenteremo un'estensione del "bend-and-break lemma", valida in dimensione qualsiasi, ottenuta in un lavoro in collaborazione con Michael McQuillan.
Tuesday,March 6, 2012- room SRVII- 10:30 - Valeria Marcucci (Pavia)
On a conjecture of Naranjo and Pirola
Abstract We will deal with the possible genus of a curve in a generic Jacobian variety. Given a birational morphism $\varphi\colon D \rightarrow J$, where $D$ is a complex smooth projective curve and $J$ is a generic Jacobian of dimension $g\geq 4$, we will show that the genus of $D$ satisfies either $g(D)=g$, or $g(D)\geq 2g-2$. This gives a positive answer to a conjecture of J.C. Naranjo and G.P. Pirola, who proved an analogous result for Prym varieties. We will then discuss whether the above result can be improved. There are some obstructions to the existence of curves of genus $2g-2$ on $J$ and this is related to a property of Prym varieties of ramified coverings.
Thursday, February 16, 2012- room 2AB/45 - 14:00 - Ernesto Mistretta (Padova)
Stabilita' di Span Duali e Indice di Clifford
Abstract Presentiamo un lavoro svolto con Lidia Stoppino: mostriamo la stabilita' di alcuni fibrati vettoriali su curve algebriche, chiamati Dual Span Bundles, o Lazarsfeld Bundles, legati a varie costruzioni in geomteria algebrica (studio di spazi di moduli di sistemi coerenti, stabilita' dei fibrati di Picard, generazione normale di fibrati vettoriali su curve). Otteniamo la stabilita' di questi fibrati, sotto condizioni legate all'indice di Clifford della curva, mettendola in relazione con altre condizioni di stabilita' di fibrati o sistemi lineari (quali la stabilita' lineare, o la stabilita' coomologica), e in alcuni casi di pendenza intera, mostriamo l'esistenza di divisori-Theta.
Thursday, February 9, 2012 - room 2AB/45 - 14:30 - Chiara Camere (Hannover)
La stabilità del fibrato tangente di P^r ristretto ad una curva o ad una superficie
Abstract Dati X una varietà proiettiva liscia ed L un fibrato in rette su X generato dalle sue sezioni globali, il morfismo di valutazione delle sezioni v:C-->P^r è ben definito e v*T è la restrizione a X del fibrato tangente T di P^r. Nel caso in cui X sia una curva di genere g>1 mostreremo che se deg L>2g-c(C)-1 allora v*T è semistabile, precisando quando si ha stabilità. Vedremo poi come tale risultato sulle curve si possa utilizzare per studiare la \mu-stabilità di v*T rispetto a L nel caso in cui X sia una superficie K3 o abeliana.
Thursday, February 2, 2012 - room 1BC/50 - 14:30 Matteo Longo (Padova)
Quaternionic Darmon points and arithmetic applications
Abstract In 2001 H. Darmon introduced the notion of''Stark-Heegner points'' on rational elliptic curves. These are local points on elliptic curves (defined over the algebraic closure of Q_p). A deep conjecture by Darmon states that they are actually global points (defined over abelian extensions of real quadratic fields). Also, they are non-torsion if and only if the special value of the derivative of the relevant complex L-function of the elliptic curve is non-zero. These conjectures generalize what is known in the case of imaginary quadratic extensions (where the role of Start-Heegner points is played by classical Heegner points). In this talk I will present a construction of Darmon-style points on elliptic curves in a more general setting than that originally considered by Darmon. I will also explain in which cases Darmon's conjectures can actually be proved and, if time permits, I will offer an application to the Birch and Swinnerton-Dyer conjecture for elliptic curves. These results have been obtained in collaboration with V. Rotger and S. Vigni.
Thursday, January 26, 2012 - room - 14:30 - Mario Edmundo (Lisbon)
Grothendieck's six operations on o-minimal sheaves (Joint work with L. Prelli)
Abstract O-minimality is the analytic part of model theory (a branch of logic) and deals with theories of ordered, hence topological, structures satisfying certain tameness properties, it generalizes real and sub-analytic geometry and formalizes Grothendieck's notion of tame topology (topologie modérée). In this context, after given the motivation and presenting some of the most important applications of o-minimality, we present the formalism of Grothendieck's six operations on o-minimal sheaves which eneralizes similar theories in real and sub-analytic geometry
Tuesday, January 17, 2012 - room 2AB/40 - 10:00 - Thomas Hudson
Un'estensione dei polinomi di Schubert alla $K$-teoria connessa mediante il cobordismo algebrico
Abstract Sia $X$ uno schema liscio. Dato un morfismo tra fibrati vettoriali sufficientemente generico, le classi fondamentali dei luoghi di degenerazione ad esso associati possono essere espresse, in quanto elementi dell'anello di Chow $CH^*(X)$, mediante i polinomi doppi di Schubert. A patto di sostituire a questi ultimi i polinomi doppi di Grothendieck, questo risultato ha un esatto corrispettivo in $K^0(X)$, l'anello di Grothendieck dei fibrati vettoriali algebrici su $X$. In questo seminario illustrer\`o la costruzione geometrica alla base di questi due risultati e come sia possibile utilizzarla nell'ambito del cobordismo algebrico $\Omega^*$, la teoria coomologica orientata universale. Spiegher\`o inoltre come, in virt\`u dell'universalit\`a di $\Omega^*$ , sia poi possibile specializzare quanto ottenuto alla $K$-teoria connessa, una teoria coomologica che generalizza sia $CH^*$ che $K^0$, fornendo un'interpretazione geometrica ai $\beta$-polinomi di Fomin e Kirillov.
Thursday, January 12, 2012 - room 2AB/45 - 14:30 - Sofia Tirabassi (Roma III)
Effettive Iitaka fibrations of irregular varieties
Abstract We present a joint work with Z. Jiang and M. Lahoz in which we prove that the 4-canonical map of varieties of maximal Albanese dimension always induces the Iitaka fibration.
Thursday, December 15, 2011 - room 2AB/45 - 14:30 - Marco Boggi (Universidad de los Andes, Bogotà)
Galois coverings of moduli spaces of curves and loci of curves with symmetry
Abstract Let $\overline{\cal M}_{g,[n]}$, for $2g-2+n>0$, be the stack of genus $g$, stable algebraic curves over $\mathbb C$, endowed with $n$ unordered marked points. Looijenga introduced the notion of Prym level structures in order to construct smooth projective Galois coverings of the stack $\overline{\cal M}_{g}$. We will introduce the notion of Looijenga level structure which generalizes Looijenga construction and provides a tower of Galois coverings of $\overline{\cal M}_{g,[n]}$ equivalent to the tower of all geometric level structures over $\overline{\cal M}_{g,[n]}$. Then, Looijenga level structures are interpreted geometrically in terms of moduli of curves with symmetry. A byproduct of this characterization is a simple criterion for their smoothness. As a consequence of this criterion, we will show that Looijenga level structures are smooth under relatively mild hypotheses. Time permitting, we will give a description of the nerve of the D--M boundary of abelian level structures and show how this construction can be used to "approximate" the nerve of Looijenga level structures. These results are then applied to elaborate a new approach to the congruence subgroup problem for the Teichmüller modular group.
Rimandato - room TBA - 10:15 - Matteo Longo (Padova)
Quaternionic Darmon points and arithmetic applications
Abstract In 2001 H. Darmon introduced the notion of "Stark-Heegner points" on rational elliptic curves. These are local points on elliptic curves (defined over the algebraic closure of Q_p). A deep conjecture by Darmon states that they are actually global points (defined over abelian extensions of real quadratic fields). Also, they are non-torsion if and only if the special value of the derivative of the relevant complex L-function of the elliptic curve is non-zero. These conjectures generalize what is known in the case of imaginary quadratic extensions (where the role of Start-Heegner points is played by classical Heegner points). In this talk I will present a construction of Darmon-style points on elliptic curves in a more general setting than that originally considered by Darmon. I will also explain in which cases Darmon's conjectures can actually be proved and, if time permits, I will offer an application to the Birch and Swinnerton-Dyer conjecture for elliptic curves. These results have been obtained in collaboration with V. Rotger and S. Vigni.
Thursday, December 1, 2011- room 2BC/60- 11:30 - Alessia Mandini (Instituto Superior Tècnico, Lisbon University)
Hyperpolygons and Moduli Spaces of Parabolic Higgs Bundles
Abstract I will describe two families of manifolds: hyperpolygon spaces and moduli spaces of stable, rank-2, holomorphically trivial parabolic Higgs bundles over $P^1$ with fixed determinant and trace free Higgs field, proving the existence of an isomorphism between them. This relationship connecting two different fields of study allows us to benefit from techniques and ideas from each of these areas to obtain new results. In particular, using the study of variation of moduli spaces of parabolic Higgs bundles over a curve, we describe the dependence of hyperpolygon spaces $X(\alpha)$ and their cores on the choice of the parameter $\alpha$, and show that, when a wall is crossed, the hyperpolygon space suffers an elementary transformation in the sense of Mukai. If time permits, I will describe how one can take advantage of the geometric description of the core components of a hyperpolygon space to obtain explicit expressions for the computation of the intersection numbers of the core components of hyperpolygon spaces. Using our isomorphism we can obtain similar formulas for the nilpotent cone components of the moduli space of rank-2, holomorphically trivial parabolic Higgs bundles over $P^1$ with fixed determinant and trace-free Higgs field. This is joint work with Leonor Godinho.
Thursday, October 13, 2011 - room 1BC/45 - 14:00 - Jilong Tong (Bordeaux)
On torsors under elliptic curves
Thursday, October 6, 2011 - room 1BC/45 - 11:30 Monday, October 10, 2011 - room 1BC/45 - 10:00 - T. Abe (Tokyo)
Nearby cycles and vanishing cycles
Wednesday, September 28, 2011 - room 2AB/40 - 15:00 - Francesco Baldassarri (Padova)
Finite morphisms of p-adic curves and radii of convergence of connections
Abstract We illustrate the theory of radii of convergence of p-adic connections, on the example of the direct image connection for a finite morphism of p-adic curves. We prove a variant of the p-adic Rolle theorem. We compare the local definition of radius of convergence used by Kedlaya with the global one I have recently introduced. This comparison is crucial to obtain our statements.
Thursday, September 22, 2011 - room 2AB/45 - 14:00 - Adriano Marmora (Univ. Strasburgo)
Product formula for $p$-adic epsilon factors
Abstract Let $X$ be proper and smooth curve on a finite field of characteristic $p$ and $\ell$ be a prime different from $p$. In 1987, Laumon proved a formula, conjectured by Deligne, which correlates the constant appearing in the functional equation of the $L$-function of an $\ell$-adic sheaf over $X$, with the product of local data (the epsilon factors) at the points of $X$. In this talk, we report on the analogue of this formula in rigid cohomology, which has been recently proved in a joint work with Tomoyuki Abe.
Wednesday, September 21, 2011 - room 1BC/50 - 10:30 - Vadim Vologodsky (Univ. of Oregon)
On the center of the ring of differential operators on a smooth variety over $Z/p^nZ$
Abstract This is a joint work with Allen Stewart. Given an associative algebra $A_0$ over $Z/pZ$ and its deformation $A_n$ over $Z/p^nZ$, I will show that under a certain non-degeneracy condition the center of $A_n$ is isomorphic to the ring of length n Witt vectors over the center of $A_0$. As an application, I will compute the center of the ring of differential operators on a smooth variety over $Z/p^nZ$ confirming a conjecture of Kaledin. My talk will be completely elementary (accessible to the second year graduate students).
Tuesday, June 28, 2011 - room 2BC/60 - 15:00 - Giovanni Cerulli Irelli (Bonn)
Uso della teoria delle rappresentazioni di quiver nello studio della geometria di degenerazioni di varietà di bandiera
Abstract In un recente lavoro in collaborazione con Evgeny Feigin e Markus Reineke abbiamo notato che le degenerazioni di varietà di bandiera studiate da Feigin sono varietà ben note nella teoria delle rappresentazioni di quiver: le cosiddette Grassmanniane quiver. Data una rappresentazione M di un quiver Q, la Grassmanniana quiver Gr_e(M) parametrizza le sottorappresentazioni di M di dimensione e. In questo seminario illustreremo come questa semplice osservazione produca potenti semplificazioni nello studio della geometria delle varietà di bandiera degenerate da Feigin.
Thursday, May 26, 2011 - room 1A/150 - 11:30 - Francois Brunault (Lyon)
Explicit p-adic regulators for K_2 of elliptic curves
Abstract We will explain how to use the local part of Kato's Euler system and the Perrin-Riou exponential map to get an explicit formula for the p-adic regulator of specific elements in $K_2$ of the modular curve $X(N)$. From this we deduce a similar formula for the modular curves $X_1(N)$ and $X_0(N)$. This construction yields, for an elliptic curve E defined over Q without complex multiplication, an explicit element in $K_2(E)$ whose p-adic regulator is proportional to the special value at 0 of the p-adic L-function associated to E.
Thursday, May 19, 2011 - room 2BC/60 - 14:00 - Cinzia Casagrande (Torino)
Sul numero di Picard dei divisori di una varietà di Fano
Abstract Parleremo di un risultato che mette in relazione il numero di Picard di una varietà di Fano e quello di un divisore nella varietà. Più precisamente, sia X una varietà di Fano complessa e liscia, e D un divisore primo in X. Allora la dimensione d del nucleo della restrizione $N^1(X) \to N^1(D)$ è al più 8. Inoltre, se d>3, X è il prodotto di una superficie di Del Pezzo con un'altra varietà di Fano. Dopo aver spiegato il risultato e il suo contesto, darò un'idea delle tecniche usate nella dimostrazione, e discuterò qualche applicazione.
Wednesday, May 4, 2011 - room 1BC/45 - 11:00 - Jorge Mozo Fernandez (Valladolid)
Results on analytic classification of germs of holomorphic foliations
Abstract We shall review the main known results concerning the analytic classification of germs of codimension one, singular holomorphic foliations in dimension two and three. In dimension two, we shall focus in the classical works of J. Martinet and J.P. Ramis, in the reduced case, and in the works of Cerveau, Moussu, Meziani, Berthier, Sad and others, in the nilpotent case. In the quasi-homogeneous case, we shall mention the work of Y. Genzmer. The state-of-art of this subject in dimension three will be explained. For, we shall recall the main concepts involved: reduction of singularities, existence of separatrices, and holonomy of the leaves, and how they are used in order to establish the results.
Thursday, April 28, 2011 - room 1A/150 - 11:30 - Francesco Bottacin
Sulla corrispondenza di Simpson
Wednesday, April 20, 2011 - room 2BC/60 - 15:00 - Shanwen Wang (EPFL)
Kato's Euler system and p-adic L function of modular form
Abstract I will sketch the method decribed in Colmez's Bourbaki talk, and explain what I had modified to make Colmez's method.
Tuesday, April 19, 2011 - room 2AB/45 - 16:15 Wednesday, April 20, 2011 - room 2BC/60 - 14:00 - Daniele Faenzi (Pau)
Configurazioni di iperpiani e fibrati logaritmici
Abstract Nella prima parte del seminario vedremo una panoramica di alcuni invarianti associati a una configurazione di iperpiani, in termini combinatorici, algebrici, e geometrici, con un particolare interesse per il fascio delle forme con poli logaritmici. Nella seconda parte presenterò un lavoro in collaborazione con D. Matei e J. Vall?s (arXiv:1011.4611). Vedremo come da questo invariante si possa, sotto certe condizioni, ricostruire completamente la configurazione (teorema di Torelli).
Friday, April 15, 2011 - room 2AB/45 - 14:30 - Denis Benois (Bordeaux)
Trivial zeros of modular forms at near critical points
Thursday, April 14, 2011 - room 2AB/45 - 12:00 - Simone Diverio (Institut de Mathématiques de Jussieu - UPMC, Paris)
Positività e iperbolicità secondo Kobayashi delle varietà complesse proiettive
Abstract Una varietà complessa compatta è iperbolica secondo Kobayashi se e solo se non ammette immagini olomorfe non costanti del piano complesso. Se tale varietà è anche proiettiva, l'iperbolicità è congetturalemente equivalente a diverse propriet? algebriche di positività del suo fibrato canonico. In questo seminario cercheremo di fare una panoramica di tali equivalenze, illustrando in particolare alcuni risultati volti a provare che una varietà proiettiva iperbolica ha fibrato canonico ampio.
Tuesday, April 5, 2011 - room 1BC/45 - 11:30 - Minhyong Kim (University College, London)
Siegel's theorem for P^1-{three points}
Wednesday, March 30, 2011 -room 2AB/45 - 12:00 - Matteo Penegini (Bayreuth Universitaet, Germany)
Nuove superfici di Beauville e gruppi finiti
Abstract In questo talk presenterò la costruzione di nuove superfici di Beauville con gruppo $PSL(2,p^e)$, o con gruppo alterno $A_n$, o con gruppo simmetrico $S_n$ e alcuni esempi con gruppi di Lie semplici. Le costruzioni si basano su metodi probabilistici di teoria dei gruppi sviluppati da Liebeck, Shalev e Marion nei casi $A_n$, $S_n$ e gruppi di Lie semplici, mentre nel caso $PSL(2,p^e)$ la costruzione si basa sull'analisi dei sottogruppi del gruppo stesso sviluppata da MacBeath. Questo lavoro è frutto di una collaborazione con S. Garion.
Thursday, March 24, 2011 - room 1BC/50 - 11:00 - Orsola Tommasi (Hannover)
La coomologia dello spazio di moduli delle varieta' abeliane di dimensione 4
Abstract La coomologia dello spazio di moduli $A_g$ delle varieta' abeliane principalmente polarizzate di dimensione $g$ e' nota solo per $g\leq 3$. In questi casi, i risultati si basano sul fatto che l'applicazione di Torelli e' dominante, nel senso che in questi casi una varieta' abeliana generale e' sempre la jacobiana di una curva. Questo permette di descrivere $A_g$ usando informazioni sugli spazi di moduli di curve, la cui coomologia e' meglio nota. In dimensione 4, l'applicazione di Torelli non e' piu' dominante e la sua immagine e' un divisore in $A_4$. Cio' nonostante, ha senso porsi il problema di quanta parte delle coomologia di $A_4$ sia determinata dalla coomologia dello spazio di moduli delle curve lisce di genere 4 e da quella degli spazi di moduli di varieta' abeliane di dimensione piu' piccola. In questo seminario, basato su una ricerca in collaborazione con Klaus Hulek (Hannover), cercheremo di rispondere a questa domanda per quanto riguarda la coomologia a coefficienti razionali della seconda compattificazione di Voronoi di $A_4$, una compattificazione toroidale che risulta particolarmente interessate per motivi geometrici. Spiegheremo in che modo l'approccio qui delineato determina la coomologia della compattificazione in tutti i gradi differenti da quello intermedio.
Thursday, March 16, 2011 - SR 7th floor B - 14:00 - Masaaki Yoshida (Kyushu University)
Hyperbolic Schwarz maps
Thursday, March 10, 2011 - room 2AB/40 - 15:00 Friday, March 11, 2011 - room 2AB/40 - 15:00 Wednesday, March 16, 2011 - SR 7th floor B - 14:00 Wednesday, April 6, 2011 - room 2BC/60 - 13:35 Thursday, April 7, 2011 - SR 7th floor B - 11:00 - Lorenzo Ramero (Lille)
Almost purezza e teoria p-adica di Hodge
Abstract La teoria p-adica di Hodge produce un isomorfismo naturale tra la coomologia étale p-adica di una variet? definita su un campo p-adico, e la coomologia di de Rham (o cristallina) della stessa variet?. Vari metodi sono attualmente disponibili per ottenere tali isomorfismi; uno di questi è dovuto a Faltings, ed è basato sul suo "teorema della almost purezza". Intendo dapprima presentare l'enunciato del teorema della almost purezza, e spiegare come lo si utilizzi per stabilire gli isomorfismi tra funtori coomologici. In seguito, discutero' le idee principali che entrano nella dimostrazione di questo teorema.
Thursday, March 3, 2011 - room 2BC/60 - 14:00 - Marc-Hubert Nicole (Marseille)
Conjectures de troncature de Traverso et raffinements de la stratification de Newton
Abstract Vers 1979, C. Traverso conjectura l'existence de bornes universelles surprenamment petites pour d?terminer un groupe de Barsotti-Tate ? isomorphisme pr?s (resp. isog?nie pr?s) ? partir de ses groupes de Barsotti-Tate tronqu?s d'?chelon n. La conjecture d'isog?nie fut prouv?e par le conf?rencier et A. Vasiu en 2006 de mani?re ?l?mentaire. Dans cet expos?, nous parlerons de la conjecture d'isomorphisme. Nous remplacerons la borne originale de Traverso par une borne correcte et optimale (dans un sens pr?cis). En particulier, cette conjecture de Traverso est vraie quand la codimension du groupe de Barsotti-Tate est ?gale ? sa dimension (exemple primordial: le groupe de Barsotti-Tate d'une vari?t? ab?lienne). Nous d?crirons aussi la variation en famille des invariants qui apparaissent naturellement dans la r?solution de ces conjectures, et qui donnent lieu ? des stratifications naturelles des strates [sic] de Newton. Travail en commun avec E. Lau et A. Vasiu.
Monday, January 31, 2011 - room 1BC/50 - 16:00 - Nicola Mazzari
On the rigid and the de Rham cycle classes
Abstract Let $V=Spec(R)$ and $R$ be a complete discrete valuation ring of mixed characteris $(0,p)$. For any flat $R$-scheme $X$ the de Rham fundamental class of the generic fiber is compatible with the rigid fundamental class of the special fiber. This is the key result of a joint work with B. Chiarellotto and A. Ciccioni. I will sketch the proof and give some applications.
Friday, January 28, 2011 - SR 7th floor B - 10:00 - Linsheng YIN (Tsinghua University, Pechino)
Index-class number formulae
Abstract In this talk, I will introduce some results on the index of units generated by the torsion points of rank one Drinfeld modules, and the index of Stickelberger ideal constructed by zeta functions in Galois group ring, which generalize Kummer-Sinnott's and Iwasawa-Sinnott's formulae respectively.
Monday, January 17, 2011 - room 2BC/50 - 14:00 Wednesday, January 19, 2011 - room 1BC/50 - 14:00 - G. Vezzosi (Firenze)
An introduction to DAG
Abstract Why derived algebraic geometry (dag)? - Overview of dag: derived stacks, derived algebraic stacks, deformation theory, base-change formula, quasi-smoothness and obstruction theories - Examples of derived stacks - Open problems - Chern character for dg-categories.
Thursday, January 13, 2011 - room 1BC/50 - 14:30 - Teresa Monteiro Fernandes (Universidade de Lisboa)
Grauert's theorem for open subanalytic sets in the real domain
Abstract One of the greatest contributions of H. Grauert to complex analysis and geometry is the celebrated theorem which asserts that any open set in a real analytic manifold admits, in any of its complexifications, a fondamental system of Stein open neighborhoods. In particular, it is the trace of a Stein open set. In this talk we explain how to obtain the subanalytic version of this theorem using a very deep result of Bierstone, Millman and Pawlucki which asserts that for any $p$, any closed subanalytic set is the zero set of a $C^p$ function. We finish giving an example and applications. (joint work with Daniel Barlet)
Monday, December 20, 2010 - room 2AB/40 - 14:30 - Olivier Brinon (Paris 13)
Surconvergence de la monodromie p-adique, application aux formes modulaires p-adiques surconvergentes
Abstract Dans ce travail en commun avec Farid Mokrane, nous prolongeons la tour d'Igusa au-dessus d'un voisinage strict du lieu ordinaire dans la variet? de Siegel de genre $r$ de niveau $N \geq 3$ sur $Q_p$ (avec p>2), ce qui permet de donner une definition des formes modulaires $p$-adiques surconvergentes.
Thursday, December 16, 2010 - room 1BC/50 - 14:30 - Thomas Ludsteck
Unipotent Schottky bundles on curves and abelian varieties
Abstract In a joint work with Carlos Florentino, we study a natural map from representations of a free (resp. free abelian) group of rank g in $GL_r (C)$, to holomorphic vector bundles of degree 0 over a compact Riemann Surface X of genus g (resp. complex torus of dimension g). These free (resp. free abelian) groups are associated with Schottky uniformizations of X and are called Schottky groups. Our main result is that this natural map induces an equivalence of categories between the category of unipotent representations of Schottky groups, as well as the category of unipotent vector bundles on X. Finally we give an application to the $p$-adic case, where we show that for an algebraic $p$-adic torus there is an equivalence of categories between the category of integral and discrete representations of the temperate fundamental group, as well as the category of homogeneous vector bundles.
Friday, December 9 - room 1BC/50 - 15:00 Thursday, December 10 - room 1BC/50 - 15:00 Monday, December 13 - room 1BC/50 - 15:00 Wednesday, December 15 - room 1BC/50 - 15:00 - Yves Andr? (ENS, Paris)
Introduction to Motives
I: Overview. II: Topics on pure motives. III: Foundations of mixed motives. IV: Periods and mixed Tate motives.
T.B.A. - Yves Andr? (ENS, Paris)
New viewpoints on Euclidean lattices (categorical, group-theoretic, and valuative aspects
Friday, November 26, 2010 - room 1A/150 - 15:30 - Gergely Zabradi (Budapest)
Exactness of the reduction on etale modules
Abstract We prove the exactness of the reduction map from etale (phi,Gamma)-modules over completed localized group rings of compact open subgroups of unipotent p-adic algebraic groups to usual etale (phi,Gamma)-modules over Fontaine's ring. This reduction map is a component of a functor from smooth p-power torsion representations of p-adic reductive groups (or more generally of Borel subgroups of these) to (phi,Gamma)-modules. Therefore this gives evidence for this functor - which is intended as some kind of p-adic Langlands correspondence for reductive groups - to be exact.
Friday, November 19 - room 1BC/50 - 14:30 Monday, November 22 - room 1BC/50 - 14:30 Friday, November 26 - room 1A/150 - 14:00 - Gerard Freixas (Jussieu, Paris)
Generalized holomorphic analytic torsion
Abstract The aim of this series of talks is to present joint work with J. I. Burgos and R. Litcanu on holomorphic analytic torsion. Analytic torsion forms are differential forms that transgress the Grothendieck-Riemann-Roch theorem to the level of differential forms. The existing constructions impose a number of restrictions. For instance, one can only define analytic torsion forms of hermitian vector bundles with respect to proper submersions of complex manifolds. We develop a formalism to extend the theory of analytic torsion forms to complexes of coherent sheaves with suitable hermitian structures and to arbitrary projective morphisms. We will divide the exposition in three talks. The rough contents will be as follows: - Lecture 1: The first talk will review the theory of Bott-Chern secondary classes and introduce the notion and basic properties of holomorphic analytic torsion forms. This includes work of Bismut and coworkers. We will try to insist on the motivating points. - Lecture 2: We will present the formalism of hermitian structures on objects of the bounded derived category of coherent sheaves. In particular we will introduce the category of complex algebraic varieties and projective morphisms with metrics. All this will reveal useful to extend the notion of analytic torsion forms to the case of coherent sheaves and arbitrary projective morphisms. - Lecture 3: In the last talk we will state an existence theorem and a classification of all possible theories of analytic torsion forms. Some applications will be presented, in relation to Grothendieck duality and Quillen metrics of degenerating families of curves.
Friday, November 12, 2010 - room 2AB/40 - 15:30 - Matteo Penegini (Bayreuth)
On surfaces with $p_g=q=2$ , $K^2=5$ and Albanese map of degree $3$
Abstract In a joint work with Francesco Polizzi, we construct a connected, irreducible component of the moduli space of minimal surfaces of general type with $p_g=q=2$ and $K^2=5$, which contains both examples given by Chen-Hacon and by myself in a previous work. This component is generically smooth of dimension $4$, and all its points parametrize surfaces whose Albanese map is a generically finite triple cover. In this talk I shall explain how we proved this result starting with the definition of Chen-Hacon surface and giving a characterization of these surfaces.
Thursday, November 4, 2010 - room 2AB/45 -14:30 - Matteo Longo (Padova)
Punti di Heegner ed aritmetica delle famiglie di Hida
Abstract Nella prima parte del seminario, dopo una panoramica su alcuni risultati legati alla congettura di Birch e Swinnerton-Dyer per curve ellittiche razionali, discuteremo generalizzazioni di questi risultati al contesto di famiglie di Hida di forme modulari. Nella seconda parte daremo un cenno alla costruzione di famiglie di punti di Heegner che, congetturalmente, controllano il rango di gruppi di Selmer legati alla rappresentazione di Galois associata ad una data famiglia di Hida.
Thursday, October 28, 2010 - room 2AB/45 - 14:30 - Ernesto C. Mistretta (Padova)
Stabilità e divisori Theta
Abstract Definiamo il divisore Theta associato ad un fibrato vettoriale con pendenza intera. L'esistenza del divisore Theta associato ad un fibrato implica la $\mu$-semistabilità del fibrato. Mostriamo che sotto opportune condizioni i nuclei delle valutazioni associate a proiezioni generiche ammettono divisore Theta e sono quindi semistabili. Le tecniche impiegate possono essere generalizzate per studiare i legami tra la stabilità lineare e la $\mu$-(semi)stabilità di nuclei di valutazioni (W.I.P. con Lidia Stoppino).
Thursday, October 21, 2010 - room 2AB/45 - 14:00 - Lidia Stoppino (Università dell'Insubria)
Stability conditions and inequalities for surfaces
Abstract Let $S$ be a complex surface with a fibration $f$ over a curve $B$. Consider a divisor $L$ on $S$, and a subsheaf $\mathcal G$ of the pushforward $ f_*\mathcal{O}_S(L)$. Note that the fibre of $\mathcal G$ on $b\in B$ represents a linear system $\mathcal G_b\subseteq H^0(L_{|f^*(b)})$ on the fibre $f^*(b)$. I will describe three different stability properties for these objects, and three methods that exploit these properties for deriving bounds on the ratio $L^2/\deg \mathcal G$. There is a yet another stability property which seems to be the key concept that unifies the three approaches: the $\textit{linear stability}$ of the linear system $\mathcal G_b$ on the general fibre $f^*(b)$. I will then describe an application of these methods to the geography of fibred surfaces. The geographical problem for fibred surfaces is to find the range of variation of the basic invariants $K_f^2$ and $\chi_f$, and constrains imposed by geometrical properties of $f$. Linear stability can be proved for suitable canonical projections on a curve of genus $\geq 2$, and this allows us to obtain a lower bound for $K^2_f/\chi_f$, that depends on the relative irregularity $q_f$ and on the Clifford index of the general fibre. These are results obtained in collaboration with Miguel Angel Barja.
Friday, October 15, 2010 - room 2BC/60 - 14:30 - Laura Desideri (Tubinga)
The Plateau problem, Fuchsian systems and the Riemann-Hilbert problem
Abstract The Plateau problem is to prove than any given closed and connected Jordan curve in R3 bounds at least one minimal surface of disk-type. The first almost complete resolutions are given in the early 1930's by Douglas and Rado. However, in 1928, Garnier published a resolution for polygonal boundary curves which seems to have been forgotten. His proof is really different from the variational method, it relies on the fact that one can associate with each minimal disk with a polygonal boundary curve a real Fuchsian second-order equation defined on the Riemann sphere. The monodromy of the equation is determined by the oriented directions of the edges of the boundary. To solve the Plateau problem, we are thus led to solve a Riemann-Hilbert problem and to use isomonodromic deformations of Fuchsian equations. I will give a sketch of the proof, and I will briefly explain how Garnier's result can be extended into Minkowski 3-space.
