Michael Temkin (Einstein Institute of Mathematics, Jerusalem)

4 Talks on Berkovich spaces and differential forms.
Thursday Oct. 4 at 15:00 room 2BC30, Friday Oct. 5 at 14:15 room 2AB45, Thursday Oct. 11 at 14:15 room SRVII, Friday Oct. 12 at 14:15 room SRVII.

Abstract and title of the talks:
1. "Metrization of differential forms on Berkovich spaces"
Quite naturally, sheaves of differential forms on Berkovich spaces admit a natural metrization, that I will construct in this lecture. Slightly surprisingly, this induces only upper semicontinuous seminorm functions on differential forms, so in the first part of the talk we will develop a general theory of metrics on sheaves of modules. This is based on my paper "Metrization of differential pluriforms on Berkovich analytic spaces".
2. "Differential forms on Berkovich curves: the residual characteristic zero case"
In this talk we will consider Berkovich curves over a field of residual characteristic zero (e.g. Pusiex series over C). In this case, I will provide a very concrete description of differential forms on curves and their reduction, and a lifting theorem. This is a joint work with I. Tyomkin (and the only work in progress in this series of talks), which aims to provide a purely algebraic approach to results of Bainbridge-Chen-Gendron-Grushevsky-Moelkler on compactification of moduli spaces of curves with a differential form.
3. "Wild covers of Berkovich curves: the different and profile functions"
In this talk I consider the case when the residual characteristic is positive, and I will describe a structure of wildly ramified covers of Berkovich curves. The first main invariant of such a covering is a so-called different function, which is closely related to norms of differential forms. A more refined invariant, the profile function, is an analogue of the Herbrand's function in the classicaltheory of ramification. This talk is based on the paper "Morphisms of Berkovich curves and the different function" joint with Cohen and Trushin, and my paper "Metric uniformizaiton of morphisms between Berkovich curves".
4. "Wild covers of Berkovich curves: the lifting problem"
The different function of a wild covering can be viewed as a tropical invariant. In this talk I will further refine it to a reduction type invariant, which is a certain bivariant differential form on residual curves. Then for minimally wild covers I will also prove a lifting theorem stating that any such residual datum satisfying a list of explicit conditions can be lifted to an actual morphism of curves. This can be seen as a generalization of a theorem of Baker-Brugalle-Payne-Rabinoff to the (minimally) wild case.

Glenn Stevens (Boston Univ.)

p-Adic Zeta Functions and the Main Conjecture of Iwasawa Theory
8 Lectures, on Wednesday, 16:30, in Room 2AB40.
First Lecture: Monday April 11, 2018

Abstract: In recent decades, p-adic analysis -- a synthesis of arithmetic and analysis -- has emerged as a fascinating and independent offspring of the union of complex analysis and arithmetic. Our course will illustrate the power of these tools by focusing on a key example -- the Main Conjecture of Iwasawa Theory -- which expresses deep and unexpected connections between special values of the Riemann Zeta function and the arithmetic of cyclotomic number fields. The importance of these ideas is attested to by the many generalizations of the Main Conjecture that have emerged over time and which continue to play a prominent role in the theory of elliptic curves, and automorphic representation theory.
Iwasawa himself gave the first proof of his Main Conjecture in cases where a conjecture of Vandiver is known to be true. The general case of Iwasawa's conjecture was first proved by Mazur and Wiles in 1984. More recently, Karl Rubin showed how to use new ideas of Kolyvagin and Thaine to give a more elementary proof based on the theory of Euler systems. In our course we will explain the proof following the methods of Iwasawa and Rubin, while developing general methods from Iwasawa theory and the theory of Euler systems.

Lorenzo Ramero (Lille)

On Huber's theory of adic spaces
Thu 18/01/2018, 15:00-16:30 room 2BC30
Thu 25/01/2018, 14:30-16:00 room 2AB45
Fri 26/01/2018, 11:00-12:30 room 2AB45

Abstract: I will present some modest generalization and some variants of Huber's theory of adic spaces. I will try to recall the relevant definitions and preliminaries
from topological algebra, but some previous exposure to rigid analytic varieties would be useful.

Tadashi Ochiai.

Iwasawa Theory
Lecture time: 11:30-13:00
Mon 22/01/2018, 1BC50
Tue 23/01/2018, 2BC30
Wed 24/01/2018, 2BC30

Abstract:
First Lecture (Mon)
Title: Cyclotomic Iwasawa Main conjecture for ideal class group
Abstract: We start by some motivation for the Iwasawa theory of class group. Then we explain some results on the algebraic side and analytic side of the theory. For the algebraic side, we recall some basic results on class group including Iwasawa class number formula. For the analytic side, we recall what is the p-adic L-function of Kubota-Leopoldt. We finish by explaining Iwasawa Main Conjecture.
Second Lecture (Tue)
Title: Cyclotomic Iwasawa Main conjecture for elliptic curves
Abstract: We discuss how Iwasawa theory for ideal class group of yesterday is generalized to elliptic curves. New point is that the theory is quite different depending on the elliptic curve is ordinary or supersingular at the fixed prime. We mainly concentrate on the ordinary case and recall the Iwasawa Main conjecture in that case.
Third Lecture (seminar talk) (Wed)
Title: Iwasawa Main conjecture for Coleman families
Abstract: We can consider Iwasawa theory for a family of modular forms. In ordinary case, we have established a generalization of Kato's result from Iwasawa Main Conjecture for a single modular to Iwasawa theory for a Hida family. We discuss Iwasawa theory for non-ordinary family (Coleman family) as a further generalization.

J. M. Fontaine (Paris-Sud).

Almost $C_p$-representations and vector bundles
Lecture time: 14:30-16:00

Abstract: Let $K$ be a finite extension of $\mathbb Q_p$ and $G_K$ the absolute Galois group. Then $G_K$ acts on the fundamental curve $X$ of $p$-adic Hodge theory and we may consider the abelian category $\mathcal{M}(G_K)$ of coherent $\mathcal{O}_X$-modules equipped with a continuous and semi-linear action of $G_K$.
An $\text{ almost $C_p$-representation of $G_K$}$ is a $p$-adic Banach space $V$ equipped with a linear and continuous action of $G_K$ such that there exists $d\in\mathbb N$, two $G_K$-stable finite dimensional sub-$\mathbb Q_p$-vector spaces $U$ of $V$, $U'$ of $C_p^d$, and a $G_K$-equivariant isomorphism
$$V/U\longrightarrow C_p^d/U'\ .$$
These representations form an abelian category $\mathcal{C}(G_K)$.
The main purpose of this series of lectures is to construct an equivalence of triangulated categories
$$D^b(\mathcal{M}(G_K))\longrightarrow D^b(\mathcal{C}(G_K))$$
and to describe possible generalisations.
Main topics:
-- Construction and main properties of the curve X, generalisations,
-- Classification of coherent $\mathcal{O}_X$-modules,
-- The category $\mathcal{M}(G_K)$,
-- Almost $C_p$-representations,
-- Effective coherent $\mathcal{O}_X[G_K]$-modules and effective almost $C_p$-representations,
-- The main theorem.
-- The étale, the pro-étale and the $v$ sites of an adic space $S$.
-- Possible generalisations.
-- The case where $S$ is the adic space associated to $C_p$. Banach-Colmez spaces.

L. Illusie.

Revisiting the de Rham-Witt complex
(Doctoral course.)
Lecture time: 14:30-16:00

Abstract: The de Rham-Witt complex was constructed by Spencer Bloch in the mid 1970's as a tool to analyze the crystalline cohomology of proper smooth schemes over a perfect field of characteristic $p >0$, with its action of Frobenius, and describe its relations with other types of cohomology, like Hodge cohomology or Serre's Witt vector cohomology. Since then many developments have occurred.
After recalling the history of the subject, I will explain the main construction and in the case of a polynomial algebra give its simple description by the so-called complex of integral forms. I will then describe the local structure of the de Rham-Witt complex for smooth schemes over a perfect field and its application to the calculation of crystalline cohomology. In the proper smooth case, I will discuss the slope spectral sequence and the main finiteness properties of the cohomology of the de Rham-Witt complex in terms of coherent complexes over the Raynaud ring. I will mention a few complements (logarithmic Hodge-Witt sheaves, Hyodo-Kato log de Rham-Witt complex, Langer-Zink relative variants), and make a tentative list of open problems.

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