# Seminars Calendar

## 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.

# Working Seminars

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