Number theory seminar

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Date / Time

Speaker

Title

Location

* 21 September 2020
17:15-18:15

Prof. Dr. David Loeffler
University of Warwick

Event Details

Title	Euler systems and special values of L-functions
Speaker, Affiliation	Prof. Dr. David Loeffler, University of Warwick
Date, Time	21 September 2020, 17:15-18:15
Location	HG G 43
Abstract	I will sketch the statement of the Bloch--Kato conjecture, which relates special values of L-functions to the cohomology of Galois representations; this extends many classical theorems and conjectures in number theory, such as the Dirichlet unit theorem for number fields, and the Birch--Swinnerton-Dyer conjecture for elliptic curves. I will then introduce Euler systems, which are the most powerful tool known for attacking this conjecture. I will describe a novel strategy developed by my coauthors and myself for constructing Euler systems, using ideas from the theory of modular and automorphic forms (the 'Langlands program'), and explain our ongoing work where we use these Euler systems to prove many new cases of the Bloch--Kato conjecture.

Euler systems and special values of L-functionsread_more

16 October 2020
14:15-15:15

Prof. Dr. Ana Caraiani
Imperial College London

Event Details

Title	Vanishing theorems for Shimura varieties
Speaker, Affiliation	Prof. Dr. Ana Caraiani, Imperial College London
Date, Time	16 October 2020, 14:15-15:15
Location	Zoom Seminar
Abstract	The Langlands program is a vast network of conjectures that connect number theory to other areas of mathematics, such as representation theory and harmonic analysis. The global Langlands correspondence can often be realised through the cohomology of Shimura varieties, which are certain moduli spaces equipped with many symmetries. In this talk, I will survey some recent vanishing results for the cohomology of Shimura varieties with mod p coefficients and mention several applications to the Langlands program and beyond. I will discuss some results that have an l-adic flavour, where l is a prime different from p, that are primarily joint work with Peter Scholze. I will then mention some results that have a p-adic flavour, that are primarily joint work with Dan Gulotta and Christian Johansson. I will highlight the different kinds of techniques that are needed in these different settings using the toy model of the modular curve.

Vanishing theorems for Shimura varietiesread_more

Zoom Seminar

* 30 October 2020
14:15-15:15

Prof. Dr. Peter Simon Jossen
ETH Zurich, Switzerland

Event Details

Title	A non-hypergeometric E-function
Speaker, Affiliation	Prof. Dr. Peter Simon Jossen, ETH Zurich, Switzerland
Date, Time	30 October 2020, 14:15-15:15
Location	Zoom Seminar
Abstract	With the goal of generalising the theorems of Hermite, Lindemann, and Weierstrass about transcendence of values of the exponential function, Siegel introduced the notion of E-function in his landmark 1929 paper "Über einige Anwendungen diophantischer Approximationen". Hypergeometric functions provide a rich class of E-functions, and Siegel asked whether in fact every E-function is a polynomial expression in hypergeometric E-functions. In a joint work with Javier Fresán, we answer Siegel's question in the negative.

A non-hypergeometric E-functionread_more

Zoom Seminar

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