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Herbstsemester 2020

Datum / Zeit Referent:in Titel Ort
* 21. September 2020
17:15-18:15 		Prof. Dr. David Loeffler
University of Warwick
Details

Number Theory Seminar

Titel Euler systems and special values of L-functions
Referent:in, Affiliation Prof. Dr. David Loeffler, University of Warwick
Datum, Zeit 21. September 2020, 17:15-18:15
Ort 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
HG G 43
16. Oktober 2020
14:15-15:15 		Prof. Dr. Ana Caraiani
Imperial College London
Details

Number Theory Seminar

Titel Vanishing theorems for Shimura varieties
Referent:in, Affiliation Prof. Dr. Ana Caraiani, Imperial College London
Datum, Zeit 16. Oktober 2020, 14:15-15:15
Ort 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. Oktober 2020
14:15-15:15 		Prof. Dr. Peter Simon Jossen
ETH Zurich, Switzerland
Details

Number Theory Seminar

Titel A non-hypergeometric E-function
Referent:in, Affiliation Prof. Dr. Peter Simon Jossen, ETH Zurich, Switzerland
Datum, Zeit 30. Oktober 2020, 14:15-15:15
Ort 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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