# Number theory seminar

## Spring Semester 2017

Note: The highlighted event marks the next occurring event and events marked with an asterisk (*) indicate that the time and/or location are different from the usual time and/or location.

Date / Time Speaker Title Location
24 February 2017
14:15-15:15
Prof. Dr. François Charles
Université Paris-Sud
HG G 43
Abstract: The Bertini irreducibility theorem states that, over an infinite field, a general hyperplane section of projective variety that is irreducible of dimension at least 2 remain irreducible. Over a finite field, this statement does not hold, but we can salvage it by replacing hyperplane sections by hypersurfaces of larger degree (joint with Poonen). We will explain how to prove the arithmetic version of this result.
10 March 2017
14:15-15:15
Prof. Dr. Solomon Friedberg
Boston College
HG G 43
Abstract: After giving necessary background, I will discuss new integrals representing tensor product L-functions of classical groups with general linear groups. These require new ingredients from the representation theory of p-adic groups (new kinds of unique models). The integrals also work for covering groups. Though there is still more work to be done, these integrals remove the main obstruction to proving the existence of endoscopic lifts for all automorphic representations without using the trace formula. This is joint work with Cai, Ginzburg and Kaplan.
31 March 2017
14:15-15:15
Prof. Dr. Yiannis Petridis
University College London
HG G 43
Abstract: Mazur, Rubin, and Stein have recently formulated a series of conjectures about statistical properties of modular symbols in order to understand central values of twists of elliptic curve L-functions. Two of these conjectures relate to the asymptotic growth of the first and second moments of the modular symbols. In joint work with Morten S. Risager we prove these on average using analytic properties of Eisenstein series twisted with modular symbols. We also prove another conjecture predicting the Gaussian distribution of normalized modular symbols ordered according to the size of the denominator of the cusps.
7 April 2017
14:15-15:15
Dr. James Borger
ANU
HG G 43
Abstract: The concepts of big Witt vectors and lambda-rings extend naturally from the category of rings to that of semirings. This implies, for example, that there are natural positivity structures on Witt vectors of rings which themselves have positivity structures---most importantly the integers and the real numbers. I'll explain this and make some enticing possible connections to the philosophy of the field with one element and Deninger's hypothetical arithmetic dynamical systems.
* 14 April 2017
No seminar (Easter Friday; vacations)
* 21 April 2017
14:15-15:15
No seminar, vacations
* 5 May 2017
No seminar: Grothendieck Seminar at FIM
12 May 2017
14:15-15:15
Prof. Dr. Abhishek Saha
Bristol University
HG G 43
Abstract: I will talk about some of my recent work with Pitale and Schmidt where we prove an explicit pullback formula that gives an integral representation for the twisted standard L-function for a holomorphic vector-valued Siegel cusp form of degree n and arbitrary level. In contrast to all previously proved pullback formulas in this situation, our formula involves only scalar-valued functions despite being applicable to L-functions of vector-valued Siegel cusp forms. Further, by specializing our integral representation to the case n = 2, we prove an algebraicity result for the critical L-values in that case (generalizing previously proved critical-value results for the full level case).
* 18 May 2017
11:15-12:15
Prof. Dr. Youness Lamzouri
York University
HG G 43
Abstract: Let $f$ be a Hecke cusp form of weight $k$ for the full modular group, and let $\lambda_f(n)\}_{n\geq 1}$ be the sequence of its normalized Fourier coefficients. In this talk, we will present recent results on the order of magnitude of the short sums $S_f(x)=\sum_{n\leq x} \lambda_f(n)$ (where $x$ varies in terms of $k$), as well as applications to the first negative Hecke eigenvalue. To explore these sums, we shall introduce a probabilistic random model for $S_f(x)$, and compare this sum to the sum of the divisor function $\tau(n)$ over smooth (or friable) numbers. Our results are $GL_2$ analogues of work of Granville and Soundararajan on character sums, and could also be generalized to other families of automorphic forms.
26 May 2017
14:15-15:15
Tiago Fonseca
Université Paris-Sud
HG G 43
Abstract: The Ramanujan equations are some algebraic differential equations satisfied by the classical Eisenstein series E_2, E_4, E_6. These equations play a pivotal role in the proof of Nesterenko's celebrated theorem on the algebraic independence of values of Eisenstein series, which gives in particular a lower bound on the transcendence degree of fields of periods of elliptic curves. Motivated by the problem of extending the methods of Nesterenko to other settings, we shall explain in this talk how to generalize Ramanujan's equations to higher dimensions via a geometric approach, and how the values of a particular solution of these equations relate with periods of abelian varieties.
* 2 June 2017
Number Theory Days in Lausanne