## Number theory seminar

### August - December 2021

03/09/2021 1400 (Online) Mihir Sheth (IISc)
Title: mod p local Langlands correspondence for GL_2
Abstract: Let F be a non-archimedean local field of residue characteristic p. The classical local Langlands correspondence is a 1-1 correspondence between 2-dimensional irreducible complex representations of the Weil group of F and certain smooth irreducible complex representations of GL_2(F). The number-theoretic applications made it necessary to seek such correspondence of representations on vector spaces over a field of characteristic p. In this talk, however, I will show that for F of residue degree > 1, unfortunately, there is no such 1-1 mod p correspondence. This result is an elaboration of the arguments of Breuil and Paskunas to an arbitrary local field of residue degree > 1.

01/10/2021 1400 (Online) Santosh Nadimpalli (IIT Kanpur)
Title: Generic representations of $p$-adic unitary groups
Abstract: In this talk, we will discuss genericity of cuspidal representations of $p$-adic unitary groups. Generic representations play a central role in the local Langlands correspondences and explicit knowledge of such representations will be useful in understanding the local Langlands correspondence in a more explicit way. After a brief review of $p$-adic unitary groups, their unipotent subgroups, Whittaker functionals and genericity of cuspidal representations in this context, we will discuss the arithmetic nature of the problem.

08/10/2021 1400 (Online) Aditya Karnataki (BICMR)
Title: Families of $(\varphi, \tau)$-modules and Galois representations
Abstract: Let $K$ be a finite extension of $\mathbb{Q}_p$. The theory of $(\varphi, \Gamma)$-modules constructed by Fontaine provides a good category to study $p$-adic representations of the absolute Galois group $Gal(\bar{K}/K)$. This theory arises from a devissage'' of the extension $\bar{K}/K$ through an intermediate extension $K_{\infty}/K$ which is the cyclotomic extension of $K$. The notion of $(\varphi, \tau)$-modules generalizes Fontaine's constructions by using Kummer extensions other than the cyclotomic one. It encapsulates the important notion of Breuil-Kisin modules among others. It is thus desirable to establish properties of $(\varphi, \tau)$-modules parallel to the cyclotomic case. In this talk, we explain the construction of a functor that associates to a family of $p$-adic Galois representations a family of $(\varphi, \tau)$-modules. The analogous functor in the $(\varphi, \Gamma)$-modules case was constructed by Berger and Colmez . This is joint work with L\'{e}o Poyeton.

22/10/2021 1400 (Online) C S Rajan (TIFR Mumbai)
Title: On equidistribution of Gauss sums of cuspidal representations of GL_d(\F_q).
Abstract: We investigate the distribution of the angles of Gauss sums attached to the cuspidal representations of general linear groups over finite fields. In particular we show that they happen to be equidistributed with respect to the Haar measure. However, for representations of $PGL_2(F_q)$, they are clustered around $1$ and $-1$ for odd $p$ and around $1$ for $p=2$. This is joint work with Sameer Kulkarni. x

29/10/2021 1400 (Online) Ritabrata Munshi (ISI Kolkata)
Title: Recent applications of delta symbol
Abstract: The delta symbol is the key in solving many different problems in the analytic theory of numbers. In recent years this has been used to solve various sub-convexity problems for higher rank L-functions. This talk will be a brief report on some new progresses. In particular, I will mention the results obtained in recent joint works with Roman Holowinsky & Zhi Qi and Sumit Kumar & Saurabh Singh.

05/11/2021 2030 (Online) Neelam Saikia (University of Virginia)
Title: Forbenius trace distributions for Gaussian hypergeometric functions
Abstract: In the 1980's, Greene defined hypergeometric functions over finite fields using Jacobi sums. The framework of his theory establishes that these functions possess many properties that are analogous to those of the classical hypergeometric series studied by Gauss, Kummer and others. These functions have played important roles in the study of Apéry-style supercongruences, the Eichler-Selberg trace formula, Galois representations, and zeta-functions of arithmetic varieties. In this talk we discuss the distributions (over large finite fields) of natural families of these functions. For the 2F1 functions, the limiting distribution is semicircular, whereas the distribution for the 3F2 functions is Batman distribution.