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Naomi Sweeting: Kolyvagin’s Conjecture and Higher Congruences of Modular Forms

22nd April, 2021

Given an elliptic curve E, Kolyvagin used CM points on modular curves to construct a system of classes valued in the Galois cohomology of the torsion points of E. Under the conjecture that not all of these classes vanish, he gave a description for the Selmer group of E. This talk will report on recent work proving new cases of Kolyvagin's conjecture. The proof builds on work of Wei Zhang, who used congruences between modular forms to prove Kolyvagin's conjecture under some technical hypotheses. We remove many of these hypotheses by considering congruences modulo higher powers of p. The talk will explain the difficulties associated with higher congruences of modular forms and how they can be overcome.

Adam Harper: Low moments of character sums

8th April, 2021

Sums of Dirichlet characters ∑n≤xχ(n) (where χ is a character modulo some prime r, say) are one of the best-studied objects in analytic number theory. Their size is the subject of numerous results and conjectures, such as the Pólya-Vinogradov inequality and the Burgess bound. One way to get information about this is to study the power moments 1/(r−1) ∑χ mod r|∑n≤xχ(n)|2q, which turns out to be quite a subtle question that connects with issues in probability and physics. In this talk I will describe an upper bound for these moments when 0≤q≤1. I will focus mainly on the number-theoretic issues arising.

Sam Mundy: Eisenstein series, p-adic deformations, Galois representations, and the group G2

1st April, 2021

I will explain some recent work on special cases of the Bloch-Kato conjecture for the symmetric cube of certain modular Galois representations. Under certain standard conjectures, this work constructs non-trivial elements in the Selmer groups of these symmetric cube Galois representations; this works by p-adically deforming critical Eisenstein series in a generically cuspidal family of automorphic representations, and then constructing a lattice in the associated family of Galois representations, all for the exceptional group G2. While I will touch on all of these aspects of the construction, I will mainly focus on the Galois side in this talk.

Hang Xue: The local Gan-Gross-Prasad conjecture for real unitary groups

25th March, 2021

A classical branching theorem of Weyl describes how an irreducible representation of compact U(n+1) decomposes when restricted to U(n). The local Gan-Gross-Prasad conjecture provides a conjectural extension to the setting of representations of noncompact unitary groups lying in a generic L-packet. We prove this conjecture. Previously Beuzart-Plessis proved the "multiplicity one in a Vogan packet" part of the conjecture for tempered L-packets using the local trace formula approach initiated by Waldspurger. Our proof uses theta lifts instead, and is independent of the trace formula argument.

Gal Dor: Monoidal Structures on GL2-Modules and Abstractly Automorphic Representations

4th March, 2021

Consider the function field F of a smooth curve over 𝔽q, with q>2. L-functions of automorphic representations of GL2 over F are important objects for studying the arithmetic properties of the field F. Unfortunately, they can be defined in two different ways: one by Godement-Jacquet, and one by Jacquet-Langlands. Classically, one shows that the resulting L-functions coincide using a complicated computation. Each of these L-functions is the GCD of a family of zeta integrals associated to test data. I will categorify the question, by showing that there is a correspondence between the two families of zeta integrals, instead of just their L-functions. The resulting comparison of test data will induce an exotic symmetric monoidal structure on the category of representations of GL2. It turns out that an appropriate space of automorphic functions is a commutative algebra with respect to this symmetric monoidal structure. I will outline this construction, and show how it can be used to construct a category of automorphic representations.

Joel Nagloo: Ax-Lindemann-Weierstrass Theorem for Fuchsian automorphic functions

21st January, 2021

Over the last decades, following works around the Pila-Wilkie counting theorem in the context of o-minimality, there has been a surge in interest around functional transcendence results, in part due to their connection with special points conjectures. A prime example is Pila's modular Ax-Lindemann-Weierstrass (ALW) Theorem and its role in his proof of the André-Oort conjecture.

In this talk we will discuss how an entirely new approach, using the model theory of differential fields, can be used to prove the ALW Theorem with derivatives for Fuchsian automorphic functions - a direct generalization of Pila’s ALW theorem. We will also explain how new cases of the André-Pink conjecture can be obtained using this new approach.

This is joint work with G. Casale and J. Freitag.

Pierre Debes: The Hilbert-Schinzel specialization property

10th January, 2021

Hilbert's Irreducibility Theorem shows that irreducibility over the field of rationals is 'often' preserved when one specializes a variable in some irreducible polynomial in several variables. I will present a version 'over the ring' for which the specialized polynomial remains irreducible over the ring of integers. The result also relates to the Schinzel Hypothesis about primes in value sets of polynomials: I will discuss a weaker 'relative' version for the integers and the full version for polynomials. The results extend to other base rings than the ring of integers; the general context is that of rings with a product formula.

Joni Teräväinen: On the Liouville function at polynomial arguments

10th December, 2020

Let λ be the Liouville function and P(x) any polynomial that is not a square. An open problem formulated by Chowla and others asks to show that the sequence λ(P(n)) changes sign infinitely often. We present a solution to this problem for new classes of polynomials P, including any product of linear factors or any product of quadratic factors of a certain type. The proofs also establish some nontrivial cancellation in Chowla and Elliott type correlation averages.

Jingwei Xiao: A Unitary Analogue of Friedberg-Jacquet Periods and Central Values of Standard L Functions on GL(2n)

3rd December, 2020

Let G be a reductive group over a number field F and H a subgroup. Automorphic periods study the integrals of cuspidal automorphic forms on G over H(F)\H(AF). They are often related to special values of certain L-functions. One of the most notable cases is when (G,H)=(U(n+1)☓U(n), U(n)), and these periods are related to central values of Rankin-Selberg L-functions on GL(n+1)☓GL(n). In this talk, I will explain my work in progress with Wei Zhang that studies central values of standard L-functions on GL(2n) using (G,H)=(U(2n), U(n)☓U(n)) and some variants. I shall explain the conjecture and a relative trace formula approach to study it. We prove the required fundamental lemma using a limit of the Jacquet-Rallis fundamental lemma and Hironaka’s characterization of spherical functions on the space of non-degenerate Hermitian matrices. Also, the question admits an arithmetic analogue.