I will discuss some recent progress in analytic number theory for polynomials over finite fields, giving strong new estimates for the number of primes in arithmetic progressions, as well as for sums of some arithmetic functions in arithmetic progressions. The strategy of proof is fundamentally geometric, and I will explain some of the geometric ideas in the proof, including how we can use the representation of the symmetric group to handle many different arithmetic functions in a uniform way.
I will discuss recent work with Harald Helfgott in which we establish roughly speaking that the graph connecting n to n ± p with p a prime dividing n is almost "locally Ramanujan". As a result we obtain improvements of results of Tao and Tao-Teravainen on logarithmic Chowla. I will discuss the main ideas in the proof and the connections with logarithmic Chowla.
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.
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.
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.
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.
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.
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.
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