Number Theory Seminar : Spring 2020

Atul Dixit (IIT Gandhinagar)

Superimposing theta structure on a generalized modular relation

By a modular relation for a certain function , we mean that which is governed by the map  but not necessarily by . Equivalently, the relation can be written in the form , where . There are many generalized modular relations in the literature such as the general theta transformation  or the Ramanujan-Guinand formula  etc. The latter, equivalent to the functional equation of the non-holomorphic Eisenstein series on , admits a beautiful generalization of the form , that is, one can superimpose theta structure on it.

Recently, a modular relation involving infinite series of the Hurwitz zeta function  was obtained. It generalizes a result of Ramanujan from the Lost Notebook. Can one superimpose theta structure on it? While answering this question affirmatively, we were led to a surprising new generalization of . We show that this new zeta function, , satisfies a beautiful theory. In particular, it is shown that  can be analytically continued to the whole complex plane except . Hurwitz's formula for  is also generalized in this setting. We also prove a generalized modular relation involving infinite series of , which is of the form . This is joint work with Rahul Kumar.

Lucia Mocz (Chicago)

Heights and p-adic Hodge Theory

We discuss connections between p-adic Hodge theory and the Faltings height. Most namely, we show how new tools in p-adic Hodge theory can be used to prove new Northcott properties satisfied by the Faltings height, and demonstrate phenomenon which are otherwise predicted by various height conjectures. We will focus primarily on the Faltings height of CM abelian varieties where the theory can be made to be computational and explicit.

Patrick Allen (Illinois)

Modularity of some  representations

Serre's conjecture, proved by Khare and Wintenberger, states that every odd two dimensional mod p representation of the absolute Galois group of the rationals comes from a modular form. This admits a natural generalization to totally real fields, but even the real quadratic case seems completely out of reach. I'll discuss some of the difficulties one encounters and then discuss some new cases that can be proved when p = 5. This is joint work with Chandrashekhar Khare and Jack Thorne.

Jack Buttcane (Maine)

The Kuznetsov formulas for GL(3)

The Kuznetsov formulas for GL(2) connect the study of automorphic forms to the study of exponential sums.  They are useful in a wide variety of seemingly unrelated problems in analytic number theory, and I will (briefly) illustrate this with a pair of examples:  First, if we consider the roots v of a quadratic polynomial modulo a prime p, then the sequence of fractions v/p is uniformly distributed modulo 1; this is the “mod p equidistribution” theorem of Duke, Friedlander, Iwaniec and Toth.  Second, the Random Wave Conjecture states that a sequence of automorphic forms should exhibit features of a random wave as their Laplacian eigenvalues tend to infinity.  I will discuss their generalization to GL(3) and applications.

Kevin Ford (Illinois)

Divisors of integers, permutations and polynomials

We describe a probabilistic model that describes the statistical behavior of the divisors of integers, divisors of permutations and divisors of polynomials over a finite field.  We will discuss how this can be used to obtain new bounds on the concentration of divisors of integers, improving a result of Maier and Tenenbaum.  This is joint work with Ben Green and Dimitris Koukoulopoulos.

Ian Whitehead (Swarthmore)

The Third Moment of Quadratic L-Functions

I will present.a smoothed asymptotic formula for the third moment of Dirichlet L-functions associated to real characters. Beyond the main term, which was known, the formula has an unexpected secondary term of size X^3/4 and an error of size X^2/3. I will give background on the multiple Dirichlet series techniques that motivated this result. And I will describe the new ideas about local and global multiple Dirichlet series that made the final, sieving step in the proof possible. This is joint work with Adrian Diaconu.

Robert Hough (Stony Brook)

The shape of low degree number fields

In his thesis, M. Bhargava proved parameterizations and identified local conditions which he used to give asymptotic counts for  quartic and quintic number fields, ordered by discriminant.  This talk will discuss results in an ongoing project to add detail to Bhargava's work by considering in addition to the field discriminant, the lattice shape of the ring of integers in the canonical embedding, and by giving strong rates with lower order terms in the asymptotics. These results build on earlier work of Taniguchi-Thorne, Bhargava-Shankar-Tsimerman and Bhargava Harron.

Shiang Tang (Illinois)

Potential automorphy of Galois representations into general spin groups

Given a connected reductive group  defined over a number field , the Langlands program predicts a connection between suitable automorphic representations of  and geometric -adic Galois representations  into the L-group of . Striking work of Arno Kret and Sug Woo Shin constructs the automorphic-to-Galois direction when  is the group  over a totally real field , and  is a cuspidal automorphic representation of  that is discrete series at all infinite places and is a twist of the Steinberg representation at some finite place: To such a , they attach geometric -adic Galois representations . In this work we establish a partial converse, proving a potential automorphy theorem, and some applications, for suitable -valued Galois representations. In this talk, I will explain the background materials and the known results in this direction before touching upon the main theorems of this work.

Zarathustra Brady (MIT)

Poisson imitators and sieve theory

I'll describe how sieve theory is actually a question about probability distributions whose low moments agree with the low moments of Poisson distributions. In particular, we can derive Selberg’s “parity problem” without using properties of the Möbius function or the Liouville function - instead, we use the fact that the alternating group forms a subgroup of the symmetric group.

Jeff Manning (UCLA)

The Wiles defect for Hecke algebras that are not complete intersections

In his work on modularity theorems, Wiles proved a numerical criterion for a map of rings R->T to be an isomorphism of complete intersections. He used this to show that certain deformation rings and Hecke algebras associated to a mod p Galois representation at non-minimal level were isomorphic and complete intersections, provided the same was true at minimal level. In addition to proving modularity theorems, this numerical criterion also implies a connection between the order of a certain Selmer group and a special value of an L-function.

In this talk I will consider the case of a Hecke algebra acting on the cohomology a Shimura curve associated to a quaternion algebra. In this case, one has an analogous map of ring R->T which is known to be an isomorphism, but in many cases the rings R and T fail to be complete intersections. This means that Wiles' numerical criterion will fail to hold. I will describe a method for precisely computing the extent to which the numerical criterion fails (i.e. the 'Wiles defect"), which will turn out to be determined entirely by local information at the primes dividing the discriminant of the quaternion algebra.

This is joint work with Gebhard Bockle and Chandrashekhar Khare.

Chen An (Duke)

Log-free zero density estimates for automorphic L-functions

One of the most important topics in number theory is the study of zeros of L-functions. Near the edge of the critical strip, one may show that the number of zeros for certain L-functions is small; such a result is called a zero density estimate. For Dirichlet L-functions, this topic is well understood by the work of Gallagher, Selberg, Jutila, etc. For families of automorphic L-functions, Kowalski and Michel show that the number of zeros near the edge of the critical strip is small on average. The proof uses a large sieve inequality with key objects called pseudo-characters. I will present my recent progress on the refinement of Kowalski-Michel's large sieve inequality, which gives rise to a better zero density estimate for automorphic L-functions.

Eric Stubley (Chicago)

Locally Split Galois Representations and Hilbert Modular Forms of Partial Weight One

The Galois representation attached to a p-ordinary eigenform is upper triangular when restricted to a decomposition group at p. A natural question to ask is under what conditions this upper triangular decomposition splits as a direct sum. Ghate and Vatsal have shown that for Galois representations coming from families of p-ordinary eigenforms, the restriction to a decomposition group at p is split if and only if the family has complex multiplication; in their proof, the weight one members of the family play a key role.

I'll talk about work in progress which aims to answer similar questions in the case of Galois representations for a totally real field which are split at only some of the primes above p. In this work Hilbert modular forms of partial weight one play a central role; I'll discuss what is known about them and to what extent the techniques of Ghate and Vatsal can be adapted to this situation.