Number Theory Seminar : Fall 2019

Patrick Allen (Illinois)

On the modularity of elliptic curves over imaginary quadratic fields

Wiles's proof of the modularity of semistable elliptic curves over the rationals uses the Langlands-Tunnell theorem as a starting point, implying that the mod 3 Galois representation attached to the elliptic curve arises from a modular form of weight one. In order to feed this into a modularity lifting theorem, one needs to use congruences between modular forms of weight one and modular forms of higher weight. Similar congruences are not known over imaginary quadratic fields and Wiles's strategy runs into problems right from the start. We circumvent this congruence problem and show that mod 3 Galois representations over imaginary quadratic fields arise from automorphic forms that are the analog of higher weight modular forms. Our argument relies on a 2-adic automorphy lifting theorem over CM fields together with a "2-3 switch" that gives a criterion for when a given mod 6 representation arises from an elliptic curve. As an application, we deduce that a positive proportion of elliptic curves over imaginary quadratic fields are modular. This is joint work in progress with Chandrashekhar Khare and Jack Thorne.

Alexander Dunn (Illinois)

Moments of half integral weight modular L–functions, bilinear forms and applications

Given a half-integral weight holomorphic newform , we prove an asymptotic formula for the second moment of the twisted L-function over all primitive characters modulo a prime. In particular, we obtain a power saving error term and our result is unconditional; it does not rely on the Ramanujan-Petersson conjecture for the form . This gives a very sharp Lindelöf on average result for L-series attached to Hecke eigenforms without an Euler product. The Lindelöf hypothesis for such series was originally conjectured by Hoffstein. In the course of the proof, one must treat a bilinear form in Salié sums. It turns out that such a bilinear form also has several arithmetic applications to equidistribution. These are a series of joint works with Zaharescu and Shparlinski-Zaharescu.

Olivia Beckwith (Illinois)

Indivisibility and divisibility of class numbers of imaginary quadratic fields

For any prime p > 3, the strongest lower bounds for the number of imaginary quadratic fields with discriminant down to -X for which the class group has trivial (resp. non-trivial) p-torsion are due to Kohnen and Ono (Soundararajan). I will discuss refinements of these classic results in which we consider the imaginary quadratic fields for which the class number is indivisible (divisible) by p and which satisfy the property that a given finite set of rational primes split in a prescribed way. We prove a lower bound for the number of such fields with discriminant down to -X which is of the same order of magnitude as in Kohnen and Ono's (Soundararajan's) results. For the indivisibility case, we rely on a result of Wiles establishing the existence of imaginary quadratic fields with trivial p-torsion in their class groups which satisfy a finite set of local conditions, and a result of Zagier which says that the Hurwitz class numbers are the Fourier coefficients of a mock modular form.

Kevin Ford (Illinois)

Large prime gaps and Siegel zeros

We show that the existence of zeros of Dirichlet L-functions very close to 1 ("Siegel zeros") implies larger prime gaps than are currently known.  We also present a heuristic argument that the existence of Siegel zeros implies gaps of larger order than , that is, larger than the Cramer conjecture.

Gregory Debruyne (Illinois & Ghent University)

Optimality of Tauberian theorems

The Wiener-Ikehara and Ingham-Karamata theorems are two celebrated Tauberian theorems which are known to lead to short proofs of the prime number theorem. In this talk, we shall investigate quantified versions of these theorems and show that these are optimal. For the optimality, rather than constructing counterexamples, we shall use an attractive functional analysis argument based on the open mapping theorem. The talk is based on work in collaboration with David Seifert and Jasson Vindas.

Jesse Thorner (University of Florida)

A new approach to bounds for L-functions

Let  be the -function of a cuspidal automorphic representation of  with analytic conductor .  The Phragmen-Lindelof principle implies the convexity bound  for all fixed , while the generalized Riemann hypothesis for  implies that .  A major theme in modern number theory is the pursuit of subconvexity bounds of the shape  for some fixed constant .  I will describe how to achieve (i) an unconditional nontrivial improvement over the convexity bound for all automorphic -functions (joint work with Kannan Soundararajan), and (ii) an unconditional subconvexity bound for almost all automorphic -functions (joint work with Asif Zaman).

Rizwanur Khan (University of Mississippi)

Non-vanishing of Dirichlet L-functions

-functions are fundamental objects in number theory. At the central point , an -function  is expected to vanish only if there is some deep arithmetic reason for it to do so (such as in the Birch and Swinnerton-Dyer conjecture), or if its functional equation specialized to  implies that it must. Thus when the central value of an -function is not a "special value", and when it does not vanish for trivial reasons, it is conjectured to be nonzero. In general it is very difficult to prove such non-vanishing conjectures. For example, nobody knows how to prove that  is nonzero for all primitive Dirichlet characters . In such situations, analytic number theorists would like to prove 100% non-vanishing in the sense of density, but achieving any positive percentage is still valuable and can have important applications. In this talk, I will discuss work on establishing such positive proportions of non-vanishing for Dirichlet -functions.

Carl Wang-Erickson (University of Pittsburgh)

The Eisenstein ideal with squarefree level

In his landmark paper "Modular forms and the Eisenstein ideal," Mazur studied congruences modulo a prime p between the Hecke eigenvalues of an Eisenstein series and the Hecke eigenvalues of cusp forms, assuming these modular forms have weight 2 and prime level N. He asked about generalizations to squarefree levels N. I will present some work on such generalizations, which is joint with Preston Wake and Catherine Hsu.

Ashay Burungale (Caltech)

An even parity instance of the Goldfeld conjecture

We show that the even parity case of the Goldfeld conjecture holds for the congruent number elliptic curve. We plan to outline setup and strategy (joint with Ye Tian).

Ghaith Hiary (Ohio State University)

Bounds on 

I survey some upper and lower bounds in the theory of the Riemann zeta function, in particular lower bounds on , the fluctuating part of the zeros counting function for the Riemann zeta function. I outline a new unconditional lower bound on , which is work in progress

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William Banks (University of Missouri)

Sums with the Mobius function twisted by characters with powerful moduli

In the talk, I will describe some recent joint work with Igor Shparlinski, in which we have combined classical ideas of Postnikov and Korobov to derive new bounds on short character sums for certain nonprincipal characters of powerful moduli.  Our results are used to bound sums with the Mobius function twisted by such characters, and we obtain new results on the size and zero-free region of Dirichlet L-functions attached to the same class of moduli.

Ilya Khayutin (Northwestern University)

Fourth Moments of Modular Forms on Arithmetic Surfaces

I will describe a method to study the fourth moment of periods of Hecke eigenforms using a second moment pre-trace formula. The second moment pre-trace formula is constructed out of the usual pre-trace formula using non-standard test functions involving all Hecke operators. It can also be understood using the theta correspondence. Our main application is to the sup-norm problem for modular forms on arithmetic surfaces. Joint work with Raphael Steiner.