# Vanderbilt Number Theory Seminar 2019-2020

Date Speaker, Title, Abstract
Aug 29 Th Aug 29, 2019 @ 1:00pm

##### Angelica Babei (Vanderbilt University) – Hilbert modular forms in Magma.

In this talk, we showcase a new Magma implementation of Hilbert modular forms (HMFs), available at https://github.com/edgarcosta/hilbertmodularforms. Computational tools include sources of HMFs (such as Eisenstein series, newforms, theta series, elliptic curves), operations on HMFs (such as multiplication, division, and Hecke operators), as well as computing sets of generators and relations for the graded ring of HMFs over quadratic fields. Joint work with Ben Breen, Sara Chari, Edgar Costa, Michael Musty, Sam Schiavone, Sam Tripp, and John Voight.

Sep 12 Th Sep 12, 2019 @ 1:00pm

##### Ian Wagner (Vanderbilt University) – Partitions and a conjecture of John Thompson.

For a finite group $$G$$, let $$K(G)$$ denote the field generated over $$\mathbb{Q}$$ by its character values. For alternating groups, G. R. Robinson and J. G. Thompson determined $$K(A_n)$$ as an explicit multiquadratic field.  Confirming a speculation of Thompson, we show that arbitrary suitable multiquadratic fields are similarly generated by the values of $$A_n$$-characters restricted to elements whose orders are only divisible by ramified primes.  We also extend  this result to suitable linear groups and show that cyclotomic fields and subfields are generated by the values of characters restricted to elements with prime power order.

Sep 19 Th Sep 19, 2019 @ 1:00pm

##### Zack Tripp (Vanderbilt University) – Bounds on multiplicities of zeros of a family of zeta functions.

In “The Pair Correlation of Zeros of the Zeta Function”, Montgomery finds the asymptotics of the pair correlation function in order to give a lower bound on the proportion of zeros that are simple (assuming the Riemann Hypothesis). We will discuss some of the necessary tools to extend his proof to pair correlation for zeros of Dedekind zeta functions of abelian extensions, and as in the Riemann zeta case, we can then use this to obtain results on multiplicities of zeros for these zeta functions. However, we also are able to relate the counts of multiplicities to Cohn-Elkies sphere-packing type bounds, allowing us to use semi-definite programming techniques to obtain better results in lower degree extensions than could be found from a direct analysis. In particular, we are able to conclude that more than 45% of the zeros are distinct for Dedekind zeta functions of quadratic number fields. This is based on joint work with M. Alsharif, D. de Laat, M. Milinovich, L. Rolen, and I. Wagner.

Sep 26 Th Sep 26, 2019 @ 1:00pm

##### Ben Breen (Dartmouth College) – Heuristics for abelian fields: totally positive units and narrow class groups.

We describe heuristics in the style of Cohen-Lenstra for narrow class groups and units in abelian extensions of odd degree. These results stem from a model for the 2-Selmer group of a number field. We conclude with computational evidence for cyclic extensions of degree n = 3,5,7.

Oct 3 Th Oct 3, 2019 @ 1:00pm

##### Vaughan Jones (Vanderbilt University) – On the Petersson inner product of cusp forms-an operator valued extension.

Oct 18 Fri Oct 18, 2019 @ 3:00pm SC1312

##### Ayla Gafni (University of Mississippi) – Extremal primes for elliptic curves without complex multiplication.

Fix an elliptic curve $$E$$ over $$\mathbb{Q}$$.  An “extremal prime” for $$E$$ is a prime $$p$$ of good reduction such that the number of rational points on $$E$$ modulo $$p$$ is maximal or minimal in relation to the Hasse bound.  In this talk, I will discuss what is known and conjectured about the number of extremal primes $$p \leq X$$ and give the first non-trivial upper bound for the number of such primes when $$E$$ is a curve without complex multiplication.  The result is conditional on the hypothesis that all the symmetric power $$L$$-functions associated to $$E$$ are automorphic and satisfy the Generalized Riemann Hypothesis.  In order to obtain this bound, we use explicit equidistribution for the Sato-Tate measure as in recent work of Rouse and Thorner, and refine certain intermediate estimates taking advantage of the fact that extremal primes have a very small Sato-Tate measure.

Oct 31 Th Oct 31, 2019 @ 1:00pm

##### Doug Hardin (Vanderbilt University) – TBA.

Nov 21 Th Nov 21, 2019 @ 1:00pm

##### Doug Hardin (Vanderbilt University) – TBA.

Jan 24 Fri Jan 24, 2020 @ 4:10pm

##### Alex Dunn (University of Illinois at Urbana-Champaign) – Moments of half integral weight modular L-functions, bilinear forms and applications.

Given a half-integral weight holomorphic newform $$f$$, 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 $$f$$. 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.

Feb 4 Tue Feb 4, 2020 @ 3:00pm

##### John Voight (Dartmouth College) – Special hypergeometric motives and their L-functions: Asai recognition.

We recognize certain special hypergeometric motives, related to and inspired by the discoveries of Ramanujan more than a century ago, as arising from Asai L-functions of Hilbert modular forms.

Feb 7 Fri Feb 7, 2020 @ 4:10pm

##### Jeffrey Lagarias (University of Michigan) – The Lerch Zeta Function and the Heisenberg Group.
The Lerch zeta function is a three-variable zeta function generalizing the Riemann zeta function, the Hurwitz zeta function and the periodic zeta function. This talk surveys its analytic properties proved in a series of papers with Winnie Li.  The last part of the talk  gives an interpretation of it as an automorphic object with respect to a four-dimensional solvable Lie group which is an extension of the real Heisenberg group.

Feb 27 Th Feb 27, 2020 @ 1:00pm

##### Marie Jameson (University of Tennessee) – Cusp forms and p-adic limits.

In 2016, S. Ahlgren and D. Samart used the theory of modular forms to give strong results expressing certain cusp forms as p-adic limits of weakly holomorphic modular forms.  Here, we explore these results and search for other examples of this behavior.