# UVA Number Theory Seminar (Fri. @ 1:00pm)

Passcode: Numerical Value of 1^3+12^3. (i.e. Ramanujan's taxi-cab number)

Spring 2021 Schedule

• February 5th: Jesse Thorner (U. Illinois, Urbana-Champaign)
An approximate form of Artin's holomorphy conjecture and nonvanishing of Artin L-functions
Abstract: The speaker will present some recent work with Robert Lemke Oliver and Asif Zaman in which we noticeably expand the region in which almost all Artin L-functions in certain families are holomorphic and nonvanishing.  This combines Galois theory, character theory, and analytic number theory.  These results are motivated by a wide range of classically flavored applications; I will focus on applications to the study of class groups of high-degree number fields.
• February 12th: Martin Raum (Chalmers U)
Divisibility questions on the partition function and their connection to modular forms
Abstract. The partition function records the number of ways an integer can be written as a sum of positive integers. Already studied by Euler, it has turned out to be a great source of inspiration in the theory of modular forms over the course of the past century. This development was ignited by Ramanujan. At at a time when it was a challenge to merely calculate values of the partition function, he anticipated divisibility properties of astonishing regularity. We will explain some of the ideas that emerged from Ramanujan's conjectures and some of their modern manifestations. Many of these are connected to modular forms and via these to Galois representations. They help us to understand in an increasingly precise sense how frequently Ramanujan's divisibility patterns and their generalizations occur.
• February 19th: Maryam Khaqan (Emory U)
Elliptic curves and moonshine
Abstract: The study of moonshine originated from a series of numerical coincidences connecting finite groups to modular forms. It has since evolved into a rich theory that sheds light on the underlying algebraic structures that these coincidences reflect. We prove the existence of one such structure, a module for Thompson's sporadic simple group, whose graded traces are specific half-integral weight weakly holomorphic modular forms. We then use this module to study the ranks of certain families of elliptic curves. In particular, this serves as an example of moonshine being used to answer questions in number theory. This talk is based on arXiv: 2008.01607, where we classify all such Thompson-modules where the graded dimension is a specific weakly-holomorphic modular form and prove more subtle results concerning geometric invariants of certain other families of elliptic curves. Time permitting, we will talk about some of these results as well.
• February 26th: Jennifer Balakrishnan (Boston U)
Rational points on curves and quadratic Chabauty
Abstract: Let C be a smooth projective curve defined over the rational numbers with genus at least 2. It was conjectured by Mordell in 1922 and proved by Faltings in 1983 that C has finitely many rational points. However, Faltings' proof does not give an algorithm for finding these points, and in practice, given a curve, provably finding its set of rational points can be quite difficult. In the case when the Mordell--Weil rank of the Jacobian of C is less than the genus, the Chabauty--Coleman method can be used to find rational points, using the construction of certain p-adic line integrals.  Nevertheless, the situation in higher rank is still rather mysterious. I will describe the quadratic Chabauty method (developed in joint work with N. Dogra, S. Müller, J. Tuitman, and J. Vonk), which can apply when the rank is equal to the genus. I will also highlight some examples of interest, from the time of Diophantus to the present day.
• March 5th: Jonathan Love (Stanford)
Explicit Rational Equivalences of Zero-cycles on Surfaces
Abstract: The Chow group of zero-cycles on a smooth projective surface X is obtained by taking the free abelian group generated by closed points on X, and declaring two elements (“zero-cycles”) to be equal if their difference is a sum of divisors of rational functions on curves in X; in this setting we say the zero-cycles are “rationally equivalent.” These Chow groups are notoriously difficult to compute; while a set of conjectures due to Bloch and Beilinson predict certain relations must hold in these groups when X is defined over a number field, there are very few non-trivial cases in which these relations have been proven to hold. In this talk, I will discuss several techniques that can be used to compute rational equivalences exhibiting some of the expected relations, in the case that X is a product of two elliptic curves over Q.
• March 12th: Liyang Yang (Cal Tech)
Average Central L-values on U(2,1)$\times$ U(1,1), Nonvanishing and Subconvexity
Abstract: In this talk, we study an average of automorphic periods on $U(2,1)\times U(1,1).$ We also compute local factors in Ichino-Ikeda formulas for these periods to obtain an explicit asymptotic expression. Combining them together we would deduce some important properties of central $L$-values on $U(2,1)\times U(1,1)$ over a family of holomorphic cusp forms: the first moment, nonvanishing and subconvexity. This is joint work with Philippe Michel and Dinakar Ramakrishnan.
• March 19th: Wei-Lun Tsai (University of Virginia)
Malle's Conjecture for GxA, with G=S3, S4, S5
Abstract: Malle's Conjecture predicts the asymptotic distribution of number fields with fixed Galois group. The conjecture remains wide open. Here we prove several infinite families of the conjecture. This is joint work with R. Masri, F. Thorne, and J. Wang.
• March 26th: Vandita Patel (U Manchester)
Perfect powers that are sums of consecutive like powers

Abstract: In this talk, we present some of the techniques used to tackle subfamilies of the Diophantine equation $(x+1)^k + (x+2)^k + ... + (x+d)^k = y^n$. We compare two very different approaches which naturally arise when considering the parity of the exponent $k$. We present all integer solutions, $(x,y,n)$ to the equation in the case $k=3, 1<d<51$ (joint work with Mike Bennett - UBC and Samir Siksek - Warwick), and a (natural) density result when $k$ is a positive even integer, showing that for almost all $d$ at least $2$, the equation has no integer solutions, $(x,y,n)$ with $n$ at least $2$ (joint work with Samir Siksek - Warwick).

• April 2nd:Michael Bennett (U British Columbia)
Values of the Ramanujan tau-function

Abstract. If a is an odd positive integer, then a result of Murty, Murty and Shorey implies that there are at most finitely many positive integers n for which tau(n)=a, where tau(n) is the Ramanujan tau-function. In this talk, I will discuss non-archimidean analogues of this result and show how the machinery of Frey curves and their associated Galois representations can be employed to make such results explicit, at least in certain situations. Much of what I will discuss generalizes readily to the more general situation of coefficients of cuspidal newforms of weight at least 4, under natural arithmetic conditions. This is joint work with Adela Gherga, Vandita Patel and Samir Siksek.

• April 9th: Jan-Willem van Ittersum (Max Planck, Bonn)
Applications of the Bloch-Okounkov theorem

Abstract. Partitions of integers and (quasi)modular forms are related in many ways. We discuss a connection made by a certain normalized generating series of functions f on partitions, called the q-bracket of f. This is in fact the beginning of a whole story: Dijkgraaf had given a physical argument, which was proven by Kaneko and Zagier, that the problem of counting certain coverings of a torus led to quasimodular forms, and the Bloch-Okounkov theorem was found as a generalisation of this. We explain this result, and indicate how certain recently studied functions on partitions, i.e., p-adic generalizations of the shifted symmetric functions and the weighted t-hook functions, naturally fit into the framework provided by the Bloch-Okounkov theorem.

• April 16th:Levent Alpoge (Columbia U)

Effective height bounds for odd-degree totally real points on some curves.
Abstract: Let \o be an order in a totally real field F. Let K be an odd-degree totally real field. Let S be a finite set of places of K. We study S-integral K-points on integral models H_\o of Hilbert modular varieties because not only do said varieties admit complete curves (thus reducing questions about such curves' K-rational points to questions about S-integral K-points on these integral models), they also have their S-integral K-points controlled by known cases of modularity, in the following way. First assume for clarity modularity of all \GL_2-type abelian varieties over K --- then all S-integral K-points on H_\o arise from K-isogeny factors of the [F:\Q]-th power of the Jacobian of a single Shimura curve with level structure (by Jacquet-Langlands transfer). By a generalization of an argument of von Känel, isogeny estimates of Raynaud/Masser-Wüstholz and Bost's lower bound on the Faltings height suffice to then bound the heights of all points in H_\o(\o_{K,S}). As for the assumption, though modularity is of course not known in this generality, by following Taylor's (sufficiently explicit for us) proof of his potential modularity theorem we are able to make the above unconditional. Finally we use the hypergeometric abelian varieties associated to the arithmetic triangle group \Delta(3,6,6) to give explicit examples of curves to which the above height bounds apply. Specifically, we prove that, for a\in \Qbar^\times totally real of odd degree (e.g. a = 1), for all L/\Q(a) totally real of odd degree and S a finite set of places of L, there is an effectively computable c = c_{a,L,S}\in \Z^+ such that all x,y\in L satisfying x^6 + 4y^3 = a^2 satisfy h(x) < c. Note that this gives infinitely many curves for each of which Faltings' theorem is now effective over infinitely many number fields.

• April 23rd:Caroline Turnage-Butterbaugh (Carleton)

Gaps between zeros of the Riemann zeta-function
Abstract: Let $0 < \gamma_1 \le \gamma_2 \le \cdots$ denote the ordinates of the complex zeros of the Riemann zeta-function function in the upper half-plane. The average distance between $\gamma_n$ and $\gamma_{n+1)$ is $2\pi / \log \gamma_n$ as $n\to \infty$. An important goal is to prove unconditionally that these distances between consecutive zeros can much, much smaller than the average for a positive proportion of zeros. We will discuss the motivation behind this endeavor, progress made assuming the Riemann Hypothesis, and recent work with A. Simonič and T. Trudgian to obtain an unconditional result that holds for a positive proportion of zeros.

• May 5th (Note Change of Date) Humberto Diaz (Washington Univ) 1:00pm-2:00pm

The Griffiths group of an Abelian threefold over a finite field
Abstract: The Griffiths group of a smooth projective variety of dimension at least 3 is an interesting and mysterious invariant. For varieties over finite fields, it is expected that the Griffiths group is torsion. This is known to be true for Abelian varieties of dimension 3. Curiously enough, it is also expected that this group is nontrivial for ordinary Abelian varieties of dimension 3. In the case of a triple power of an ordinary elliptic curve, it is actually possible to produce explicit torsion cycles of arbitrarily large order, thanks to a result of Schoen. In this talk, I will discuss some possible generalizations of Schoen’s result.

• May 7th:Alex Dunn (Cal Tech)

Moments of half integral weight modular L–functions, bilinear forms and applications
Abstract:  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. 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 Lindelof on average result for L-series attached to Hecke eigenforms without an Euler product. The Lindelof hypothesis for such series was originally conjectured by Hoffstein. In the course of the proof, one must treat a bilinear form in Salie sums. It turns out that such a bilinear form also has several arithmetic applications to equidistribution. This is a series of joint works with Zaharescu and Kerr-Shparlinski-Zaharescu.