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Number Theory seminar abstracts, Term 2 2017-18

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A Galois counting problem, by Sam Chow

We count monic quartic polynomials with prescribed Galois group, by box height. Among other things, we obtain the order of magnitude for $D_4$ quartics, and show that non-$S_4$ quartics are dominated by reducibles. Weapons include determinant method estimates, the invariant theory of binary forms, the geometry of numbers, and diophantine approximation. Joint with Rainer Dietmann.



Effective bounds for singular units, by Yuri Bilu
A singular modulus is a j-invariant of a CM elliptic curve. It is known that it is always an algebraic integer. In 2015 Habegger proved that at most finitely many singular moduli are algebraic units. It was a special case of his more general ``Siegel Theorem for Singular Moduli''. Unfortunately, this result was not effective, because Siegel's zero was involved (through Duke's equidistribution theorem).
In the present work we obtain an explicit bound: if $\Delta$ is an imaginary quadratic discriminant such that the corresponding singular moduli are units, then $|\Delta|<10^{15}$.
Joint work with Philipp Habegger and Lars K├╝hne.
Extremal primes of non-CM elliptic curves, by Ayla Gafni

Fix an elliptic curve $E/\mathbb{Q}$. 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 up to $X$, and give the first non-trivial upper bound for the number of such primes in the non-CM setting. In order to obtain this bound, we count primes with certain arithmetic characteristics and combine those results with the Chebotarev density theorem. This is joint work with Chantal David, Amita Malik, Neha Prabhu, and Caroline Turnage-Butterbaugh.



Mass equidistribution for half-integral weight modular forms, by Steve Lester

In this talk I will discuss the distribution of $L^2$ mass of half integral weight cusp forms. For integral weight Hecke cusp forms, Holowinsky and Soundararajan have shown that the mass of these forms equidistributes with respect to hyperbolic measure in the limit as the weight tends to infinity. Their method uses sieve bounds for multiplicative functions and weak subconvexity estimates for L-functions. I will discuss the analogues of some of their methods in the half-integral weight setting, which have led to new results on the distribution of mass of these forms. This is joint work with Maksym Radziwill.



On chaos measures, statistics of Riemann zeta, and random matrices, by Eero Saksman

We try to describe how complex multiplicative chaos measures emerge as description of functional statistics of the Riemann zeta function on the critical line. We also consider statistics on the mesoscopic scale and relate it to that of random matrices. The talk is based on joint work with Christian Webb (Aalto University, Helsinki)



Potential automorphy over CM fields and applications, by James Newton

I will discuss joint work with Allen, Calegari, Caraiani, Gee, Helm, Le Hung, Scholze, Taylor and Thorne that establishes potential automorphy results for certain compatible systems of Galois representations over CM fields. This has applications to the Sato-Tate conjecture for elliptic curves over CM fields and the Ramanujan conjecture for weight zero cohomological automorphic representations of GL(2) over CM fields.

Potential modularity of abelian surfaces, by Toby Gee
I will discuss joint work in progress with George Boxer, Frank Calegari, and Vincent Pilloni, in which we prove that all abelian surfaces over totally real fields are potentially modular. We also prove that infinitely many abelian surfaces over Q are modular.


Quartic forms in 30 variables, by Pankaj Vishe

Given an integral homogeneous polynomial F, when it contains a rational point is a key problem in Diophantine Geometry. A variety is supposed to satisfy Hasse Principle if it contains a rational zero in the absence of any local obstructions. We will prove that smooth Quartic (deg F=4) hypersurfaces satisfy the Hasse Principle as long as they are defined over at least 30 variables. The key tool here is employing a revolutionary idea of Kloosterman in our setting.This is a joint work with Oscar Marmon (U Lund).



Root numbers of abelian varieties, by Matthew Bisatt

Given an abelian variety $A$ over a number field, the famed Birch and Swinnerton-Dyer conjecture connects the rank of $A$ to its L-function; unfortunately very little is known about this conjecture. Instead, we focus on a corollary known as the parity conjecture: this connects the rank modulo 2 to the global root number of $A$, which is defined independently of the L-function.

In this talk, I will discuss how to compute the global root number of an abelian variety, giving an example with the Jacobian of a hyperelliptic curve. Moreover, I will apply the results to find a genus two hyperelliptic curve with simple Jacobian whose global root number is invariant under quadratic twist.



Unlikely intersections and the Chabauty-Kim method over number fields, by Netan Dogra

Let K be a number field. By theorems of Siegel and Faltings, if X is a hyperbolic curve over K, then X has only finitely many integral points. Recently, Kim had developed a method which can sometimes give new proofs of these results when K=Q by locating the integral points inside the zero sets of certain p-adic analytic functions, generalising previous work of Chabauty. In this I will explain how to extend some of these results to general K by proving certain 'unlikely intersection' results for the zeroes of these functions.