NTS ABSTRACT

Return to NTS Spring 2016

Sep 8

Arunabha Biswas

Limiting values of higher Mahler Measure and cyclotomic polynomials.

We consider the k-higher Mahler measure m_k(P) of a Laurent polynomial P as the integral of log^k |P| over the complex unit circle. In number theory, Lehmer's conjecture and the appearance of higher Mahler measures in L-functions are the main sources of motivation for studying various properties of m_k(P). Beyond number theory, Mahler measure has connections with topological entropies of dynamical systems and polynomial knot invariants. In this talk I shall present (1) an explicit formula for the value of |m_k(P)|/k! as k approaches infinity, (2) some asymptotic results regarding m_k(P) and (3) a scheme to approximate special values of a class of L-functions.

Sep 15

Naser T. Sardari

Discrete Log problem for the algebraic group PGL_2.

We consider the problem of finding the shortest path between a given pair of vertices in the LPS Ramanujan graphs $X_{p,q}$ where $p$ is a fixed prime number and $q$ is an integer. We give a polynomial time algorithm in $\log(q)$ which returns the shortest path between two diagonal vertices under a standard conjecture on the distribution of integers representable as sum of two squares and assuming one can factor quickly. Numerically, for a typical pair of vertices corresponded to diagonal elements the minimal path has a length about $3\log(q)+ O(1)$ while provably, there are pairs of points with distance at least $4\log(q)+ O(1)$ . For a general pair of vertices, we write it as a product of three Euler angels and as a result for a typical pair we find a path with distance $9 \log(q)+ O(1)$.

Sep 22

Alex Smith

Statistics for 8-class groups and 4-Selmer groups

Assuming the grand Riemann hypothesis, we verify that the set of quadratic imaginary fields has the distribution of 8-class groups predicted by the Cohen-Lenstra heuristic. To do this we prove that, in families of quadratic fields parameterized by a single prime p, the 8-class rank is determined by the Artin symbol of p in a certain extension of the rationals. Using Chebotarev's density theorem, we find that the distribution of 8-class ranks in most of these small families is given by the Cohen-Lenstra heuristic. We can bundle these small families together to get the full result, with GRH necessary to control error bounds in this process. By analogous means, we also find the distribution of 4-Selmer groups in the quadratic twist family of an elliptic curve with full 2-torsion.

Sep 29

Steve Lester

Quantum unique ergodicity for half-integral weight automorphic forms

Given a smooth compact Riemannian manifold (M, g) with no boundary an important problem in Quantum Chaos studies the distribution of L^2 mass of eigenfunctions of the Laplace-Beltrami operator in the limit as the eigenvalue tends to infinity. For M with negative curvature Rudnick and Sarnak have conjectured that the L^2 mass of all eigenfunctions equidistributes with respect to the Riemannian volume form; this is known as the Quantum Unique Ergodicity (QUE) Conjecture. In certain arithmetic settings QUE is now known. In this talk I will discuss the analogue of QUE in the context of half-integral weight automorphic forms. This is based on joint work Maksym Radziwill.

Oct 6

Nicole Looper

Arboreal Galois representations of higher degree polynomials and Odoni's Conjecture

Since the mid-1980s, when the study of arboreal Galois representations first began, most results have concerned the representations induced by quadratic rational maps. In the higher degree case, by contrast, very little has been known. I will discuss some recent results pertaining to higher degree polynomials over number fields. This will include a partial solution to a conjecture made by R.W.K. Odoni in 1985.

Oct 13

Ling Long

Potentially GL(2)-type Galois representations associated to noncongruence modular forms

Abstract: Among all finite index subgroups of the modular group SL(2,Z), majority of them cannot be described by congruence relations and are known as noncongruence subgroups. This talk is about modular forms for noncongruence subgroups, in particular their corresponding motivic Galois representations constructed by Scholl, which are expected to correspond to automorphic representations according to the Langlands philosophy. Using recent developments in the automorphy lifting theorem, we obtain some automphy and potential automorphy results for potentially GL(2)-type Galois representations associated to noncongruence modular forms and discuss their applications.

Oct 20

Jack Klys

The distribution of p-torsion in degree p Galois fields

The Cohen-Lenstra heuristics are a series of conjectures about the distributions of the class groups of number fields. They were extended by Gerth to the case of the p-torsion subgroup when p divides the degree of the field. Recently Fouvry and Kluners verified Gerth's conjecture for p=2 by computing the distribution of the 4-rank of class groups of quadratic fields. We will talk about our generalization of this result to the p-rank of class groups of degree p Galois fields. We will also discuss potential applications of these methods to computing distributions of extensions of quadratic fields with fixed non-abelian Galois group (joint work in progress with Brandon Alberts), which is a case of the non-abelian Cohen-Lenstra heuristics.

Oct 27

Vlad Serban

Infinitesimal p-adic Manin-Mumford and an application to Hida theory

Let $G$ be an abelian variety or a product of multiplicative groups $\mathbb{G}_m^n$ and let $C$ be an embedded curve. The Manin-Mumford conjecture (a theorem by work of Lang, Raynaud et al.) states that only finitely many torsion points of $G$ can lie on $C$ unless $C$ is in fact the translate of a subgroup of $G$. I will show how these purely algebraic statements extend to suitable analytic functions on open $p$-adic unit poly-disks. These disks occur naturally as weight spaces parametrizing families of $p$-adic automorphic forms for $GL(2)$ over a number field $F$. When $F=\mathbb{Q}$, the "Hida families" in question play a crucial role in the study of modular forms. When $F$ is imaginary quadratic, I will explain how our results imply that Bianchi modular forms are sparse in these $p$-adic families.

Nov 10

Siddarth Sankaran (University of Manitoba)

Twisted Hilbert modular surfaces, arithmetic intersections and the Jacquet-Langlands correspondence.

This is joint work with Gerard Freixas, in which we compute and compare arithmetic intersection numbers on Shimura varieties attached to inner forms of GL(2) over a real quadratic field. In the first part of the talk, we'll compute the degree of the top arithmetic Todd class on a quaternionic Hilbert modular surface in terms of derivatives of L-functions. We will then relate this quantity to the arithmetic volume of a Shimura curve, by using the Jacquet-Langlands correspondence and an arithmetic Grothendieck-Riemann-Roch formula. Finally, time permitting, I'll discuss some ongoing joint work with Freixas and Dennis Eriksson on the non-compact case.

Nov 17

Katherine Stange

Visualizing the arithmetic of imaginary quadratic fields

Let $K$ be an imaginary quadratic field with ring of integers $\mathcal{O}_K$. The Schmidt arrangement of $K$ is the orbit of the extended real line in the extended complex plane under the Mobius transformation action of the Bianchi group $\operatorname{PSL}(2,\mathcal{O}_K)$. The arrangement takes the form of a dense collection of intricately nested circles. Aspects of the number theory of $\mathcal{O}_K$ can be characterised by properties of this picture: for example, the arrangement is connected if and only if $\mathcal{O}_K$ is Euclidean. I'll explore this structure and its connection to Apollonian circle packings. Specifically, the Schmidt arrangement for the Gaussian integers is a disjoint union of all primitive integral Apollonian circle packings. Generalizing this relationship to all imaginary quadratic $K$, the geometry naturally defines some new circle packings and thin groups of arithmetic interest. In particular, I'll generalize the local-global conjecture for Apollonian circle packings.

Dec 8

Vlad Matei

Counting low degree covers of the projective line over finite fields

In joint work with Daniel Hast and Joseph Gunther we count degree 3 and 4 covers of the projective line over finite fields. This is a geometric analogue of the number field side of counting cubic and quartic fields. We take a geometric perspective, by using a vector bundle parametrization of these curves which is different from the recent work of Manjul Bhargava, Arul Shankar, Xiaoheng Wang "Geometry of numbers methods over global fields: Prehomogeneous vector spaces" in which the authors extend the geometry of numbers methods to global fields. Our count is just for $S_3$ and $S_4$ covers, and we put the rest of the curves in our error term. For trigonal curves we partially recover the asymptotic proved by Yongqiang Zhao in his Phd Thesis by using a similar approach.

Dec 15

Efrat Bank

Primes in short intervals on curves over finite fields

We prove an analogue of the Prime Number Theorem for short intervals on a smooth proper curve of arbitrary genus over a finite field. Our main result gives a uniform asymptotic count of those rational functions, inside short intervals defined by a very ample effective divisor E, whose principal divisors are prime away from E. In this talk, I will discuss the setting and definitions we use in order to make sense of such count, and will give a rough sketch of the proof. This is a joint work with Tyler Foster.