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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

Nov 3

Nov 10

Twisted Hilbert modular surfaces, arithmetic intersections and the Jacquet-Langlands correspondence.
Siddarth Sankaran (University of Manitoba)
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

Dec 1

Dec 8

Dec 15

Efrat Bank
Primes in short intervals on curves over finite fields
abstract coming soon