Department of Mathematics and Computer Science
Number Theory and Combinatorics Seminar
Spring 2016
Talks are at noon on Monday in room C630 of University Hall
For more information, or to receive an email announcement of each week's seminar,
contact Nathan Ng < ng AT cs DOT uleth DOT ca > or Dave Morris <>.
                The next talk:

June 28

at noon
in C630
Tim Trudgian
(Australian National University)
Grosswald's conjecture on primitive roots
Very little is known about the distribution of primitive roots of a prime $p$. Grosswald conjectured that the least primitive root of a prime p is less than $\sqrt{p} - 2$ for all $p> 409$. While this is certainly true for all $p$ sufficiently large, Grosswald's conjecture in still open. I shall outline some recent work which resolves the conjecture completely under the Generalised Riemann Hypothesis and which almost resolves the conjecture unconditionally.

June 28

at 2pm
in C630
Vijay Patankar
(Jawaharlal Nehru University)
Given an elliptic curve $E$ over a number field $K$. The Frobenius field attached to $E$ at a prime $p$ is the splitting field of the characteristic polynomial of the Frobenius endomorphism acting on the $\ell$-adic Tate module of $E$ ($\ell$ a prime different from $p$) over the rationals. Thus, the splitting field is either of degree $1$ or degree $2$ over the rationals. Let $E_1$ and $E_2$ be elliptic curves defined over a number field $K$, with at least one of them without complex multiplication. We prove that the set of places $v$ of $K$ of good reduction such that the corresponding Frobenius fields are equal has positive upper density if and only if $E_1$ and $E_2$ are isogenous over some extension of $K$. For an elliptic curve $E$ defined over a number field $K$, we show that the set of finite places of $K$ such that the Frobenius field at $v$ equals a fixed imaginary quadratic field $F$ has positive upper density if and only if $E$ has complex multiplication by $F$. Time permits we will provide a sketch of a result about two dimensional $\ell$-adic Galois representations that we will need using an algebraic density theorem due to Rajan.
Talks in the series this semester:
(Click on any title for more info, including the abstract. Then click on it again to hide the info.)

Date Speaker Title

Jan 11 everyone Open problem session

Jan 25 Francesco Pappalardi On never primitive points on elliptic curves
(Universitŕ Roma Tre)

Jan 27 Francesco Pappalardi The distribution of multiplicatively dependent vectors
(Universitŕ Roma Tre)

Feb 1 Micah Milinovich Fourier Analysis and the zeros of the Riemann zeta-function
(University of Mississippi)

Feb 8 Alexey Popov Operator Algebras with reduction properties

Feb 22 Nathan Ng Linear combinations of zeros of L-functions

Feb 29 Rob Craigen Survey of Negacyclic Weighing Matrices
(University of Manitoba)

Mar 7 Alia Hamieh Determining Hilbert modular forms by the central values of Rankin-Selberg convolutions

Mar 14 Joy Morris Automorphisms of circulant graphs

Mar 21 Arnab Bose Investigations on some Exponential Congruences

Apr 4 Brandon Fuller CCA groups and graphs

Apr 11 Asif Zaman The least prime ideal in the Chebotarev Density Theorem
(University of Toronto)

June 17 Ram Murty Twin Primes
(Queen's University)

June 28 Tim Trudgian Grosswald's conjecture on primitive roots
(Australian National University)

June 28 Vijay Patankar Pairs of elliptic curves and their Frobenius fields
(Jawaharlal Nehru University)
Past semesters: Fall 2007 Fall 2008 Fall 2009 Fall 2010 Fall 2011 Fall 2012 Fall 2013 Fall 2014 Fall 2015

Spring 2008 Spring 2009 Spring 2010
Spring 2012 Spring 2013 Spring 2014 Spring 2015