Distributions of ranks and Selmer groups of elliptic curves

Speaker: 

Wei Ho

Institution: 

University of Michigan

Time: 

Tuesday, May 10, 2016 - 2:00pm to 3:00pm

Location: 

RH 340P

In the last several years, there has been significant theoretical progress on understanding the average rank of all elliptic curves over Q, ordered by height, led by work of Bhargava-Shankar. We will survey these results and the ideas behind them, as well as discuss generalizations in many directions (e.g., to other families of elliptic curves, higher genus curves, and higher-dimensional varieties) and some corollaries of these types of theorems. We will also describe recently collected data on ranks and Selmer groups of elliptic curves (joint work with J. Balakrishnan, N. Kaplan, S. Spicer, W. Stein, and J. Weigandt).

Eisenstein modular symbols and p-adic L-functions

Speaker: 

Ander Steele

Institution: 

UC Santa Cruz

Time: 

Tuesday, March 8, 2016 - 2:00pm to 3:00pm

Host: 

Location: 

RH 340P

The overconvergent modular symbols of Stevens provide a natural framework for computing p-adic L-functions of newforms, but the modular symbols (and p-adic L-functions) attached to ordinary Eisenstein series are essentially trivial. Working with a larger space of pseudo-distributions, we construct non-trivial Eisenstein symbols and compute their p-adic L-functions. As a corollary, we compute the p-adic L-function of the "evil twin" Eisenstein series of critical slope. If time permits, I'll discuss work in progress on computing the symmetric square p-adic L-function at Eisenstein points on the eigencurve, as well as applications.

On a motivic method in Diophantine geometry

Speaker: 

Majid Hadian

Institution: 

Caltech

Time: 

Tuesday, March 29, 2016 - 2:00pm to 3:00pm

Host: 

Location: 

RH 340P

By studying the variation of motivic path torsors associated to a variety, we show how certain non-density assertions in Diophantine geometry can be reduced to problems concerning K-groups. Concrete results then follow based on known (and conjectural) vanishing theorems.

Simple groups stabilizing polynomials

Speaker: 

Skip Garibaldi

Institution: 

CCR La Jolla

Time: 

Tuesday, January 26, 2016 - 2:00pm to 3:00pm

Host: 

Location: 

RH 340P

The classic Linear Preserver Problem asks to determine, for a polynomial function f on a vector space V, the linear transformations g of V such that fg = f. In case f is invariant under a simple algebraic group G acting irreducibly on V, we prove that the subgroup of GL(V) stabilizing f often has identity component G and we give applications realizing various groups, including the largest exceptional group E8, as automorphism groups of polynomials and algebras. We show that starting with a simple group G and an irreducible representation V, one can almost always find an f whose stabilizer has identity component G and that no such f exists in the short list of excluded cases. The main results are new even in the special case where the field is the complex numbers, and have implications for Hasse principles for polynomials over number fields.  This talk is about joint work with Bob Guralnick. 

Zeta functions of Z_p-towers of curves

Speaker: 

Daqing Wan

Institution: 

UCI

Time: 

Tuesday, November 3, 2015 - 2:00pm to 3:00pm

Location: 

RH 340P

We explore possible stable properties of the sequence of
zeta functions associated to a geometric Z_p-tower of curves over
a finite field of characteristic p, in the spirit of Iwasawa theory.
Several fundamental questions and conjectures will be discussed,
and some supporting examples will be given. This introductory talk
is accessible to graduate students in number theory and arithmetic
geometry.

A Heuristic for Boundedness of Elliptic Curves

Speaker: 

Jennifer Park

Institution: 

University of Michigan

Time: 

Tuesday, December 1, 2015 - 2:00pm to 3:00pm

Location: 

RH 340P

I will discuss a heuristic that predicts that the ranks of all but finitely many elliptic curves defined over Q are bounded above by 21. This is joint work with Bjorn Poonen, John Voight, and Melanie Matchett Wood.

Cohen-Lenstra heuristics and random matrix theory over finite fields

Speaker: 

Jason Fulman

Institution: 

USC

Time: 

Tuesday, November 10, 2015 - 2:00pm to 3:00pm

Location: 

RH 340P

Two permutations are conjugate if and only if they have the same cycle structure, and two complex unitary matrices are conjugate if and only if they have the same set of eigenvalues. Motivated by the large literature on cycles of random permutations and eigenvalues of random unitary matrices, we study  conjugacy classes of random elements of finite classical groups. For the case of GL(n,q), this amounts to studying rational canonical forms. This leads naturally to a probability measure on the set of all partitions of all natural numbers. We connect this measure to symmetric function theory, and give algorithms for generating partitions distributed according to this measure. We describe analogous results for the other finite classical groups (unitary, symplectic, orthogonal). We were excited to learn that (at least for GL(n,q)), exactly the same random partitions arise in the “Cohen-Lenstra heuristics” of number theory.

Sub-exponential algorithms for ECDLP?

Speaker: 

Michiel Kosters

Institution: 

UCI

Time: 

Tuesday, October 20, 2015 - 2:00pm to 3:00pm

Location: 

RH 340P

In this talk we will discuss various recent claims of algorithms which solve certain instances of the elliptic curve discrete logarithm problem (ECDLP) over finite fields in sub-exponential time. In particular, we will discuss approaches which use Groebner basis algorithms to solve systems coming from summation polynomials. The complexity of these approaches relies on the so-called first fall degree assumption. We will raise doubt to this first fall degree assumption and hence to the claimed complexity.

Galois theory, automorphic forms, and number fields with prescribed ramification

Speaker: 

Brian Hwang

Institution: 

Caltech

Time: 

Tuesday, April 28, 2015 - 2:00pm to 3:00pm

Location: 

RH 340P

A classical problem in Galois theory is a strong variant of
the Inverse Galois Problem: "What finite groups arise as the Galois
group of a finite Galois extension of the rational numbers, if you
impose the additional condition that the extension can only ramify at
finite set of primes?" This question is wide open in almost every
nonabelian case, and one reason is our lack of knowledge about how to
find number fields with prescribed ramification at fixed primes. While
such fields are often constructed to answer arithmetic questions,
there is currently no known way to systematically construct such
extensions in full generality.

However, there are some inspiring programs that are gaining ground on
this front. One method, initiated by Chenevier, is to construct such
number fields using Galois representations and their associated
automorphic representations via the Langlands correspondence. We will
explain the method, show how some recent advances in these subfields
allow us to gain some additional control over the number fields
constructed, and indicate how this brings us closer to our goal. As a
application, we will show how one can use this knowledge to study the
arithmetic of curves over number fields.

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