# Southern California Number Theory Day

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## Time:

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Southern California Number Theory Day, 10am - 5pm

Saturday, November 13, 2010 - 10:00am

RH 101

Southern California Number Theory Day, 10am - 5pm

Nick Alexander

UCI

Tuesday, May 18, 2010 - 2:00pm

RH 306

We construct explicit genus 2 hyperelliptic curves whose Jacobian varieties have complex multiplication and are defined over an explicit algebraic number field. For these Jacobians, we give a formula for the number of points on their reductions modulo primes of good reduction. The construction and results can be viewed as a dimension 2 generalization of results of H. Stark. These formulas have application to cryptography and the CM method in dimension 2.

Professor Dihua Jiang

University of Minnesota

Tuesday, April 13, 2010 - 2:00pm

RH 306

Endoscopy structures of automorphic forms was one of the

basic structures discovered through the Arthur-Selberg trace formula method to establish the Langlands functoriality for classical groups.

In this talk, we will discuss my recent work on characterization of the endoscopy structure in terms of the order of pole at s=1 of certain L-functions, and in terms of a family of periods of automorphic forms, which was discovered jointly with David Ginzburg. At the end, I may discuss how to contruct the

endoscopy transfer by integral operators, which is

a joint work with Ginzburg and Soudry.

Professor Mike Fried

Montana State U-Billings, Emeritus UCI

Tuesday, April 6, 2010 - 2:00pm

RH 306

Variables Separated Equations and Finite Simple Groups: Davenport's

problem is to figure out the nature of two polynomials over a number

field having the same ranges on almost all residue class fields of the

number field. Solving this problem initiated the monodromy method.

That included two new tools: the B(ranch)C(ycle)L(emma) and the

Hurwitz monodromy group. By walking through Davenport's problem with

hindsight, variables separated equations let us simplify lessons on

using these tools. We attend to these general questions:

1. What allows us to produce branch cycles, and what was their effect

on the Genus 0 Problem (of Guralnick/Thompson)?

2. What is in the kernel of the Chow motive map, and how much is it

captured by using (algebraic) covers?

3. What groups arise in 'nature' (a 'la a paper by R. Solomon)?

Each phrase addresses formulating problems based on equations. We seem

to need explicit algebraic equations. Yet why, and how much do we lose/

gain in using more easily manipulated surrogates for them? To make

this clear we consider the difference in the result for Davenport's

Problem and that for its formulation over finite fields, using a

technique of R. Abhyankar.

Professor Lenny Fukshansky

Claremont College

Tuesday, March 2, 2010 - 2:00pm

RH 306

A lattice of rank N is called well-rounded (abbreviated WR) if its minimal vectors span R^N. WR lattices are extremely important for discrete optimization problems. In this

talk, I will discuss the distribution of WR lattices in R^2, specifically concentrating

on WR sublattices of Z^2. Studying the structure of the set C of similarity classes of

these lattices, I will show that elements of C are in bijective correspondence with

certain ideals in Gaussian integers, and will construct an explicit parametrization of

lattices in each such similarity class by elements in the corresponding ideal. I will

then use this parameterization to investigate some basic analytic properties of zeta

function of WR sublattices of Z^2.

Tommy Occhipinti

University of Arizona

Tuesday, February 2, 2010 - 2:00pm

RH 306

It is a fascinating result of Ulmer that the elliptic curve y^2=x^4+x^3+t^d attains arbitrarily large rank over $\bar{F_q}(t)$ as d varies over the positive integers. In this talk we will provide some new examples of this phenomenon and provide an overview of previous work in this area, particularly that of Ulmer and Berger.

Mr. Zev Klagsbrun

UCI

Tuesday, January 19, 2010 - 2:00pm

RH 306

Mr. Damek Davis

UCI

Tuesday, January 26, 2010 - 2:00pm

RH 306

We will discuss some of the local theory of rigid-analytic spaces including Tate's algebra, affinoid algebras, Washnitzer's algebra and dagger algebras. After we provide enough motivation we will discuss the results of research completed by myself and Professor Daqing Wan. The results of this research form a basis for generalizing Washnitzer's algebra.

Chris Davis

Max Planck Institute

Tuesday, January 5, 2010 - 2:00pm

RH 306

To an algebraic variety over the complex numbers, we can associate a complex analytic space. When the result is a smooth complex manifold, we can compute its de Rham cohomology. I would like to discuss some ways to compute this cohomology directly from our algebraic variety, and how these methods can be adapted to more general varieties. None of the material I will present is original. The results are due to many people, especially Grothendieck.