L-functions of p-adic characters

Speaker: 

Daqing Wan

Institution: 

UC Irvine

Time: 

Thursday, April 19, 2012 - 3:00pm to 4:00pm

Location: 

RH440R

Our main question is the p-adic meromorphic continuation of
the L-function attached to a p-adic character for the rational
function field over a finite field of characteristic p. In this talk,
I will explain a new and (hopefully) transparent approach to this
problem. (This is ongoing joint work with Chris Davis).

Constructing Higher Order Elements in Finite Fields

Speaker: 

Qi Cheng

Institution: 

University of Oklahoma

Time: 

Wednesday, March 21, 2012 - 10:00am to 11:00am

Host: 

Location: 

440R

Every finite field has many multiplicative generators. However,
finding one in polynomial time is an important open problem .
In fact, even finding elements of high order has not been solved
satisfactorily. In this paper, we present an algorithm that for any
positive integer $c$ and prime power $q$, finding an element of
order $\exp(\Omega(\sqrt{q^c}) ) $ in the finite field $\F_{q^{(q^c-1)/(q-1)}}$
in deterministic time $(q^c)^{O(1)}$. We also show that there are
$\exp(\Omega(\sqrt{q^c}) ) $ many weak keys for the discrete logarithm problems
in those fields with respect to certain bases.

The divisibility of the Tate-Shafarevich group of an elliptic curve in the Weil-Chatelet group

Speaker: 

Mirela Ciperiani

Institution: 

University of Texas, Austin

Time: 

Thursday, February 16, 2012 - 3:00pm

Location: 

RH 440R

In this talk I will report on progress on the following two questions, the first posed by Cassels in 1961 and the second considered by Bashmakov in 1974. The first question is whether the elements of the Tate-Shafarevich group are innitely divisible when considered as elements of the Weil-Chatelet group. The second question concerns the intersection of the Tate-Shafarevich group with the maximal divisible subgroup of the Weil-Chatelet group. This is joint work with Jakob Stix.

Abelian varieties with big monodromy

Speaker: 

David Brown

Institution: 

Emory University

Time: 

Thursday, April 5, 2012 - 3:00pm

Location: 

RH 440R

Serre proved in 1972 that the image of the adelic Galois representation
associated to an elliptic curve E without complex multiplication has open
image; moreover, he also proved that for an elliptic curve over Q the
index of the image is always divisible by 2 (and in particular never
surjective). More recently, Greicius in his thesis gave criteria for
surjectivity and gave an explicit example of an elliptic curve E over a
number field K with surjective adelic representation. Soon after, Zywina,
building on earlier work of Duke, Jones, and others, proved that the
adelic image `random' elliptic curve is maximal.

In this talk I will explain recent joint work with David Zywina in which
we generalize these theorems and prove that a random abelian variety in a
family with big monodromy has maximal image of Galois. I'll explain what
big monodromy and maximal mean an explain the analytic and geometric
techniques used in previous work and the new geometric ideas -- in
particular, Nori's method of semistable approximation-- needed to
generalized to higher dimension.

Fast Toric Algorithms Over Local Fields

Speaker: 

Professor Maurice Rojas

Institution: 

Texas A&M University

Time: 

Thursday, April 12, 2012 - 3:00pm

Location: 

RH 440R

We show how to efficiently count exactly the number of solutions of a system of n polynomials in n variables over certain local fields L, for a new class of polynomials systems. The fields we handle include the reals and the p-adic rationals. The polynomial systems amenable to our methods are made up of certain A-discriminant chambers, and our algorithms are the first to attain polynomial-time in this setting. We also discuss connections to Baker's refinement of the abc-Conjecture, Smale's 17th Problem, and tropical geometry. The results presented are, in various combinations, joint with Martin Avendano, Philippe Pebay, Korben Rusek, and David C. Thompson.

Balanced subgroups

Speaker: 

Professor Carl Pomerance

Institution: 

Dartmouth College

Time: 

Thursday, February 9, 2012 - 3:00pm

Location: 

RH 440R

For an integer n > 2, the unit group modulo n has an even number
of elements, with half of them having representatives in (0,n/2)
and the other half having representatives in (n/2,n). It is
"balanced". Say a subgroup H of this unit group is "balanced"
if each coset of H is evenly split between the bottom half and
the top half. Suppose g>1 is a fixed integer. We are concerned with
the distribution of numbers n coprime to g for which the
cyclic subgroup in the unit group mod n is balanced.
This has an application to the statistical study of the rank
of the Legendre curve over function fields. (Joint work with
Douglas Ulmer.)

On the parity conjecture for Selmer groups of modular forms

Speaker: 

Dr. Liang Xiao

Institution: 

University of Chicago

Time: 

Thursday, January 19, 2012 - 2:00pm

Location: 

RH 306

The parity conjecture is a weak version of Birch-Swinnerton-Dyer
Conjecture or more generally, Beilinson-Bloch-Kato Conjecture. It is
conjectured that the vanishing order of the L-function at the central
point has the same parity as the dimension of the Bloch-Kato Selmer
group. I will explain an approach to this conjecture for modular
forms by varying the modular forms in a p-adic family. This is a joint
work with Kiran Kedlaya and Jay Pottharst.

Complexity in coding theory

Speaker: 

Professor Daqing Wan

Institution: 

UCI

Time: 

Thursday, November 3, 2011 - 3:00pm

Location: 

RH 440R

Among the most important complexity problems in
coding theory are the maximun likelihood decoding
and the computation of the minimun distance.
In this talk, we explain a self-contained, quick and
transparent proof of the NP-hardness of these
problems based on the subset sum problem over
finite fields.

On nonvanishing of the central value of the Rankin-Sleberg L-functions

Speaker: 

Professor Dihua Jiang

Institution: 

University of Minnesota

Time: 

Thursday, November 17, 2011 - 3:00pm

Location: 

RH 440R

After introducing the basic theory of automorphic forms and L-functions, we will discuss the characterization
of the nonvanishing of the central value of certain Rankin-Selberg L-functions in terms of periods of automorphic
forms. This is part of the global Gan-Gross-Prasad conjecture, which was first announced in early 1990's by
Gross and Prasad and was reformulated by Gan, Gross and Prasad in 2010. Our results were accummulated in
a series of my three papers (2004, 2005, 2009), joint with Ginzburg and Rallis and a more recent paper (2010)joint with Ginzburg and Soudry.

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