Formal Dirichlet Series and Zeta Functions of Schemes

Speaker: 

Professor Jesse Elliott

Institution: 

Cal State Univ, Channel Islands

Time: 

Thursday, December 1, 2005 - 3:00pm

Location: 

MSTB 254

The set of multiplicative arithmetic functions over a ring R
(commutative with identity) can be given a unique functorial ring
structure for which (1) the operation of addition is Dirichlet
convolution and (2) multiplication of completely multiplicative
functions coincides with point-wise multiplication. This existence of
this ring structure can be derived from the existence of the ring of
``big'' Witt vectors, and it yields a ring structure on the set of
formal Dirichlet series that are expressible as an Euler product. The
group of additive arithmetic functions over R also has a naturally
defined ring structure, and there is a functorial ring homomorphism
from the ring of multiplicative functions to the ring of additive
functions that is an isomorphism if R is a Q-algebra. An application
is given to zeta functions of schemes of finite type over the ring
of integers.

On the unramified spectrum of spherical varieties over p-adic fields

Speaker: 

Yiannis Sakellaridis

Institution: 

Stanford University

Time: 

Thursday, October 27, 2005 - 3:00pm

Location: 

MSTB 254

Varieties with an action of a reductive group such that the Borel subgroup has an open orbit are called spherical. Spherical varieties are a unifying theme behind many analytic techniques in the theory of automorphic forms, such as the relative trace formula and integral representations of L-functions. After briefly surveying these methods - for which a general and systematic theory is missing - in order to justify this claim, I prove a general result on the representation theory of spherical varieties for split groups over p-adic fields: Irreducible quotients of the "unramified summand" of Cc&#8734(X) (where X is the spherical variety) are "roughly" parametrized by the quotient of a complex torus by a finite reflection group. This generalizes the classical parametrization of the "unramified spectrum" of G by semisimple conjugacy classes in its Langlands dual, and is compatible with recent results of D.Gaitsgory & D.Nadler which assign a "Langlands dual group" to every spherical variety. The main tool in the proof is an action, defined by F.Knop, of the Weyl group of G on the set of Borel orbits on X.

Pairings in Cryptography

Speaker: 

Professor Alice Silverberg

Institution: 

UCI

Time: 

Wednesday, May 18, 2005 - 3:00pm

Location: 

MSTB 256

This talk will serve as an introduction to the use of pairings
(especially Weil pairings on elliptic curves or abelian varieties)
in cryptography. We will mention some open questions that have
practical interest for cryptographers and should be more fully
explored by number theorists. We also show how abelian varieties
and the Weil restriction of scalars can (sometimes) be used to
"compress" points on elliptic curves.

Computing the zeta function of a $\Delta$-regular hypersurface

Speaker: 

John Voight

Institution: 

UC Berkeley

Time: 

Friday, April 22, 2005 - 4:00pm

Location: 

MSTB 254

The number of solutions to a set of polynomial equations defined over a
finite field of $q=p^a$ elements is encoded by its zeta function, which is a
rational function in one variable. A question of fundamental interest is
how to compute this function efficiently. We describe a method to solve
this problem (due to Wan) using a $p$-adic trace formula of Dwork. We
examine how well this method works in practice on some explicit examples
coming from a certain class of varieties known as $\Delta$-regular
hypersurfaces. Our experience suggests several potential avenues of further
research.

Hard Problems of Algebraic Geometry Codes

Speaker: 

Professor Qi Cheng

Institution: 

University of Oklahoma

Time: 

Wednesday, June 8, 2005 - 3:00pm

Location: 

MSTB 256

The minimum distance is one of the most important combinatorial
characterizations of a code. The maximum likelihood decoding problem is
one of the most important algorithmic problems of a code. While these
problems are known to be hard for general linear codes, the techniques
used to prove their hardness often rely on the construction of artificial
codes. In general, much less is known about the hardness of the specific
classes of natural linear codes. In this paper, we show that both problems
are NP-hard for algebraic geometry codes. We achieve this by reducing a
well-known NP-complete problem to these problems using a randomized
algorithm. The family of codes in the reductions have positive rates, but
the alphabet sizes are exponential in the block lengths.

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