# Number Theory Seminar Abstracts-Thursdays

Title: Galois groups of tamely ramified p-extensions.
Abstract: Explicit presentations of the Galois group of the maximal
p-extension of Q unramified outside a finite set S of primes not containing p, have eluded
researchers for many years. This talk will report on work by the speaker, by
Labute, and others describing these groups and their properties, with striking
consequences and connections to algebraic topology, root-discriminant
problems, deficiency zero problems, and the Fontaine-Mazur conjecture.

Title:  Deterministically finding points on elliptic curves over finite
fields.
Abstract:  In this talk we'll consider the computational problem of
finding points on elliptic curves over finite fields.  This has
cryptographic applications; for example, the ElGamal encryption system
requires a message to be encoded as a point on an elliptic curve.  An easy
probabilistic polynomial-time algorithm exists to solve the problem.
Given an elliptic curve E[F_p]: y^2 = x^3 + ax + b with p prime and > 3,
pick a random x.  If the Legendre symbol (x^3+ax+b / p ) is 1, take the
square root to get the point (x,y) in E.  Otherwise try another random x.
A similar probabilistic algorithm exists for elliptic curves over
finite fields of characteristic 2.  In this talk we'll give a
deterministic polynomial-time algorithm for finding points on elliptic
curves with j-invariant 0 over finite fields of characteristic 2.

Thurs. Sept. 22---Ben Brubaker (Stanford)

Title: Recent Progress on Dirichlet Series Associated to Weyl Groups

Abstract: In a series of papers with Dan Bump and others, we prove the meromorphic continuation and functional equations for a family of
Dirichlet series in several variables. The groups of functional equations are Weyl groups. Moreover, these Dirichlet series are
conjectured to arise from the Whittaker coefficients of certain Eisenstein series attached to simple Lie groups with the
associated Weyl group. In this talk, we'll discuss what sorts of Dirichlet series arise, giving concrete examples, and then
survey the arithmetic applications (both past and future) which can/may be obtained from them. (To the indoctrinated: Efforts
will be made to discuss different, but related, topics from those arising in the Bretton Woods workshop talks.)

Tuesday  Oct. 18---Hui Xue (Michigan)
The title is: "Central values of L-functions over CM fields"
Abstract: We will show an explicit formula  for the central value of certain
Rankin $L$-function $L(\pi\otimes{\chi})$, where $\pi$ is an automorphic
representation associated to a Hilbert newform $f$ and ${\chi}$  is a unitary
anticyclotomic  Hecke character of a CM field. As corollaries some arithmetic
properties of central values can be obtained. The proof uses explicit
computations which involve theta correspondences.