Dedekind Zeta functions at s=-1 and the Fitting ideal of the tame kernel in a relative quadratic extension

Speaker: 

Jonathan Sands

Institution: 

Univ. of Vermont and UCSD

Time: 

Tuesday, February 13, 2007 - 12:00pm

Location: 

MSTB 254

Brumer's conjecture states that Stickelberger elements combining values of L-functions at s=0 for an abelian extension of number fields E/F should annihilate the ideal class group of E when it is considered as module over the appropriate group ring. In some cases, an ideal obtained from these Stickelberger elements has been shown to equal a Fitting ideal connected with the ideal class group. We consider the analog of this at s=-1, in which the class group is replaced by the tame kernel, which we will define. For a field extension of degree 2, we show that there is an exact equality etween the Fitting ideal of the tame kernel and the most natural higher Stickelberger ideal; the 2-part of this equality is conditional on the Birch-Tate conjecture.

Growth of ranks of elliptic curves in Galois extensions of number fields

Speaker: 

Karl Rubin

Institution: 

UCI

Time: 

Thursday, November 9, 2006 - 3:00pm

Location: 

MSTB 254

Suppose E is an elliptic curve defined over a number field k, K/k is a quadratic extension, p is an odd prime, and L is a p-extension of K that is Galois over K. Let c be an element of order 2 in Gal(L/k), and H the subgroup of all elements of G := Gal(L/K) that commute with c. Under very mild hypotheses the Parity Conjecture (combined with a little representation theory) predicts that if the rank of E(K) is odd, then the rank of E(L) is at least [G:H]. For example, if L/k is dihedral and the rank of E(K) is odd, then the rank of E(L) should be at least [L:K].

In this talk I will discuss recent joint work with Barry Mazur, where we prove an analogue of this result with "rank" repaced by "p-Selmer rank".

Moment zeta functions, I

Speaker: 

Professor Daqing Wan

Institution: 

UCI

Time: 

Tuesday, January 9, 2007 - 2:00pm

Location: 

MSTB 254

This will be a series of two introductory lectures on the
distribution of closed points on a scheme of finite type
over the integers. Both general properties and important
examples will be discussed, with an emphasis on p-adic
variation for zeta functions over finite fields.

Brumer's Conjecture for Extensions of Degree 2p

Speaker: 

Barry Smith

Institution: 

UCSD

Time: 

Tuesday, January 30, 2007 - 12:00pm

Location: 

MSTB 254

An explicit expression for the L-function evaluator associated to an abelian extension K/k of number fields of degree 2p will be discussed. Instances will be given where this expression can be used to determine pieces of the ideal class group of K that are annihilated by this L-function evaluator.

Error-correcting codes on low rank surfaces

Speaker: 

Marcos Zarzar

Institution: 

University of Texas

Time: 

Thursday, May 25, 2006 - 3:00pm

Location: 

MSTB 254

Motivated by the construction made by Goppa on curves, we present some error-correcting codes on algebraic surfaces. A surface whose Neron-Severi group has rank 1 has a "nice" intersection property that allows us the construction of a good code. We will verify this on specific examples. Surfaces with many points and rank 1 are not easy to find. We were able, though, to find also surfaces with low rank and many points, and these gave us good codes too. Finally, we present a decoding algorithm for these codes. It is based on the realization of the code as a LDPC code, and it is inspired on the Luby-Mitzenmacher algorithm.

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