Characteristic polynomials of automorphisms of hyperelliptic curves

Speaker: 

Everett Howe

Institution: 

CCR - La Jolla

Time: 

Thursday, April 17, 2008 - 3:00pm

Location: 

MSTB 254

Let alpha be an automorphism of a hyperelliptic curve C of genus g,
and let alpha' be the automorphism of P^1 induced by alpha.
Let n be the order of alpha and let n' be the order of alpha'.
We show that the triple (g,n,n') completely determines the
characteristic polynomial of the automorphism alpha^* of the
Jacobian of C, unless n is even, n=n', and (2g+2)/n is even,
in which case there are two possibilities. We give explicit
formulas for the characteristic polynomial in all cases.

Hayes's conjecture on the values of equivariant Artin L-functions at s=0

Speaker: 

Dr. Barry Smith

Institution: 

UCI

Time: 

Thursday, February 7, 2008 - 3:00pm

Location: 

MSTB 254

I will introduce a recent conjecture of Hayes concerning the value at s=0 of the
equivariant Artin L-function associated with an Abelian extension K/k of number fields. It
proposes a relationship between certain unramified Kummer extensions of K and the
denominators of the coefficients of this L-function value. The conjecture can be viewed as a
new generalization of the classical analytic class number formula.

T-adic L-functions of p-adic exponential sums

Speaker: 

Professor Daqing Wan

Institution: 

UCI

Time: 

Thursday, January 24, 2008 - 3:00pm

Location: 

MSTB 254

The T-adic L-function is a unversal L-function which
interpolates classical L-functions of all p-power order
exponential sums associated to a polynomial f(x) defined
over a finite field. We study its T-adic analytic properties
(analytic continuation and its T-adic Newton polygon).
The T-adic Newton polygon provides a uniform improvement
to previous results on p-adic Newton polygon of exponential sums
in the non-ordinary case. This is joint work with Chunlei Liu.

Elliptic divisibility sequences

Speaker: 

Marco Streng

Institution: 

Leiden University

Time: 

Thursday, January 17, 2008 - 2:50pm

Location: 

MSTB 254

Elliptic divisibility sequences arise as sequences of
denominators of the integer multiples of a rational point on an elliptic
curve. Silverman proved that almost every term of such a sequence has a
primitive divisor (i.e. a prime divisor that has not appeared as a
divisor of earlier terms in the sequence). If the elliptic curve has
complex multiplication, then we show how the endomorphism ring can be
used to index a similar sequence and we prove that this sequence also
has primitive divisors. The original proof fails in this context and
will be replaced by an inclusion-exclusion argument and sharper
diophantine estimates.

Prime densities for linear recurrent sequences

Speaker: 

Professor Peter Stevenhagen

Institution: 

Leiden University

Time: 

Thursday, January 17, 2008 - 2:00pm

Location: 

MSTB 254

Given an integer sequence X={x_n}_n, a natural question is to
`quantify' the number of primes dividing at least one non-zero
term of the sequence. For most naturally occurring sequences this is a
hard question, and usually we only have conjectures.
We will show that in the case of second order linear recurrent sequences,
the set of prime divisors has a natural density that, at least in principle,
can be computed exactly.

Vanishing and non-vanishing critical values of elliptic L-functions

Speaker: 

Hershy Kisilevsky

Institution: 

Concordia University

Time: 

Thursday, February 21, 2008 - 3:00pm

Location: 

MSTB 254

I will discuss some results on the vanishing and non-vanishing of
critical values of L-functions and their derivatives, both experimental and
theoretical. I will present an example of a computational "elliptic Stark
point" in a cyclic quintic extension of the rationals.

Drinfeld modular forms and Hecke characters

Speaker: 

Gebhard Boeckle

Institution: 

Essen

Time: 

Thursday, December 6, 2007 - 3:00pm

Location: 

MSTB 254

In a similar way as in the case of elliptic modular forms, one can attach
strictly compatible systems (SCS) of Galois representations to Drinfeld
modular forms. Unlike in the classical situations, these are abelian. Goss
had asked whether they would arise from Hecke characters. Adapting to the
function field setting a correspondence of Khare between SCS of mod p
Galois representations and Hecke characters, this can indeed be shown to
be the case. If time permits, I shall also give some examples and discuss
some open questions regarding these Hecke characters.

Dynamic modular curves

Speaker: 

Michelle Manes

Institution: 

USC

Time: 

Thursday, November 15, 2007 - 3:00pm

Location: 

MSTB 254

Consider a rational map φ on the projective line, from which we form a (discrete) dynamical system via iteration, and let K be a number field. A fundamental question in arithmetic dynamics is the uniform boundedness conjecture of Morton and Silverman, which states that there is a constant independent of φ (depending only on its degree) giving an upper bound for the number of K-rational preperiodic points of φ. This is a deep conjecture, and no specific case of it is known. I have proposed a specific version of the conjecture: that in the case of a degree-2 rational map and K = Q, the upper bound is 12.

In this talk, which assumes no previous knowledge of arithmetic dynamics, I will describe why this question is so difficult and sketch work that has been done to date, including giving justification for my refined uniform boundedness conjecture. The techniques used so far, which have clear limitations, involve constructing algebraic curves parameterizing maps $\phi$ together with points of period n for varying n (so-called dynamic modular curves).

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