School of Mathematics and Statistics |
Department of Mathematics and Statistics |

**NUMBER THEORY SEMINAR**

**Fall 2006/ Winter 2007**

In this talk I will discuss the resonance method of Soundararajan which is used in detecting

large values of L-functions and character sums. In particular, I will discuss an application of

this method to finding extreme values of the derivative of the zeta function evaluated at the

zeros of zeta.

Let $K$ be a real quadratic number field and let $p$ be a prime number inert in $K$. We denote

by $K_p$ the completion of $K$ at $p$. Using periods of modular functions we construct a family

of $\ZZ$-valued measures on $\PP^1(\QQ_p)$. We then use those measures to do $p$-adic integration

and construct $p$-adic invariants in $K_p^{\times}$. Those $p$-adic invariants are conjectured to be

global $p$-units in abelian extensions of $K$. The truth of this conjecture would entail an explicit

class field theory for $K$. We also construct $p$-adic $L$-functions for which we relate their first

derivative at $s=0$ with our $p$-adic invariants. This is an analogue of the classical Kronecker limit formula.

I present a factoring algorithm that factors N=UV, where U < V, provably in O(N^(1/3+epsilon))

time. I also discuss the potential for improving this to a sub-exponential algorithm. Along the way,

I consider the distribution of solutions (x,y) to xy=N modulo a, using estimates for Kloosterman sums.

Mean-value theorems (or moments) for the Riemann zeta-function and other families of L-functions are a

central problem in analytic number theory, but relatively few have been proved. The recent introduction of

random matrix theory into the field led to a greater understanding of the structure of such moments conjecturally,

and this in turn has led to a very general but detailed estimate for various moments called the Ratios Conjecture.

Here we prove some mean value formulas which confirm predictions of the Ratios Conjecture in a number of cases.

integer by certain quaternary quadratic forms. This is joint work with A. Alaca, S. Alaca and M. Lemire.

then I will explain the basics of $2$-descent on curves of genus $2$. The

principal homogeneous spaces of the Jacobian of the curve that arise are

often awkward to analyze---in particular, it is quite difficult to prove

that they have no rational points. I will describe some interesting curves

on the quotient of the homogeneous space by its natural involution and show

how to use them in some special cases to prove that the quotient has no

rational points. From this I obtain an infinite family of twists of a

curve of genus $2$ all with nontrivial Tate-Shafarevich group. This is

joint work with Ronald van Luijk.

Wilhelm Ljunggren proved many remarkable theorems on the number

of integer solutions to certain families of quartic Diophantine equations.

However, many fundamental problems remained unsettled after his

investigations. In joint work with Shabnam Akhtari and Alain Togbe,

some of the outstanding problems from Ljunggren's work have been

completely solved. For instance, we prove the best possible result

that for any positive integers $a,b$ the quartic equation $aX^4-bY^2=1$

has at most two solutions in positive integers (X,Y).

Other results of this type will also be discussed.

theory has been the non-vanishing of L-functions. Many of the

central ideas for proving the non-vanishing of an L-function date

back to Selberg's proof that a positive proportion of the zeros of the

Riemann zeta function lie on the half-line. In this talk I will discuss

some non-vanishing results which imply that the Fermat equation

A^4+B^2=C^p for p >7 a prime has no non-trivial solutions. This is

joint work with Michael Bennett and Jordan Ellenberg.

which is a generalisation to the case of Siegel spaces of arbitrary

degree, of the classical theorem of Schneider on the modular invariant

j(\tau). Given a point \tau of the Siegel space parametrizing a

principally polarised abelian variety A defined over

the algebraic closure of the rationals, we obtain a lower bound for the distance between

\tau and algebraic points \beta of the Siegel space, in terms of the

geometrical data of the problem. To achieve this,

we establish a simultaneous measure of linear independance for periods of

abelian integrals, using Baker's method.

to the context of a sequence of polynomials taking small values on the

initial terms of a fixed geometric sequence of complex numbers.

of multiplicative group varieties and describe how they are natural

generalizations of the classical Lehmer problem. Applications to other

arithmetic geometry problems will be discussed as well as the latest results in

the direction of theses conjectures. If time permits, we shall devote a few

words to the abelian variety situation.

Nathan Ng

nng362 at science.uottawa.ca

Tel. 613-562-5800 ext 3515

or

Damien Roy

droy at uottawa.ca

Tel. 613-562-5800 ext 3504