


LFunctions and Algebraic Curves
Org: YuRu Liu, David McKinnon and Michael Rubinstein (Waterloo) [PDF]
 AMIR AKBARY, Department of Mathematics and Computer Science, University of
Lethbridge, 4401 University Drive West, Lethbridge, AB
T1K 3M4
Mean Values of Product of Modular LFunctions
[PDF] 
Let f be a newform of even weight k, level M and
character y and let g be a newform of even weight l,
level N and character h. We give a generalization of a theorem
of Elliott, regarding the average values of Dirichlet Lfunctions,
in the context of twisted product Lfunctions L_{f,c}(s_{0})[`(L_{g,c}(s_{0}))] associated to f and g and Dirichlet
characters c.
 IMIN CHEN, Simon Fraser University
Diophantine equations via Galois representations
[PDF] 
I will describe recent work on applying the method of Galois
representations and modular forms to further cases of the generalized
Fermat equation.
 ALINA COJOCARU, Princeton University, Dept. Mathematics, Fine Hall,
Washington Road, Princeton, NJ 08544, USA
Uniform results for Serre's Theorem for elliptic curves
[PDF] 
Let E be an elliptic curve defined over Q. For a rational
prime q, let r_{q} be the mod q Galois representation of E.
A classical result of Serre, proven in 1972, asserts that if E is
without complex multiplication, then there exists a constant C(E) > 0
such that r_{q} is surjective for any q > C(E). Serre asked
whether the constant C(E) is absolute (i.e., it does not
depend on E). In my presentation I will discuss function field and
oneparameter average analogues of Serre's question.
This is joint work with Chris Hall (Univ. of Texas at Austin).
 BRIAN CONREY, AIM

 JOHN FRIEDLANDER, University of Toronto
Exceptional characters and the distribution of primes
[PDF] 
We study some of the extremely strong statements about the
distribution of prime numbers which follow from the (admittedly
unlikely) assumption of the existence of exceptional Dirichlet
characters.
This is joint work with Henryk Iwaniec.
 ALEX GHITZA, McGill University
A numerical exploration of mod p Hecke eigensystems
[PDF] 
In his recent preprint on Serre's conjecture in level 1, Khare asks
about the number of mod p Galois representations unramified outside
p, or equivalently (thanks to his work), the number of level 1
mod p Hecke eigensystems. We report on some computational work on
this problem and its generalizations.
 STEVE GONEK, University of Rochester, Rochester, NY 14627
A Statistical Model for the Riemann Zeta Function
[PDF] 
The recent random matrix characteristic polynomial models for the
Riemann zeta function and other Lfunctions have allowed us to
predict answers to a variety of questions previously considered
intractable. However, random matrices carry no arithmetical
information, so one generally has to insert this component of the
answer in an ad hoc manner. I will present a new model
developed with C. Hughes and J. Keating that avoids this problem and
illustrate its applications.
 HARALD HELFGOTT, Université de Montréal
Growth and generation in SL_{2}(Z/p)
[PDF] 
Some groups can be generated slowly. Some groups can't. In some
groups, sets can grow slowly. In some other groups, sets can't. What
goes wrong when additive combinatorics goes nonabelianand how can
this "wrongness" help us where automorphic forms have not?
 ERNST KANI, Queen's University
Lfunctions of certain quotient varieties
[PDF] 
In this talk I plan to discuss the HasseWeil Zetafunction of
(singular) quotient varieties Y = X/G, where G is a finite group
acting on a smooth variety X/Q. Of particular interest here is the
case that X is a modular product surface of level N,
i.e., X = X(N) ×X(N), where X(N)/Q is the modular
curve of level N.
 SHINYA KOYAMA, Zeta Institute
The double Riemann zeta function
[PDF] 
The double Riemann zeta function (zÄz)(s) is defined
by a double Euler product over pairs of primes (p,q). Any
nontrivial zero r of (zÄz)(s) is given by a sum
of zeros of z(s). Namely, there exists a pair of zeros r_{1}
and r_{2} such that r = r_{1}+r_{2}. The aim of this talk
is to introduce a basic theory of the double Riemann zeta function,
and discuss its possible applications.
The first possibility is to enlarge the zerofree region of
z(s). We obtain an explicit form of the (p,q)Euler factors
and show that the double Euler product is absolutely convergent in
Â(s) > 2. Conjecturally it should be convergent in Â(s) > 3/2,
which implies that z(s) is zerofree in Â(s) > 3/4. Thus any
improvement of our current result would give a new result toward the
RH.
The second application is to improve the ratio N_{0}(T)/N(T). Since a
zero r = r_{1}+r_{2} is simple only if both r_{1} and
r_{2} are simple, an estimate of the ratio of simple zeros for
(zÄz)(s) can possibly improve the ratio N_{0}(T)/N(T)
for z(s).
 JUNGJO LEE, Queen's University
Dirichlet Series and Hyperelliptic Curves, Part 2
[PDF] 
We will continue our investigation of attaching Dirichlet series to
hyperelliptic and superelliptic curves and deduce various arithmetic
results from analytic properties of these series.
This is joint work with M. Ram Murty.
 ADAM LOGAN, University of Liverpool (England)
Descent by Richelot isogeny on the Jacobians of plane quartics
[PDF] 
We discuss theoretical and practical aspects of Richelot isogenies on
the Jacobians of curves of genus 3, including their application to
calculating invariants of the curves.
 GREG MARTIN, University of British Columbia
Inequities in the ShanksRényi Prime Number Race
[PDF] 
Let p(x;q,a) denote the number of primes up to x that are
congruent to a modulo q. Inequalities of the type p(x;q,a) > p(x;q,b) are more likely to hold if a is a nonsquare modulo q
and b is a square modulo q (the socalled "Chebyshev Bias" in
comparative prime number theory). However, the tendencies of the
various p(x;q,a) (for nonsquares a) to dominate p(x;q,b)
have different strengths. A related phenomenon is that the six
possible inequalities of the form p(x;q,a_{1}) > p(x;q,a_{2}) > p(x;q,a_{3}), with a_{1}, a_{2}, a_{3} all nonsquares modulo q, are not
equally likely; some orderings are preferred over others. For given
values q,a,b,..., these tendencies can be quantified and
computed, but only using laborious numerical integration of functions
involving zeros of the appropriate Dirichlet Lfunctions. In this
talk we describe a framework for explaining which nonsquares a are
most dominant for a given square b, for example, based only on
elementary properties of the congruence classes a modulo q rather
than the complicated computations just mentioned. These elementary
properties, on the other hand, do derive from consideration of
differences in the distributions of zeros of various Lfunctions,
for example those corresponding to odd and even characters.
 KUMAR MURTY, University of Toronto, 100 St. George Street, Toronto, ON
M5S 3G3
On the order of vanishing of Lfunctions at the central
critical point
[PDF] 
We formulate a pair correlation hypothesis for certain general
Lfunctions and discuss the implication of such a hypothesis for the
order of vanishing at the central critical point.
 DOUG ULMER, University of Arizona
Abelian varieties of large analytic rank over function fields
[PDF] 
I will discuss a simple linear algebra fact which allows one to
produce lots of Lfunctions with large order zeroes at the critical
point. Two sample applications:
(1) for every p and every g there exist geometrically
simple, nonisotrivial abelian varieties of dimension g over
Fp(t) with arbitrarily large analytic rank; and
(2) if E is any elliptic curve over Fq(t) with
jinvariant not in Fq, then E obtains arbitrarily large
analytic rank over extensions of the form Fq(u), where t is a
rational function of u.

