


Lfunctions, Automorphic Forms and Representation Theory
Org: Amir Akbary (Lethbridge) and Clifton Cunningham (Calgary) [PDF]
 JEFFREY ACHTER, Colorado State University, Fort Collins, CO 80523, USA
Monodromy representations and function field class numbers
[PDF] 
Given an elliptic curve over a finite field, one might ask for the
chance that it has a rational point of order l. More generally,
what is the chance that a curve drawn from a family over a finite
field has a point of order l on its Jacobian?
The answer is encoded in the ladic monodromy representation of
the family in question. In this talk, I'll discuss recent work on
this representation for various families of curves, and use it to
prove a CohenLenstratype result for class groups of function fields.
 IMIN CHEN, Simon Fraser University
Diophantine equations via Galois representations
[PDF] 
Recently, the use of Galois representations attached to elliptic
curves has been used to resolve several cases of the generalized
Fermat equation. In this talk, I will discuss the method and some
further cases which can be analyzed at least partially, including the
equation a^{2}+ b^{2}p = c^{r}, where r = 3 or 5. Although a complete
resolution is not yet possible, a computational criterion can be
obtained for r = 3, based on previous work by BennettSkinner and
Kraus. For r = 5, I outline a possible strategy using a combination
of quadratic Qcurves and elliptic curves over Q.
 GERALD CLIFF, University of Alberta, Dept. of Math. and Stat. Sci.,
Edmonton, AB
Ktypes of local Weil representations
[PDF] 
Let F be a nonarchimedian local field with ring of integers R,
maximal ideal P, and residue field k of odd characteristic. Let
W be the Weil representation of the symplectic group Sp(2n,F),
corresponding to a character c of the additive group of F.
Suppose that the conductor of c is the fractional ideal P^{l}.
If l is even, the restriction of W to the maximal compact
subgroup Sp(2n,R) is known to be a direct sum
Å_{m=0}^{¥} T_{m}, where each T_{m} is can be regarded as a
representation of Sp(2n,R/P^{2m}); this uses the lattice model of
the Schrödinger representation of the Heisenberg group. We show
that there is an analogous decomposition in the case that l is
odd. Each T_{m} arises as a direct summand of a Weillike
representation of Sp(2n,R/P^{2m+1}). In particular, T_{0} is the
Weil representation of Sp(2n,k).
This is joint work with David McNeilly.
 CHANTAL DAVID, Concordia University
Nonvanishing of cubic twists of elliptic curves
[PDF] 
We will present a proof of a quantitative nonvanishing result for
cubic twists of an elliptic curve E over the field K = Q(Ö{3}). Our proof is based on the classical approach of
Iwaniec (for the case of quadratic twists of Lfunctions). A
similar result was also obtained by Brubaker, Friedberg and Hoffstein
by studying some multiple Dirichlet series. We obtain that at least
X^{2/3e} twisted Lfunctions do not vanish at the
critical point. If one could improve the large sieve inequality of
HeathBrown for cubic characters by removing a residual term, the
proportion of nonvanishing would be at least X^{1e},
which is the optimal result that can be obtained without mollifying.
Joint work with D. Milicevic and G. Ricotta.
 LASSINA DEMBELE, University of Calgary
Examples of automorphic forms on the unitary group U(3)
[PDF] 
In this talk, we will present examples of automorphic forms on the
unitary groups in three variables attached to various quadratic
extensions E/F. We will then study their corresponding Galois
representations. Namely, we will determine the set of exceptional
primes which corresponds to the padic family attached to each of
the forms. Those sets which are very important for arithmetic
applications, have been determined for classical modular forms by
works of Serre, Ribet and others. But there is no known result in the
case of the U(3).
 WENTANG KUO, University of Waterloo, Waterloo, Ontario
A generalization of the SatoTate Conjecture
[PDF] 
The original SatoTate Conjecture concerns the angle distribution of
the eigenvalues arisen from nonCM elliptic curves. In this talk, we
formulate an analogue of the SatoTate Conjecture for generic
automorphic forms of GL_{n} and conjecture their angle distributions.
Under a reasonable hypothesis, we can prove that the expected
distribution is indeed true.
 YURU LIU, University of Waterloo, Waterloo, Ontario
On ErdösPomerance's conjecture for the Carlitz module
[PDF] 
For a Î Z, m Î N with (a,m)=1, let
l_{a}(m) be the order of a in (Z/mZ)^{*}. Let
w( l_{a}(m) ) be the number of distinct prime divisors
of l_{a}(m). A conjecture of Erdös and Pomerance states that if
a > 1, then the quantity

w 
æ è

l_{a}(m) 
ö ø

 
1
2

(loglogm)^{2} 



distributes normally. The problem remains open until today. A
conditional proof of it was obtained recently by Murty and Saidak. Li
and Pomerance also provided an alternative proof of the same result.
In this talk, we formulate an analogous question for the Carlitz
module and provide an unconditional proof of it.
This is a joint work with W. Kuo.
 PAUL MEZO, Carleton University, 1125 Colonel By Drive, Ottawa, ON
K1S 5B6
Twisted trace PaleyWiener theorems
[PDF] 
One method of comparing automorphic Lfunctions arising from
different groups is the comparison of trace formulas. Occasionally
there is a group automorphism which presents itself in the comparison
of "twisted" trace formulas. To implement the automorphism
properly, one must characterize twisted characters at the archimedean
places. We describe recent progress in this characterization, which
is joint work with P. Delorme.
 FIONA MURNAGHAN, University of Toronto
Tame supercuspidal representations
[PDF] 
We will discuss recent results (joint with Jeff Hakim) concerning
criteria for equivalence of tame supercuspidal representations. These
criteria are expressed in terms of the Gdata used in J.K. Yu's
construction of tame supercuspidal representations. Then we will
indicate how the criteria for equivalence can be used to give a new
parametrization of tame supercuspidal representations in some cases.
 KUMAR MURTY, Toronto

 RAM MURTY, Queen's University
The LangTrotter conjecture
[PDF] 
Given a normalized Hecke eigenform f of weight k and level N,
let a_{n}(f) denote its nth Fourier coefficient. The generalized
LangTrotter conjecture predicts that for k > 3, and a given value
c, the number of n such that a_{n}(f) = c, is finite. If c is
odd and N=1, we will prove this conjecture. By connecting this to
Serre's epsilon conjecture, we resolve it for all levels of the form
2^{a} N_{0} with N_{0} = 1, 3, 5, 15 or 17.
This is joint work with V. Kumar Murty.
 NATHAN NG, University of Ottawa
Discrete mean values of the Riemann zeta function
[PDF] 
I will survey the theory of the discrete moments of the Riemann zeta
function and I will indicate how they differ from the ordinary
continuous moments of the zeta function. In addition, I will
highlight several number theoretic applications that can be derived
from knowing asymptotics for the discrete moments. In particular, we
will show that there exist large and small values of z¢(r)
where r denotes a nontrivial zero of the Riemann zeta function.
 RACHEL PRIES, Colorado State University, Fort Collins, CO
Twists of representations of fundamental groups in positive
characteristic
[PDF] 
We study a twisting action on representations of fundamental groups of
affine varieties in positive characteristic. For affine curves, we
study how this twisting affects invariants of the representation.
This yields applications to the subject of Galois covers of curves in
positive characteristic.
 HADI SALMASIAN, Queen's University, Kingston, ON
Small degenerate principal series and exceptional dual pairs
[PDF] 
I will describe a general theorem which connects unitary
representations of reductive groups and Kirillov's orbital theory for
nilpotent groups. We show how this result applies to the study of
certain dual pairs in exceptional groups which basically generalize
the (noncompact) GL(1)GL(n) duality.

