James Arthur, University of Toronto:
"On the trace formula and the classification of representations."
The talk will be an introduction to the trace formula, as it applies to
the automorphic representation theory of classical matrix groups. There
has been a recent breakthrough in the form of a proof by Ngo of the
fundamental lemma, which builds on earlier work of a number of
mathematicians. This allows us to construct the long conjectured
correspondence of test functions on different groups. The resulting
comparison of trace formulas leads to many interesting questions, whose
resolution yields a classification of the representations of
classical groups in terms of those of general linear groups.
Cristina Ballantine, Holy Cross:
"Biregular expanders and the Ramanujan Conjecture."
The relationship between regular expander graphs and the Ramanujan Conjecture
is well understood and has lead to the definition and construction of
asymptotically optimal regular expanders called Ramanujan graphs. In this talk
we will show that biregular graphs obtained from the Bruhat-Tits building of a
group whose representations satisfy the Ramanujan conjecture are indeed
optimal biregular expander graphs.
Jonathan Bayless, Dartmouth College (grad):
"An analogue of Carmichael's conjecture."
Carmichael's conjecture that there is no unique preimage for the
Euler function is well-known and widely believed. We show that the analogue
for the function U, which maps an integer n to the class of Abelian groups
isomorphic to the unit group mod n, is false and give a lower bound on the
number of counterexamples up to x.
Jennifer Beineke, Western New England College:
"Modified Atkinson's Formulas for the Mean Square of the
Riemann Zeta Function."
In 1949, Atkinson determined an explicit formula for the error term
in the asymptotic equation of the second moment of the Riemann zeta
function. Jutila then developed a modified version of Atkinson's
formula for the mean square of the Riemann zeta function on the line
Re(s) = 1/2. We will describe an extension of Jutila's formula to
other values of s. This result requires an approximate functional
equation for a product of shifted zeta functions, and a smooth
version of the Oppenheim summation formula.
Robert Benedetto, Amherst College:
"Another n-point abc Conjecture."
We will interpret the abc conjecture as a statement about four
points in the projective line. For function fields of characteristic
zero, where the $abc$ conjecture is a theorem, we will consider
generalizations corresponding to n points in the projective line.
If time permits, we will also discuss analogous conjectures for number
fields.
Elliot Benjamin, University of Maine:
"The Infinite 2-Class Field Tower Conjecture And 2-Class Number Lower Bounds,
For Imaginary Quadratic Number Fields."
Let k be an imaginary quadratic number field such that the 2-class
group of k is elementary of rank 4, and let k_1 denote the Hilbert 2-class
field of k.
Utilizing the Kuroda class number formula and lower bounds on a unit index
of k_1, we show that if exactly one negative prime discriminant divides the
discriminant d_k of k and d_k is not congruent to 4 mod 8, then the
2-class number of k_1 is greater than or equal to 219.
An application of this
result is given to the infinite 2-class field tower conjecture for imaginary
quadratic number fields.
Ji Bian, University of Rochester (grad):
"The Pair Correlation of the Zeros of ξκ(s)."
In this talk, we apply Montgomery's method to study the pair
correlation of the zeros of ξκ(s), the κth
derivative of Riemann's xi-function.
We show that the zeros tend to even out when we take
high derivatives, a fact predicted by a general theorem of Farmer and
Rhoades. The rate at which this happens as a function of κ has
potential applications to the Class Number problem. We explore this and
also obtain new results on the size of small gaps between the zeros of
ξκ(s) and on the percentage of simple zeros of ξκ(s).
Michael Bush, Smith College:
"Orbits of mild groups."
Mild pro-p groups have many nice properties and often arise in number theory as
the Galois groups associated to maximal p-extensions with ramification
restricted to a finite set of primes S. In this talk, I will review some
earlier joint work (with J. Labute) where we classify such extensions when S
contains 4 primes (plus some other restrictions) and describe work in progress
to extend these results to sets containing 5 or more primes.
Gabriel Chenevert, McGill University (grad):
"On the modularity of geometric Galois representations."
When the Serre conjecture is applied to an odd, 2-dimensional Galois
representation occurring in the cohomology of an arithmetic variety,
a lack of understanding of the geometry of the variety at primes of bad
reduction yields an inefficient bound for the level of the predicted
modular form. An extension of the Faltings-Serre method, which applies
even when no modularity results are known, allows to circumvent this
problem. An explicit example related to cubic exponential sums will be
discussed.
John Cullinan, Bard College:
"Algebraic Properties of a Family of Jacobi Polynomials."
The Jacobi Polynomials are a two-parameter family of
orthogonal polynomials that have many number-theoretic applications.
For example, they contain as special cases the Chebyshev, Gegenbauer,
and Legendre polynomials, the irreducibility properties of which are
not fully known. More recently, Jacobi polynomials have proven to be
intimately connected with the arithmetic of elliptic curves.
We will show that there are infinitely many one-parameter subfamilies
of polynomials Pn(x,t)
with the property that if n≥8
then, with the exception of finitely many
t0∈Q,
the polynomial Pn(x,t0)
is irreducible over Q
with Galois group Sn.
Henri Darmon, McGill University:
"Stark units and Eisenstein series."
I will discuss a conjectural relationship between Stark units
and periods of Eisenstein series. This is a report on joint work with
Pierre Charollois.
Jean-Marie De Koninck, Laval University:
"On the set of Wieferich primes."
A prime number p is called a Wieferich prime if
2p-1≡1(mod p2).
Let W stand for the set of Wieferich primes and Wc for its
complement in the set of all primes. In 1988, Silverman showed that it follows
from the abc conjecture that
(*) |{p≤ x:p∈Wc}|>> log(x)
We show that this lower bound is a consequence of a weaker hypothesis. In fact,
letting λ(n):=log(n)/logγ(n)
(where γ(n):=Πp|np)
stand for the index of composition of n,
we prove that if the index of
composition of 2n-1 remains "small" as n increases, then (*)
holds and
that if it is not "too small" for infinitely many n's, then
|W|=+∞.
This is joint work with Nicolas Doyon.
Brooke Feigon, University of Toronto:
"Averages of central L-values of Hilbert modular forms"
I will describe an application of the relative trace formula to obtain
exact formulas for central values of certain twisted quadratic base change
L-functions averaged over Hilbert modular forms of a fixed weight and
level. This talk is based on joint work with David Whitehouse.
Sharon Frechette, Holy Cross:
"Weyl group Multiple Dirichlet Series"
Given a reduced root system Φ of rank r, we
may attach a multiple Dirichlet series in r complex variables,
whose group of functional equations is isomorphic to the Weyl
group of Φ. These multiple Dirichlet series contain
interesting arithmetic information. I will discuss joint work
with Jennifer Beineke and Ben Brubaker, on the multiple Dirichlet
series associated to the root system Φ = B2.
Alex Ghitza, Colby College:
"Lifting from positive characteristic"
In the study of automorphic forms in characteristic p, it is often
helpful to know when it is possible to lift the form to characteristic
zero. For example, it is known that all modular forms of weight at
least 2 can be lifted. I will discuss this problem in the context of
Siegel modular forms and explain how it relates to the vanishing of
certain cohomology groups.
Steve Gonek, University of Rochester:
"Finite Euler Products and the Riemann Hypothesis."
slides for the talk
We begin by discussing approximations of the Riemann zeta-function
by truncations of its Dirichlet series and then of
its Euler product. We then construct a parameterized family
of non-analytic approximations to the zeta-function.
Every function in the family has all but a
finite number of its zeros on the critical line.
We show that when the parameter is not too large,
the functions have roughly the same number of zeros as
the zeta-function, their zeros are all simple, and they repel.
In fact, if the Riemann Hypothesis is true, the zeros of these
functions converge to those of the zeta-function as the parameter increases,
and between zeros of the zeta-function the functions in the family
tend, in modulus, to twice that of the zeta-function.
For these and other reasons they may be regarded as models of the Riemann
zeta-function. The structure of the functions explains the simplicity and
repulsion of their zeros and suggests a mechanism that might be responsible
for the corresponding properties of the zeros of the zeta-function.
Eyal Goren, McGill University: "Ramanujan graphs
- some new constructions and applications."
Ramanujan graphs are an interesting family of graphs with
strong expansion property ("rumors spread very fast" along such graphs).
There is much interest in such graphs for various technological
applications and because their construction seems to rely on rather deep
results from number theory, arithmetic geometry and representation
theory. In this talk, I shall describe some applications and results
concerning particular Ramanujan graphs (related to arithmetic geometry)
and some outstanding questions. I shall also describe some interesting
numerical results concerning bi-regular Ramanujan graphs. This joint
work with Kristin Lauter and Denis Charles from Microsoft Research, and
with Rosalie Belanger-Rioux and Ioan Filip from McGill University.
Paul Gunnells, University of Massachusetts:
"Weyl Group Multiple Dirichlet Series"
Multiple Dirichlet series are generalizations of L-functions
involving several complex variables. While the functional equation
of a usual L-series is an involution s → 1-s, a multiple
Dirichlet series satisfies a group of functional equations that
intermixes all the variables.
In this talk we describe a construction of such series attached to
Dynkin diagrams, where the resulting group of functional equations is
the associated Weyl group. These series are expected to be
Fourier-Whittaker coefficients of metaplectic Eisenstein series.
This is joint work with Gautam Chinta.
Benjamin Howard, Boston College:
"Generalized Gross-Zagier theorems and applications to Hida theory."
Modular forms can be p-adically interpolated in Hida
families, and a conjecture of Greenberg predicts that all but
finitely many forms in a Hida family have the same analytic rank:
either 0 or 1 depending on the sign in the functional equation of the
p-adic L-function of the family. I will discuss a method, based on
Heegner points and generalized Gross-Zagier theorems, of verifying
Greenberg's conjecture for any given Hida family.
Rafe Jones, University of Wisconsin:
"Arboreal Galois Representations."
Given a rational function R(z) of degree d with integer
coefficients, denote by R^n the n-fold composition of R with itself. The
Galois group obtained by adjoining to Q the solutions to R^n(z) = 0 for
all n naturally injects into Aut(T), the automorphism group of the
complete d-ary rooted tree. This gives a so-called arboreal Galois
representation of the absolute Galois group of Q. One may pose the same
questions about this representation as in the classical case of l-adic
Galois representations, in particular on the size of the image and on
properties of Frobenius conjugacy classes. In this talk we discuss recent
work on some of the analogous questions in the arboreal case.
Hershy Kisilevsky, Concordia University:
"Cubic points on elliptic curves and Rational points on K3 Surfaces".
Rational points on certain K3-surfaces control families of cubic ponts on
rational elliptic curves. We discuss several methods of finding them.
Peter Kleban, University of Maine Physics:
"Intervals between Farey fractions in the limit of infinite level."
The modified Farey sequence consists, at each level k, of 2k+1
rational fractions. We consider Ik(e),
the total length of (one set of) alternate intervals between
Farey fractions that are new (i.e. appear for the first time) at level k.
We first prove, using elementary methods, that
liminf(k→∞)Ik(e)=0
and conjecture that in fact
limk→∞Ik(e)=0.
This simple geometrical property of the Farey fractions turns out to be surprisingly subtle,
with no apparent simple interpretation. The conjecture is equivalent to
limk→∞Sk=0,
where Sk is the sum over the inverse squares of the new denominators
at level k.
Our result makes use of bounds for Farey fraction intervals in terms of their "parent"
intervals at lower levels.
We then argue, by use of the continued fraction representation, that the
conjecture may be proven by use of the dynamical properties of the Farey map.
Manfred Kolster, McMaster University:
"Remarks on reflection principles for number fields."
In the talk we will consider cohomological analogues of the
classical reflection principles (Scholz, Leopoldt, Gras), which follow
easily from Poitou-Tate duality.
John Labute, McGill University:
"Tame Galois Groups."
Let p be a rational prime. Let K be a number field, S a
finite set of primes of K not dividing p and let KS(p) be
the maximal Galois p-extension of K unramified outside S.
Until recently, not much was known about the structure of the
Galois group GS(p)=Gal(KS(p)/K). We now know that this
Galois group is quite often a duality group of cohomological
dimension 2 and strict cohomological dimension 3.
Claude Levesque, Laval University:
"Some fundamental systems of units."
We will exhibit a fundamental system of units
for a certain family of parametrized quartic fields.
Micah Milinovich, University of Rochester (grad):
"Upper bounds for Discrete Moments of the
Riemann Zeta-Function."
Assuming the Riemann Hypothesis, we
establish upper bounds for discrete moments of the
Riemann zeta-function and its derivative at or near
the zeros of ζ(s) that are close to the
conjectured order of magnitude. These results follow
from a general value distribution lemma that provides
upper bounds the frequency of large values of
ζ(s) near its zeros. Our proof is based upon a
recent method of Soundararajan that provides analogous
bounds for continuous moments of the Riemann
zeta-function as well as moments of other families of
L-functions at the central point.
Carlos Moreno, Baruch College, CUNY:
"Galois rings, Exponential sums, and Hensel's Lemma."
The passage from finite fields to p-adic fields
is often accomplished through a series of approximations which
are encoded in the standard steps in Hensel's Lemma or in the
construction of truncated Witt vectors. We analyse a number of instances
from error correcting codes, exponential sums (mostly Kloosterman sums),
and harmonic analysis of how the intermediary steps in Hensel's lemma
transform the problems under consideration into linear ones.
Ram Murty, Queen's University: "Is Euler's constant
transcendental?"
We will discuss special values of the digamma function.
As a consequence, we are able to say something about
the transcendence of generalized Euler constants,
one of which is Euler's constant.
Allison Pacelli, Williams College:
"The 3-Rank of the Class Group in Number Fields and Function Fields."
Given integers m and n, it is often a difficult problem to
find number fields or function fields of fixed degree m with high
n-rank. Some general lower bounds on the rank are known, but it is
possible to do better in special cases. We consider 3-rank in quadratic
number fields and function fields. We also look at the related question
of fields with 3-rank zero, that is, fields with class number
indivisible by 3. Much less is known in this case, but we give explicit
infinite families of number fields and function fields with 3-rank zero.
This is joint work with Michael Rosen.
Thomas Pietraho, Bowdoin College:
"Cells in unequal parameter Hecke algebras."
Kazhdan-Lusztig cells play a crucial role in the
representation theory of reductive Lie groups. We
will discuss their analogues for unequal parameter
Hecke algebras, the related combinatorics, and applications
in representation theory.
Paul Pollack, Dartmouth College (grad):
"Arithmetic properties of polynomial specializations over finite fields."
Several classical problems in number theory could be resolved if we
understood the frequency with which polynomials simultaneously assume prime
values. Schinzel's Hypothesis H and its quantitative refinements by
Hardy-Littlewood and Bateman-Horn present us with a plausible heuristic
understanding of this phenomenon, but their conjectures have proved difficult
to attack. We discuss some recent results towards an analogue of Schinzel's
Hypothesis H in the setting of polynomials over finite fields, together with
some applications of these results. Fundamental to this work is a link between
the cycle types of random permutations and the factorization types of certain
polynomial specializations. This last result implies that for certain ranges of
the parameters, one can understand not only simultaneous prime values, but many
other statistics (e.g., smoothness) of simultaneous polynomial specializations.
Carl Pomerance, Dartmouth College:
"Pseudopowers."
An x-pseudopower to base g is a positive integer which is not a power
of g yet is so modulo p for all primes p≤x.
We improve an upper bound for the least such number due to
E.~Bach, R.~Lukes, J.~Shallit, and H.~C.~Williams.
The method is based on a combination of
some bounds of exponential sums with new results
about the average behavior of the multiplicative order
of g modulo prime numbers. This paper is joint work with
Sergei Konyagin and Igor Shparlinski.
Jonathan Sands, University of Vermont:
"Dedekind Zeta functions at s=-1 and the Fitting ideal of the
tame kernel in a relative quadratic extension."
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
between 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.
John Voight, University of Vermont:
"Fundamental domains for finitely generated Fuchsian groups."
We exhibit an algorithm which efficiently computes a Dirichlet domain
for a finitely generated Fuchsian group G. As a consequence, we are
able to compute the invariants of G, including an explicit finite
presentation for G with a solution to the word problem, and
efficiently evaluate automorphic forms arising from Shimura curves.
Siman Wong, University of Massachusetts:
"Class groups of quadratic function fields."
We will review known results on quadratic global fields
with large class rank, and we will describe some new
construction of quadratic function fields with large
class rank using elliptic curves.
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List of Participants
James Arthur, University of Toronto
Cristina Ballantine, Holy Cross
Jonathan Bayless, Dartmouth College (grad)
L. Beaudet, Laval University (grad)
Jennifer Beineke, Western New England College
Robert Benedetto, Amherst College
Elliot Benjamin, University of Maine
Ji Bian, University of Rochester (grad)
D. Bonneau, Laval University (grad)
David Bradley, University of Maine
Henrik Bresinsky, University of Maine
Michael Bush, Smith College
Bryden Cais, McGill University
Gabriel Chenevert, McGill University (grad)
John Cullinan, Bard College
Henri Darmon, McGill University
David Dummit, University of Vermont
Jean-Marie De Koninck, Laval University
Brooke Feigon, University of Toronto
Cameron Franc, McGill University (grad)
Sharon Frechette, Holy Cross
Alex Ghitza, Colby College
Steve Gonek, University of Rochester
Eyal Goren, McGill University
Hester Graves, University of Michigan (grad)
Paul Gunnells, University of Massachusetts
Malcolm Harper, Champlain Regional College
Benjamin Howard, Boston College
Nathan Jones, CRM
Rafe Jones, University of Wisconsin
Hershy Kisilevsky, Concordia University
Peter Kleban, University of Maine Physics
Andrew Knightly, University of Maine
Manfred Kolster, McMaster University
P. Letendre, Laval University (grad)
John Labute, McGill University
Claude Levesque, Laval University
Marc Masdeu, McGill University (grad)
Riad Masri, CRM
Micah Milinovich, University of Rochester (grad)
Carlos Moreno, Baruch College, CUNY
Ali Ozluk, University of Maine
Ram Murty, Queen's University
Allison Pacelli, Williams College
Thomas Pietraho, Bowdoin College
Paul Pollack, Dartmouth College (grad)
Andrew Pollington, NSF
Carl Pomerance, Dartmouth College
E. Pronovost, Laval University (grad)
Jonathan Sands, University of Vermont
Shahab Shahabi, McGill University (grad)
Chip Snyder, University of Maine
J. Soucy, Laval University (grad)
Enrique Trevino, Dartmouth College (grad)
John Voight, University of Vermont
Siman Wong, University of Massachusetts
Yu Zhao, McGill University (grad)
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