Reinier Bröker, Brown University:
"Computing modular polynomials"
The classical modular polynomial Φn parametrizes elliptic curves together
with a cyclic isogeny of degree n. These polynomials are important in
many algorithms using elliptic curves, but their incredibly large size makes
it very hard to compute them. In the 1980's, computing Φ11 was considered
a major computational effort, and at the end of the 1990's the world record
was n = 359.
In this talk, I will present a new algorithm to compute Φn that has an
almost optimal running time. The algorithm is based on special properties of
certain non-maximal orders in imaginary quadratic fields. The algorithm
easily handles large values of n, and our new record is n = 5003.
Alina Bucur, IAS:
"Point counting on p-fold covers of P1(Fq)"
The number of points for such a curve is dictated by the
trace of the Frobenius endomorphism. For curves of fixed genus g and
the number of points in the base field going to infinity, the
Katz-Sarnak philosophy predicts that the trace of Frobenius is
distributed like the trace of a certain ensemble of unitary matrices,
dictated by the monodromy group of the family of curves. We will talk
about the other side of the picture: what happens when the base field
is fixed, but the genus grows?
Michael Bush, Smith College:
"Galois groups of unramified 3-extensions of imaginary quadratic fields"
The structure of the Galois group of the maximal unramified
p-extension of an imaginary quadratic field is restricted in various ways. I
will report on work in progress to catalogue all such finite groups of small
order when p = 3 and some infinite families that have been discovered.
Bryden Cais, McGill University:
"Hida families for GL2 and p-adic Hodge theory"
In the 1980's, Hida constructed p-adic analytic families of
ordinary Galois representations via a detailed study of Hecke algebras and
group cohomology. Shortly after this, Mazur and Wiles gave a geometric
interpretation of the associated families of Galois representations by
realizing them in the etale cohomology groups of towers of modular curves.
In accordance with the philosophy of p-adic Hodge theory, one expects
that there should be a corresponding geometric construction of p-adic
families of ordinary modular forms via de Rham cohomology. In this
talk, we will explain such a construction; as a consequence, we obtain
a new and purely geometric approach to Hida theory. Using recent
progress in integral p-adic Hodge theory, we will elucidate how our
construction can be used to recover that of Mazur-Wiles.
Hugo Chapdelaine, Université Laval:
"Computation of the Galois groups of a certain family of polynomials
via group permutation theory."
David Cox, Amherst College:
"Galois theory according to Galois"
I will discuss how Lagrange and Galois thought about Galois
theory. Then I will explore Galois's astonishing insight into the structure
of solvable groups.
John Cullinan, Bard College:
"Ramification in iterated towers for rational functions."
Let φ(x) be a rational function of degree d>1 defined
over a number field K and let Φn(x,t) = φ(n)(x)-t
∈ K(x,t) where φ(n)(x) is the nth
iterate of φ(x). We give a formula for the discriminant
Dn,φ(t) of the numerator
of Φn(x,t) and show that, if φ(x) is postcritically
finite, for each specialization t0 of t to K, there exists a finite
set St0 of primes of K such that for all n, the primes
dividing Dn,φ(t0) are contained in St0.
Henri Darmon, McGill University:
"Non-vanishing for quadratic twists of L-series and a formula of Rubin"
Jean-Marie De Koninck, Université Laval:
"The distance between smooth numbers"
Given an integer B≥ 2, an integer n≥ 2 is said to be
B-smooth if all its prime factors are ≤B. Let P(n) stand
for the largest prime factor of n≥2, with P(1)=1. For each
integer n≥2, let δ(n) be the distance to the nearest
P(n)-smooth number, that is to the nearest integer whose largest
prime factor is no larger than that of n, with δ(1)=1. We
study the properties of the function δ(n) and in particular
the behavior of the sums Σn≤x δ(n)
and Σn≤x 1/δ(n). Generalizations are considered.
Eyal Goren, McGill University:
"Denominators of Class Invariants"
In a recent work with K. Lauter (Microsoft Research) we
obtained a bound on the height of the class invariants constructed by
de Shalit and myself, and on the height of Igusa class polynomials.
This has applications to explicit class field theory and for the
computational complexity of constructing curves of genus two with a
given number of points (such as one wants for cryptographic
applications). I will state the result and explain the key features of
the proof.
Dick Gross, Harvard University:
"Self-dual and conjugate-dual Galois representations"
I will define self-dual and conjugate-dual representations of local Weil groups,
and discuss what we know about their epsilon factors.
I will also describe the relationship
between these representations and Langlands parameters for the classical groups.
Rafe Jones, College of the Holy Cross:
"Galois groups of rational functions with non-trivial automorphisms"
The preimages of a rational point under iteration of a rational function f(x)
form a tree in a natural way. The absolute Galois group of Q acts on this
tree, in much the same way as it acts on the l-adic Tate module of an
elliptic curve defined over Q. I will discuss the nature of this action in
the case that f is a quadratic rational function that commutes with a
non-trivial Mobius transformation. This is the analogue of the case of an
elliptic curve with complex multiplication. In particular, I'll give
conditions that show the image of the action is as large as possible. This is
joint work with Michelle Manes.
Benjamin Hutz, Amherst College:
Uniform Boundedness in Arithmetic Dynamics
We consider the iterates of a morphism of projective space. Morton and
Silverman conjectured that the number of rational preperiodic points
(points with finite orbit) is bounded depending only on the degree of
the map, the dimension of the space, and the degree of the number field.
The bound in the case of degree two polynomial maps on the projective
line over the rational numbers has been extensively considered. Poonen
has conjectured that the bound is nine. In this talk we provide strong
computation evidence for Poonen's conjecture and examine the case of
quadratic extensions. We also discuss how the bound may grow as we
allow the dimension of the projective space to increase.
Cap Khoury, University of Michigan:
"Spherical functions and quasi-split unitary groups"
Given a reductive group G with maximal compact subgroup
K, we say that a complex-valued function is spherical if it
is an eigenfunction for the natural action of the Hecke algebra
H(G,K). Various classes of spherical function have been much
studied in the number theory and representation theory literature,
including Whittaker functions and the more general Whittaker-Shintani
functions. In this talk, I will comment briefly on the relevance of
these spherical functions for the Langlands Program and mention some
results about spherical functions for quasi-split p-adic unitary
groups.
Manfred Kolster, McMaster University:
"The Coates-Sinnott Conjecture"
As an analog of the classical Stickelberger Theorem Coates and Sinnott
formulated in 1974 a conjecture about annihilation of higher Quillen
K-groups by certain Stickelberger elements lying in ZG, where G is the
Galois group of an abelian extension of the rationals, and proved it for
K2. We will survey the results known for more general relative abelian
extensions of number fields with some focus on the 2-primary part of this
conjecture.
Stephen Lichtenbaum, Brown University:
"Special values of Zeta-functions at s = 1"
If V is a smooth projective variety over a finite field, then the
leading term of the zeta-function of V at s = 1 is given by the Euler
characteristic of the additive group on V divided by the Euler characteristic
of the multiplicative group on V, assuming classical conjectures. We willl
discuss this formula and its analogue in the arithmetic case.
Benjamin Linowitz, Dartmouth College:
"Selectivity in Quaternion Algebras"
Let K be a number field, Ω be an order in a quadratic extension of K and
A be a quaternion
algebra defined over K which is not totally definite. Chinburg and Friedman
determined the maximal
orders of A admitting an embedding of Ω. Moreover, they showed that the
proportion of isomorphism
classes of maximal orders admitting such an embedding is either 0, 1/2 or 1.
We discuss a generalization
of these results to non-maximal orders of A.
Steven Miller, Williams College:
"Tests of the L-Functions Ratios Conjecture"
Recently Conrey, Farmer and Zirnbauer conjectured formulas for the
averages over a family of ratios of products of shifted L-functions. Their
L-functions Ratios Conjecture predicts both the main and lower order terms
for many problems, ranging from n-level correlations and densities to
mollifiers and moments to vanishing at the central point. There are now many
results showing agreement between the main terms of number theory and random
matrix theory; however, there are very few families where the lower order
terms are known. These terms often depend on subtle arithmetic properties of
the family, and provide a way to break the universality of behavior. The
L-functions Ratios Conjecture provides a powerful and tractable way to
predict these terms. We test a specific case here, that of the 1-level
density for the symplectic family of quadratic Dirichlet characters arising
from even fundamental discriminants d<X. For test functions supported in
(-1/3, 1/3) we calculate all the lower order terms up to size O(|X|-1/2 +
ε) and observe perfect agreement with the conjecture. Thus for this
family and suitably restricted test functions, we completely verify the
Ratios Conjecture's prediction for the 1-level density. If time permits we
will discuss other families.
Kumar Murty, University of Toronto:
"The Euler-Kronecker constant of a number field"
Ihara has introduced an invariant γK of a number field
K which seems to control the
existence of primes of small norm. In this talk, we will discuss various
properties of this invariant and its
relation to zeros of L-functions.
Patrick Rault, SUNY Geneseo:
"On uniform bounds for rational points on rational curves and thin sets."
We use rational parametrizations to make progress on an open question
about counting rational points on plane curves. Heath-Brown proved
that for any ε>0 the number of rational points of height at
most B on a degree d plane curve is
Oε,d(B2/d+ε) (the implied constant depends on
ε and d). It is known that Heath-Brown's theorem is sharp
apart from the ε, but in certain cases the bound has been
improved to ε=0. The open question is whether or not the
bound with ε=0 holds in general. We shed additional light on
this open problem by giving, in certain cases, an improved upper bound
which is inversely proportional to a positive power of the resultant
of the curve.
Adriana Salerno, Bates College:
"Rational points and hypergeometric functions"
Hypergeometric functions seem to be ubiquitous in mathematics,
particularly when counting rational points over finite fields. In this
talk, we will show that the number of points over a finite field Fq on
a certain family of varieties is a linear combination of hypergeometric
functions. We use results by Koblitz and Gross to find explicit
relationships. These formulas could be useful for computing Zeta functions
in the future and in extending some classical results by Bernard Dwork.
Jonathan Sands, University of Vermont:
"Embedding Biquadratic Extensions in D8 Extensions and the
Tame Kernel as a Galois Module"
We show that the question of whether a totally real biquadratic extension E of a
number field F can be extended to a D8-extension provides a criterion for how the Galois
group acts on K2(E). A positive answer allows us to prove a close connection between the
tame kernel (K2 of the ring of integers of E) and a Stickelberger ideal defined
analytically.
Mark Sheingorn, CUNY:
"Closed Horocyclic Orbits on
Γ(1)\H
"
We study certain closed horocyclic orbits on
Γ(1)\H
defined using the pencil of horocycles whose
common point (in the words of the Nielsen-Fenchel manuscript) is
∞. The goal of this work is to obtain explicit answers to
questions of this sort: If we are exactly here, now, going in
this direction when precisely shall we be precisely there,
going in that direction?.
Caleb Shor, Western New England College:
"Codes over certain rings with square cardinality, lattices, and theta
functions"
Let b>0 be a square-free integer congruent to 3 mod 4 and
OK the ring of integers of the imaginary quadratic field
K=Q(√-b). Let p be a prime. If p does not divide b, then
the ring R:=OK/pOK is isomorphic to
Fp2 or Fp x Fp.
Let C be a code
over R. Given such a code, one can create a lattice
Λb(C) over K. One can then construct the corresponding theta
function of such a lattice.
In 2005, working with p=2, K. S. Chua found an example of two
non-equivalent codes that have the same theta function for b=7 and
different theta functions for larger values of b. In this talk,
motivated by Chua's example, we will consider the situation for general
primes p. In particular, we will see how to represent these theta
functions in terms of some basic theta series, see connections between these
theta functions and weight enumerator polynomials, and see recent results
related to the question of whether two non-equivalent codes can have the same
theta function for some or all values of b.
Lauren Thompson, Dartmouth College:
"Heights of Divisors of xn -1"
The height of a polynomial with integer coefficients is the largest
coefficient in absolute value. Many papers have been written on the
subject of bounding heights of cyclotomic polynomials. One result, due to
H. Maier, gives a best possible upper bound of nψ(n) for almost
all n, where ψ(n) is any function that approaches infinity as
n → ∞. We will discuss the related problem of bounding
the maximal height over all polynomial divisors of xn - 1 and give an
analogue of Maier's result in this scenario.
Enrique Trevino, Dartmouth College:
"Explicit Bounds for the Burgess Bound for Character Sums"
Burgess wrote a series of papers giving inexplicit bounds for short character sums where
the Polya-Vinogradov inequality is trivial. The Burgess bound is useful to compute
L(1,χ) and give bounds for the least quadratic nonresidue. Iwaniec and Kowalski provide
explicit bounds. Booker gives better bounds for quadratic characters in restricted ranges.
I widen the range in Booker's results without cost to his constants and I also get explicit
bounds for any non-quadratic character.
John Voight, University of Vermont:
"Tables of modular elliptic curves over totally real fields: a progress report"
We give a progress report on the use of algorithms to compute spaces
of Hilbert modular forms to enumerate modular elliptic curves by
conductor over certain totally real fields.
Gary Walsh, University of Ottawa:
"On a Diophantine Problem of Michael Bennett"
Bennett posed the problem of solving the Diophantine
equation
(x2-1)/(y2-1)=(z2-1)2.
We will present a solution
to this and variants of this equation, along with a seemingly difficult
related unsolved problem that arises from considering such variants.
David Whitehouse, MIT:
"Periods and central values of quadratic base change L-functions"
An important result of Waldspurger relates the central value of quadratic base
change L-functions for GL(2) to period integrals over tori. Subsequently this result was
reproved by Jacquet using the relative trace formula. We will explain some progress on
extending Waldspurger's result to higher rank via a generalization of Jacquet's approach.
Siman Wong, University of Massachusetts:
"Class number indivisibility of quadratic function fields"
We will review facts about class numbers of quadratic
function fields, and we will discuss new results by way of
quadratic forms over function fields.
Andrew Yang, Dartmouth College:
"On the low-lying zeros of Dedekind zeta functions associated to cubic number fields."
The Katz-Sarnak philosophy asserts that to any "naturally defined family" of
L-functions, there should be an associated symmetry group which determines the distribution
of the low-lying zeros of those L-functions. We consider the family of Dedekind zeta
functions of cubic number fields, and we predict that the associated symmetry group is
symplectic. To analyze the low-lying zeros of this family, we start by using (as is
standard in these types of problems) a variant of the explicit formula used by Riemann to
study the Riemann zeta function. This reduces the problem to understanding the
distribution of how rational primes split in cubic fields of absolute discriminant X, as X
tends to infinity. This can be analyzed by using the work of H. Davenport and H. Heilbronn
on the asymptotics of the number of cubic fields as the absolute discriminant tends to
infinity. The final ingredient is a recent result of K. Belabas, M. Bhargava, and C.
Pomerance on power-saving error terms in the count of cubic fields considered by Davenport
and Heilbronn.
|
List of Participants
Joe Arsenault, University of Maine (grad)
Adam Barker-Hoyt, University of Maine (grad)
Jonathan Bayless, Husson University
David Bradley, University of Maine
Henrik Bresinsky, University of Maine
Reinier Bröker, Brown University
Tim Brown, University of Maine (undergrad)
Alina Bucur, IAS
Michael Bush, Smith College
Bryden Cais, McGill University
Hugo Chapdelaine, Université Laval
David Cox, Amherst College
John Cullinan, Bard College
Hédi Daboussi, Université Paris-Sud, Orsay
Henri Darmon, McGill University
Jean-Marie, DeKoninck, Université Laval
Luca Goldoni, University of Trento, Italy (grad)
Eyal Goren, McGill University
Fernando, Gouvêa, Colby College
Avram Gottschlich, Dartmouth College (grad)
Dick Gross, Harvard University
Pushpa Gupta, University of Maine
Ramesh Gupta, University of Maine
Benjamin Hutz, Amherst College
John Jackson, University of Maine (grad)
Rafe Jones, College of the Holy Cross
Michael "Cap" Khoury, University of Michigan
Andrew Knightly, University of Maine
Manfred Kolster, McMaster University
Philippe Lemieux, Université Laval (grad)
Patrick Letendre, Université Laval (grad)
Claude Levesque, Université Laval
Stephen Lichtenbaum, Brown University
Benjamin Linowitz, Dartmouth College (grad)
John MacCormack, University of Maine (grad)
Isaac Michaud, University of Maine (grad)
Steven Miller, Williams College
Mostafa Mache, Université Laval (grad)
Kumar Murty, University of Toronto
Virginia Nees-Hatlan, University of Maine
Ali Özlük, University of Maine
Andrew Pollington, NSF
Eric Pronovost, Université Laval (grad)
Patrick Rault, SUNY Geneseo
Mathew Rogers, University of Illinois
Adriana Salerno, Bates College
Jonathan Sands, University of Vermont
Mark Sheingorn, CUNY
Caleb Shor, Western New England College
Lloyd Simons, St. Michaels College
Chip Snyder, University of Maine
Lola Thompson, Dartmouth College (grad)
Enrique Trevino, Dartmouth College (grad)
John Voight, University of Vermont
Gary Walsh, University of Ottawa
Jonathan Webster, Bates College
David Whitehouse, MIT
Christian Wilson, University of Maine (grad)
Michael Wijaya, Dartmouth College (grad)
Siman Wong, University of Massachusetts
Andrew Yang, Dartmouth College
|
