2009 Maine/Québec Number Theory Conference

University of Maine, Orono
October 3-4, 2009
Andrew Knightly and Chip Snyder, Organizers

Full List of Participants

Saturday, October 3, 2009

Click a name to view the abstract for the talk
Time Speaker Title Room
8:00-8:30am Light Breakfast, Neville Hall (NV) Lobby
8:35-8:45am Welcoming Remarks
David Bradley, Chair of Mathematics and Statistics
Virginia Nees-Hatlan, Associate Dean
8:50-9:40 Benedict Gross
Harvard University
Self-dual and conjugate-dual Galois representations (paper) 100 NV
10:00-10:50 Kumar Murty
University of Toronto
The Euler-Kronecker constant of a number field 100 NV
11:10-11:30 Eyal Goren
McGill University
Denominators of Class Invariants 100 NV
Gary Walsh
University of Ottawa
On a Diophantine Problem of Michael Bennett 101 NV
11:40-12:00 Mark Sheingorn
Closed Horocyclic Orbits on Γ(1)\H 100 NV
Hugo Chapdelaine
Université Laval
Computation of the Galois groups of a certain family of polynomials via group permutation theory. 101 NV
12:15-1:45 Lunch, Memorial Union (on campus)
2:00-2:20 Stephen Lichtenbaum
Brown University
Special values of Zeta-functions at s = 1 100 NV
Jean-Marie De Koninck
Université Laval
The distance between smooth numbers 101 NV
2:30-2:50 Alina Bucur
Point counting on p-fold covers of P1(Fq) 100 NV
Rafe Jones
College of the Holy Cross
Galois groups of rational functions with non-trivial automorphisms (slides) 101 NV
3:00-3:20 Steven J. Miller
Williams College
Tests of the L-Functions Ratios Conjecture (slides, paper1, paper2 ) 100 NV
Siman Wong
University of Massachusetts
Class number indivisibility of quadratic function fields 101 NV
3:30-3:50 Andrew Yang
Dartmouth College
On the low-lying zeros of Dedekind zeta functions associated to cubic number fields. 100 NV
Caleb Shor
Western New England College
Codes over certain rings with square cardinality, lattices, and theta functions 101 NV
4:00-4:20 David Whitehouse
Periods and central values of quadratic base change L-functions 100 NV
Michael Bush
Smith College
Galois groups of unramified 3-extensions of imaginary quadratic fields (slides) 101 NV
4:30-5:20 David Cox
Amherst College
Galois theory according to Galois 100 NV
6:30 Dinner- Pat's Pizza, Orono

Sunday, October 4, 2009

Time Speaker Title Room
8:00-8:30am Light Breakfast, Neville Lobby
8:30-8:50 Adriana Salerno
Bates College
Rational points and hypergeometric functions (slides) 100 NV
Patrick Rault
SUNY Geneseo
On uniform bounds for rational points on rational curves and thin sets. 101 NV
9:00-9:20 Manfred Kolster
McMaster University
The Coates-Sinnott Conjecture
(lecture notes)
100 NV
Cap Khoury
University of Michigan
Spherical functions and quasi-split unitary groups 101 NV
9:30-9:50 Jonathan Sands
University of Vermont
Embedding Biquadratic Extensions in D8 Extensions and the Tame Kernel as a Galois Module (preprint) 100 NV
John Cullinan
Bard College
Ramification in iterated towers for rational functions. (slides) 101 NV
10:00-10:20 John Voight
University of Vermont
Tables of modular elliptic curves over totally real fields: a progress report 100 NV
Benjamin Linowitz
Dartmouth College
Selectivity in Quaternion Algebras (slides) 101 NV
10:30-10:50 Reinier Bröker
Brown University
Computing modular polynomials
100 NV
Lauren Thompson
Dartmouth College
Heights of Divisors of xn -1 (slides) 101 NV
11:00-11:20 Bryden Cais
McGill University
Hida families for GL2 and p-adic Hodge theory 100 NV
Enrique Trevino
Dartmouth College
Explicit Bounds for the Burgess Bound for Character Sums (slides) 101 NV
11:30-11:50 Henri Darmon
McGill University
Non-vanishing for quadratic twists of L-series and a formula of Rubin 100 NV
Benjamin Hutz
Amherst College
Uniform Boundedness in Arithmetic Dynamics (slides) 101 NV
12:00pm Concluding Remarks 100 NV

Abstracts (by last name)

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
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