# Maine/Québec Number Theory Conference

University of Maine, Orono, 2007
Andrew Knightly and Chip Snyder, Organizers

Full List of Participants

## Saturday, September 29, 2007

Click a name to view the abstract for the talk
 Time Speaker Title Room 7:45-8:15am Refreshments, Neville Hall (NV) Lobby 8:15-8:30am Welcoming Remarks: Provost Edna Szymanski 100NV 8:30-9:30 James Arthur University of Toronto On the trace formula and the classification of representations   Arthur's papers 100 NV 9:45-10:45 Steve Gonek University of Rochester Finite Euler Products and the Riemann Hypothesis   slides for the talk 100 NV 11:00-11:30 Ram Murty Queen's University Is Euler's constant transcendental? 100 NV 11:40-12:10 Henri Darmon McGill University Stark units and Eisenstein series 100 NV 12:20-12:50 John Labute McGill University Tame Galois Groups 100 NV 1:00-2:20 Lunch, Market Square 2:30-3:00 Manfred Kolster McMaster University Remarks on reflection principles for number fields 100 NV Carl Pomerance Dartmouth College Pseudopowers 101 NV 3:10-3:30 Jean-Marie De Koninck Laval University On the set of Wieferich primes 100 NV Sharon Frechette College of the Holy Cross Weyl group Multiple Dirichlet Series 101 NV 3:40-4:00 Michael Bush Smith College Orbits of mild groups 100 NV Cristina Ballantine College of the Holy Cross Biregular expanders and the Ramanujan Conjecture 101 NV 4:10-4:40 Carlos Moreno Baruch College (CUNY) Galois rings, Exponential sums, and Hensel's Lemma. 100 NV 4:50-5:10 Allison Pacelli Williams College The 3-Rank of the Class Group in Number Fields and Function Fields 100 NV Thomas Pietraho Bowdoin College Cells in unequal parameter Hecke algebras 101 NV 5:20-5:50 Robert Benedetto Amherst College Another n-point abc Conjecture 100 NV Eyal Goren McGill University Ramanujan graphs - some new constructions and applications 101 NV 6:00-6:20 Elliot Benjamin University of Maine The Infinite 2-Class Field Tower Conjecture and 2-Class Number Lower Bounds, for Imaginary Quadratic Number Fields 100 NV Brooke Feigon University of Toronto Averages of central L-values of Hilbert modular forms 101 NV 7:00-- BANQUET- New Moon Café

## Sunday, September 30, 2007

 Time Speaker Title Room 8:00-8:30am Refreshments 8:30-9:00 Hershy Kisilevsky Concordia University Cubic points on elliptic curves and Rational points on K3 Surfaces 100 NV 9:10-9:30 Claude Levesque Laval University Some fundamental systems of units 100 NV Alex Ghitza Colby College Lifting from positive characteristic 101 NV 9:40-10:10 Paul Gunnells University of Massachusetts Weyl Group Multiple Dirichlet Series 100 NV Benjamin Howard Boston College Generalized Gross-Zagier theorems and applications to Hida theory 101 NV 10:20-10:40 Jennifer Beineke Western New England College Modified Atkinson's Formulas for the Mean Square of the Riemann Zeta Function 100 NV John Cullinan Bard College Algebraic Properties of a Family of Jacobi Polynomials 101 NV 10:50-11:20 Jonathan Sands University of Vermont Dedekind Zeta functions at s=-1 and the Fitting ideal of the tame kernel in a relative quadratic extension 100 NV Siman Wong University of Massachusetts Class groups of quadratic function fields 101 NV 11:30-11:50 Rafe Jones University of Wisconsin Arboreal Galois Representations 100 NV John Voight University of Vermont Fundamental domains for finitely generated Fuchsian groups 101 NV 12:00-12:20 Micah Milinovich (grad) University of Rochester Upper bounds for Discrete Moments of the Riemann Zeta-Function 100 NV Jonathan Bayless (grad) Dartmouth College An analogue of Carmichael's conjecture 101 NV 12:25-12:45 Ji Bian (grad) University of Rochester The Pair Correlation of the Zeros of ξκ(s). 100 NV Paul Pollack (grad) Dartmouth College Arithmetic properties of polynomial specializations over finite fields 101 NV 12:50-1:10 Gabriel Chenevert (grad) McGill University On the modularity of geometric Galois representations 100 NV Peter Kleban University of Maine Physics Intervals between Farey fractions in the limit of infinite level 101 NV 1:15pm Concluding Remarks 100 NV

## Abstracts (by last name)

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 t0Q, 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.

## List of Participants

James Arthur, University of Toronto
Cristina Ballantine, Holy Cross
Jennifer Beineke, Western New England College
Robert Benedetto, Amherst College
Elliot Benjamin, University of Maine
Ji Bian, University of Rochester (grad)
Henrik Bresinsky, University of Maine
Michael Bush, Smith College
Bryden Cais, McGill University
John Cullinan, Bard College
Henri Darmon, McGill University
David Dummit, University of Vermont
Jean-Marie De Koninck, Laval University
Brooke Feigon, University of Toronto
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
John Labute, McGill University
Claude Levesque, Laval University
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