# 2015 Maine/Québec Number Theory Conference

The University of Maine, Orono
October 3-4, 2015

Financial Support
The National Science Foundation
The University of Maine Office of the Vice President for Research
The University of Maine College of Liberal Arts and Sciences
University of Maine Department of Mathematics and Statistics

Full List of Participants

## Saturday, October 3, 2015

Click a name to view the abstract for the talk
 Time Room Speaker Title 8:00-8:20am Coffee/Tea, Neville Hall (NV) Lobby 8:20-8:25 am 101 NV Welcoming Remarks Nigel Pitt, Chair of Mathematics and Statistics Emily Haddad, Dean of the College of Liberal Arts and Sciences 8:25-9:10 101 NV Michael Zieve University of Michigan Arithmetic of prime-degree functions 9:20-9:40 100 NV Claude Levesque Université Laval On Thue equations 101 NV David Krumm Colby College A local-global principle in the dynamics of quadratic polynomials 108 NV Jeffrey Hatley Union College Elliptic curves with maximally disjoint division fields.   paper 9:50-10:10 100 NV Christian Maire Besançon / CRM Cohomological dimension, number fields and ramification 101 NV Bianca Thompson Smith College A very elementary proof of a conjecture of B. and M. Shapiro for cubic rational functions 108 NV Harris Daniels Amherst College Torsion Subgroups of Rational Elliptic Curves Over the Compositum of All Cubic Fields 10:20-11:05 100 NV Jeffrey Hoffstein Brown University Shifted multiple Dirichlet series and second moments of GL(2) L-series 101 NV2x20 Andrew Schultz Wellesley College Deformations of the Weyl character formula via ice models Jonathan Sands University of Vermont Derivatives of L-functions and annihilation of ideal class groups.   slides 11:20-12:05 100 NV2x20 Jan Vonk McGill University Stable models of Hecke operators Li-Mei Lim Bard College at Simon's Rock Counting Square Discriminants.  paper slides 101 NV David Rohrlich Boston University Almost abelian Artin representations Lunch, Memorial Union (on campus) 1:25-2:10 101 NV Shou-Wu Zhang Princeton University Faltings heights and Zariski density of CM abelian varieties 2:20-3:05 100 NV2x20 Harry Altman University of Michigan Well-Ordering Phenomena in Computation of Natural Numbers Steven Miller/Kevin Yang Williams C. / Harvard U. Biases in Moments of Satake Parameters and Models for $L$-function Zeros.   slides 101 NV Henri Darmon McGill University Generalised Kato classes 3:15-3:30 100 NV Patrick Letendre Université Laval New upper bounds for the number of divisors function.   slides 101 NV Michael Chou University of Connecticut Torsion of Rational Elliptic Curves over Quartic Galois Number Fields 108 NV David Burt/Blaine Talbot Williams C. / U. Chicago Large Gaps Between Zeros of $L$-Functions Associated to GL(2) Cusp Forms.   slides 3:30-3:55 Tea 3:55-4:10 100 NV Daniel Nichols University of Massachusetts Analog of the Collatz Conjecture in Polynomial Rings of Characteristic 2.   slides 101 NV Gautier Ponsinet Université Laval Functional equation for multi-signed Selmer groups 108 NV Sarah Manski Kalamazoo College A Ramsey Theoretic Approach to Finite Fields and Quaternions.   slides 4:20-4:40 100 NV Taylor Dupuy Hebrew U. / U. Vermont Examples of Varieties of General Type over Function Fields Whose Rational Points are Not Dense 101 NV Dimitris Koukoulopoulos Université de Montréal Sums of Euler products and statistics of elliptic curves 4:50-5:35 100 NV2x20 Dubi Kelmer Boston College Counting lattice points in a random lattice Antonio Lei Université Laval Asymptotic behaviour of the Shafarevich-Tate groups of modular forms 101 NV Chantal David Concordia University One-level density in one-parameter families of elliptic curves with non-zero average root number 6:45 Dinner: High Tide Restaurant, 5 South Main Street, Brewer. Bus leaves the hotel at 6:20 pm.

## Sunday, October 4, 2015

 Time Room Speaker Title 8:15-8:30am Coffee/Tea, Neville Lobby 8:30-8:50 100 NV Hugo Chapdelaine Université Laval A new proof of the functional equation of a special class of real analytic Eisenstein series 101 NV Ari Shnidman Boston College Counting cusps via isogeny volcanoes 108 NV Adriana Salerno Bates College Multiple zeta values: A combinatorial approach to structure 9:00-9:20 100 NV Naomi Tanabe Dartmouth College Determining Hilbert modular forms by central values of Rankin-Selberg convolutions   slides 101 NV Kenneth McMurdy Ramapo College of New Jersey Elliptic curves with non-abelian entanglements.   slides 108 NV Paul Kinlaw Husson University Repeated values of Euler's function slides 9:30-9:50 100 NV Thomas Hulse Colby College Averaging Average Orders with Shifted Series.  slides 101 NV Christelle Vincent University of Vermont Computing equations of hyperelliptic curves whose Jacobian has CM 108 NV Zhenguang Gao Framingham State University Triangular numbers in the Jacobsthal family 10:00-10:45 100 NV Ju-Lee Kim MIT Asymptotic behavior of supercuspidal characters and Sato-Tate equidistribution for families 101 NV2x20 Álvaro Lozano-Robledo University of Connecticut On the minimal degree of definition of $p$-primary torsion subgroups of elliptic curves  slides Liubomir Chiriac University of Massachusetts Vanishing Traces of Frobenius in Artin-type Representations 10:55-11:15 100 NV Karen Taylor Bronx Community College Hyperbolic Fourier Expansions of Modular Forms on the Full Modular Group 101 NV Leo Goldmakher Williams College Mock characters 11:25-12:10 101 NV Carl Pomerance Dartmouth College The ranges of some familiar arithmetic functions.   slides

## Abstracts (by last name)

Harry Altman, University of Michigan
Well-Ordering Phenomena in Computation of Natural Numbers
In this talk we will consider the complexity of computing natural numbers in two very simple models of computation. One, known simply as "integer complexity", is the smallest number of 1's needed to write a number using any combination and multiplication; for a number $n$, we denote this $\|n\|$. The other, the addition chain length of $n$, $l(n)$, is the smallest number of additions needed to make $n$ starting from 1, where numers may be reused freely once they are made. For each we define a notion of "defect" by subtracting off a logarithmic lower bound -- for integer complexity, we consider $\|n\|-3\log_3(n)$, and for addition chain length, we consider $l(n)-\log_2(n)$. Surprisingly, if we consider the set of all integer complexity defects, this turns out to be a well-ordered set, of order type $\omega^\omega$; and if we consider the set of all addition chain defects, this is also a well-ordered set of order type $\omega^\omega$. In this talk we will discuss more detailed forms of these results as well as some speculative extensions of them, including to other models of computation.

David Burt (Williams College) and Blaine Talbut (University of Chicago)
Large Gaps Between Zeros of $L$-Functions Associated to GL(2) Cusp Forms.
The spacing of gaps between zeros of $L$-functions encode arithmetic information. The first nontrivial bound on the normalized spacing of large gaps between zeros of $L$-functions of cuspidal newforms on GL(2) was made last year by Barrett et al. We look to improve upon this bound using a method of Hall; similar techniques have successfully improved lower bounds for gaps between zeros of the Riemann Zeta function and Dirichlet $L$-functions. We report on progress on the first step, deriving a shifted twisted second moment for such $L$-functions, where the twist is by an arbitrary, sufficiently short Dirichlet polynomial. This is joint work with Owen Barrett, Steven J. Miller and Caroline Turnage-Butterbaugh.

Hugo Chapdelaine, Université Laval
A new proof of the functional equation of a special class of real analytic Eisenstein series.
Let $E(z,s)$ be a real analytic Eisenstein with a unitary integral weight. Here $z$ varies over the Poincare upper half-plane (the symmetric space associated to the number field $\mathbb{Q}$) and $s$ varies over the right half-plane $Re(s)>1$. In this talk we will explain how one may prove the analytic continuation and the functional equation of the function $s \mapsto E(z,s)$ using the analytic Fredholm theorem for a certain family of compact operators associated to the hyperbolic Laplacian. This result automatically implies the analytic continuation and the functional equation of the coefficients of the Fourier series of $z\mapsto E(z,s)$. In particular, one obtains a new proof of a functional equation of a class of partial zeta functions constructed by the author some years ago. Initially, the functional equation was proved using the Poisson summation formula. In this sense, it is fair to say the classical role played by the Poisson summation formula is being replaced by some spectral analysis and the analytic Fredholm theorem. This set of ideas take its roots from the pioneering work of Lax and Philips who applied scattering theory to the study of automorphic forms. All this work generalizes to an arbitrary number field $K$.

Liubomir Chiriac, University of Massachusetts
Vanishing Traces of Frobenius in Artin-type Representations
In this talk we address Serre's lacunarity question of Frobenius traces in Artin-type representations. Assuming the irreducibility of the adjoint action we give a bound on the density of the vanishing traces, which is independent of the dimension of the representation. In addition, we construct an infinite family of representations of finite groups with an irreducible adjoint action.

Michael Chou, University of Connecticut
Torsion of Rational Elliptic Curves over Quartic Galois Number Fields
The classification of the torsion subgroup of elliptic curves over $\mathbb{Q}$ was determined by Mazur. The classification over quadratic number fields was completed due to work of Kamienny, Kenku, and Momose. However, over cubic fields the classification is already incomplete. In this talk we discuss a refined version of this problem: let $E$ be an elliptic curve defined over $\mathbb{Q}$ and $K$ be a number field of degree $d$; what groups appear as $E(K)_{\text{tors}}$? In particular, we will present a classification over all quartic galois number fields $K$ and show how the techniques used may be applied to other fields.

Harris Daniels, Amherst College
Torsion Subgroups of Rational Elliptic Curves Over the Compositum of All Cubic Fields
Let $E/\mathbf{Q}$ be an elliptic curve and let $\mathbf{Q}(3^\infty)$ be the compositum of all cubic extensions of $\mathbf{Q}$. In this talk we show that the torsion subgroup of $E(\mathbf{Q}(3^\infty))$ is finite and determine 20 possibilities for its structure, along with a complete description of the $\overline{\mathbf{Q}}$-isomorphism classes of elliptic curves that fall into each case. We provide rational parameterizations for each of the 16 torsion structures that occur for infinitely many $\overline{\mathbf{Q}}$-isomorphism classes of elliptic curves, and a complete list of $j$-invariants for each of the 4 that do not. This is joint work with Álvaro Lozano-Robledo, Filip Najman, and Andrew Sutherland.

Henri Darmon, McGill University
Generalised Kato classes
I will discuss the conjectured properties of the eponymous objects of the title and their connection with the p-adic Birch-Swinnerton-Dyer conjecture in the spirit of Kato and Perrin-Riou.

Chantal David, Concordia University
One-level density in one-parameter families of elliptic curves with non-zero average root number
This is joint work with Sandro Bettin and Christophe Delaunay. We present in this talk a (conjectural) formula for the one-level density of general one-parameter families of elliptic curves, in term of $n$, the rank of $E$ over $\mathbb{Q}(t)$ and the average root number $W_E$ over the family. In the general case, $W_E$ is zero, and the one-level density is given by orthogonal symmetries as predicted by the conjectures of Katz and Sarnak. In the exceptional cases where $W_E \neq 0$, we find that the statistics are given by a weighted sum of even orthogonal and odd orthogonal symmetries. The most dramatic and counter-intuitive cases occur when $W_E = \pm 1$. In that case, the one-level density exhibits even orthogonal symmetries when $(-1)^n W_E = 1$ and odd orthogonal symmetries when $(-1)^n W_E = -1$, and there is a shift of the symmetries (between orthogonal odd and orthogonal even) when $n$ is odd.
We also build several one-parameter families of elliptic curves with $W_E \neq 0$, and which exhibit the shifts of the symmetries.

Taylor Dupuy, Hebrew University / University of Vermont
Examples of Varieties of General Type over Function Fields Whose Rational Points are Not Dense
The Lang-Bombieri-Noguchi conjecture says that a variety of general type over Q has nondense Q-points. In this talk we exhibit some examples of varieties satisfying an analogous conjecture over function fields (in characteristic zero). These examples cannot be embedded into abelian varieties and hence provide examples of Lang-Bombieri-Noguchi outside Mordell-Lang (Falting's Theorem). This is joint work with Daniel Litt.

Zhenguang (Jeff) Gao, Framingham State University
Triangular numbers in the Jacobsthal family
Using congruences, second-order Diophantine equations, and linear algebra, we identify Jacobsthal and Jacobsthal-Lucas numbers that are also triangular numbers. The conclusion is that there are 5 triangular Jacobsthal numbers and there is only 1 triangular Jacobsthal-Lucas number.

Leo Goldmakher, Williams College
Mock characters
I will discuss recent work (joint with Jean-Paul Allouche) on a class of functions which are, in some sense, as close as possible to being Dirichlet characters without *actually* being Dirichlet characters. The talk will contain results, conjectures, and manipulatives.

Jeffrey Hatley, Union College
Elliptic curves with maximally disjoint division fields
This talk will discuss a recent paper of the same title, written jointly with Harris Daniels and James Ricci. We exhibit an infinite two-parameter family of elliptic curves whose direct product Galois representation has maximal image. This maximality is closely related to the arithmetic of the number fields generated by the elliptic curves' torsion points.

Jeffrey Hoffstein, Brown University
Shifted multiple Dirichlet series and second moments of GL(2) L-series
I'll explain what shifted multiple Dirichlet series are. I'll also describe recent joint work with Min Lee in which we apply the meromorphic properties of shifted multiple Dirichlet series to the problem of expressing the mean square of $GL(2)$ $L$-series averaged over characters as a main term plus an error term.

Thomas Hulse, Colby College
Averaging Average Orders with Shifted Series.
Here we consider Dirichlet series where the coefficients are related to the average order of Fourier coefficients of holomorphic cusp forms. By considering the spectral decomposition of certain shifted sums, we are able to obtain a meromorphic continuation of these Dirichlet series which in turn yields information about the cancellation of partial sums of these Fourier coefficients. Joint work with Chan Ieong Kuan, David Lowry-Duda and Alexander Walker.

Dubi Kelmer, Boston College
Counting lattice points in a random lattice
Consider the counting function counting the number of lattice points of a bounded norm as a function on the space of unimodular lattices. By considering local averages of this function, and in particular averages on all shears of a lattice, it is possible to give optimal bounds for the remainder that hold on average for any compact set in the space of lattices.

Julee Kim, MIT
Asymptotic behavior of supercuspidal characters and Sato-Tate equidistribution for families
We establish properties of family of automorphic representations as we vary prescribed supercuspidal representations at a given finite set of primes. We prove the limit multiplicity property with error terms. Thereby we obtain a Sato-Tate equidistribution for the Hecke eigenvalues. The main problem is to show that the orbital integrals of matrix coefficients of supercuspidal representations with increasing formal degree on a connected reductive $p$-adic group tend to zero uniformly for every noncentral semisimple element. I will start with a brief survey on Sato-Tate conjecture and many examples. This is a joint work with Sug Woo Shin and Nicholas Templier.

Paul Kinlaw, Husson University
Repeated values of Euler's function
We consider solutions of $\varphi(n)=\varphi(n+1)$ and $\sigma(n)=\sigma(n+1)$. Both equations are conjectured to have infinitely many solutions. Work of Erdös, Pomerance and Sárközy shows that the sum of reciprocals of solutions is convergent.
We will discuss recent joint work with Jonathan Bayless, including explicit bounds on the counting functions of smooth numbers as well as numbers with $k$ distinct prime factors. We use these results as tools to put explicit numerical bounds on the sum of reciprocals of solutions of $\varphi(n)=\varphi(n+1)$ and $\sigma(n)=\sigma(n+1)$. We show that there are infinitely many $n$ such that $\varphi(n)=\varphi(n+k)$ for some $k<\sqrt[3]{n}$, and the same for $\sigma$.

Dimitris Koukoulopoulos, University of Montréal
Sums of Euler products and statistics of elliptic curves
I will present a new approach to several statistical questions about elliptic curves over finite fields, such as the average Lang-Trotter conjecture and the vertical Sato-Tate conjecture. The starting point is a theorem of Gekeler that provides a probabilistic reinterpretation of Deuring's theorem about the number of elliptic curves in a given isogeny class. In the heart of our approach lies a general technical theorem about averages of Euler products. As a corollary of this general result, we obtain new proofs of various results, some already known and some of which are new. One of the new results is the vertical Sato-Tate conjecture for very short intervals. This is joint work with Chantal David and Ethan Smith.

David Krumm, Colby College
A local-global principle in the dynamics of quadratic polynomials
Let $K$ be a number field and let $f\in K[x]$ be a polynomial. For any nonnegative integer $n$, let $f^n$ denote the $n$-fold composition of $f$ with itself. We say that an element $\alpha\in K$ is periodic for $f$ if there exists a positive integer $n$ such that $f^n(\alpha)=\alpha$. In that case, the least such $n$ is called the period of $\alpha$. It is clear that if $f$ has a point of period $n$ in $K$, then it has a point of period $n$ in every extension of $K$; in particular, for every place $v$ of $K$, $\,f$ has a point of period $n$ in the completion $K_v$. In this talk we will discuss whether the converse holds: if $f$ has a point of period $n$ in every completion of $K$, must it then have a point of period $n$ in $K$?

Antonio Lei, Université Laval
Asymptotic behaviour of the Shafarevich-Tate groups of modular forms
Let $E$ be an elliptic curve with good ordinary reduction at a prime $p$. Mazur has studied the $p$-primary part of the the Shafarevich-Tate group of $E$ over the $\mathbb{Z}_p$-cyclotomic extensions of a number field. In particular, he showed that there is an asymptotic formula for the size of these groups in terms of the Iwasawa invariants of the Selmer group of $E$. This has been generalized to supersingular primes by Kobayashi and Sprung using plus and minus Selmer groups. In this talk, I shall discuss a generalization of these results to modular forms using the machinery of Wach modules developped by Loeffler, Zerbes and myself.

Patrick Letendre, Université Laval
New upper bounds for the number of divisors function.
Let $\tau(n)$ stand for the number of divisors of the positive integer $n$. We obtain new upper bounds for $\tau(n)$ in terms of $n$ and the number of distinct prime factors of $n$. This is joint work with Jean-Marie De Koninck (Laval University).

Claude Levesque, Université Laval
On Thue equations.
We will survey some joint results with Michel Waldschmidt on Thue equations.

Li-Mei Lim, Bard College at Simon's Rock
Counting Square Discriminants
Hee Oh and Nimish Shah prove that the number of integral binary quadratic forms whose coefficients are bounded by a quantity $X$, and with discriminant a fixed square integer $d$, is $cX\log X+O(X(\log X)^{3/4})$. This result was obtained by the use of ergodic methods. Here we use the method of shifted convolution sums of Fourier coefficients of certain automorphic forms to obtain a sharpened result of a related asymptotic, obtaining a second main term and an error of $O(X^{1/2})$.

Álvaro Lozano-Robledo, University of Connecticut
On the minimal degree of definition of $p$-primary torsion subgroups of elliptic curves
In this talk, we discuss the minimal degree $[K(T):K]$ of a $p$-subgroup $T\subseteq E(\overline{K})_\text{tors}$ for an elliptic curve $E/K$ defined over a number field $K$. Our results depend on the shape of the image of the $p$-adic Galois representation $\rho_{E,p^\infty}:\operatorname{Gal}(\overline{K}/K)\to \operatorname{GL}(2,\mathbf{Z}_p)$. However, we are able to show that there are certain uniform bounds for the minimal degree of definition of $T$. When the results are applied to $K=\mathbb{Q}$ and $p=2$, we obtain a divisibility condition on the minimal degree of definition of any subgroup of $E[2^n]$ that is best possible. This is joint work with Enrique González-Jiménez (UAM).

Christian Maire, Besançon / CRM / University of Montréal
Cohomological dimension, number fields and ramification
In this talk I will discuss on pro-$p$-extensions of number fields for which the cohomological dimension is finite. I will show how to produce some (very different) arithmetic situations where the dimension is at most 2.

Sarah Manski (Kalamazoo College)
A Ramsey Theoretic Approach to Finite Fields and Quaternions.
Ramsey theory concerns itself with how large a set needs to be for a certain structure to arise. We concentrate on sets avoiding 3-term geometric progressions. Previous work studied this problem in the integers and number fields; we resolve analogous problems for polynomials over finite fields and in the Hurwitz Quaternions. New features emerge in the function field case -- the proofs are distinctly combinatorial, a feature not seen in other cases. We take advantage of the combinatorics arising from finite characteristic through counting $q$-smooth elements and the number of irreducible polynomials. In the Hurwitz Quaternions, the loss of commutativity greatly complicates the arguments and affects the limiting behavior. We construct maximally sized sets of Hurwitz quaternions that avoid geometric progressions up to units and bound their densities, while adjusting for unique properties of the ring. The proofs involve a mix of the algebra of the quaternions with an analysis of the resulting infinite products. This work is joint with Megumi Asada, Eva Fourakis, Eli Goldstein, Gwyn Moreland, Nathan McNew and Steven J. Miller.

Kenneth McMurdy, Ramapo College of New Jersey
Elliptic curves with non-abelian entanglements.
Let $K$ be a number field. An elliptic curve $E/K$ is said to have a non-abelian entanglement if there are relatively prime positive integers, $m_1$ and $m_2$, such that $K(E[m_1])\cap K(E[m_2])$ is a non-abelian Galois extension of K. In this talk, we will discuss our ongoing efforts to classify, using explicit methods, all infinite families of elliptic curves $E/K$, for a fixed $K$, with non-abelian entanglements. This problem is closely related to that of determining when the image of $\rho_E$ in $\operatorname{GL}_2(\hat{Z})$ is maximal, and to the study of correction factors for various conjectural constants for elliptic curves over $\mathbf{Q}$. This is joint work with Nathan Jones.

Steven Miller (Williams) and Kevin Yang (Harvard)
Biases in Moments of Satake Parameters and Models for $L$-function Zeros.
We report on two related projects concerning zeros of $L$-functions. The Katz-Sarnak philosophy states that the statistics of zeros of families of $L$-functions agree with those of eigenvalues of classical compact Lie groups in the limit of large conductor (resp. dimension). In 2006, Miller discovered a discrepancy in these statistics for elliptic curve $L$-functions of finite conductor; in 2012 Duenez, Huynh, Keating, Miller and Snaith explained this behavior through their Excised Orthogonal Ensemble, which depends on an effective matrix size which incorporates the discreteness of central values of the $L$-functions. We investigate a similar effective matrix size construction for quadratic Dirichlet $L$-functions and quadratic twists of $L$-functions given by symmetric squares of holomorphic cusp forms, both symplectic families. We find a negative effective matrix size, indicating that for bounded conductor, such families of $L$-functions statistically resemble the group SO(odd) for suitable dimension, and also provide a formal justification for improved statistics provided by the excised ensemble.
The second project involves the moments of the Satake parameters of $L$-functions; these values play an important role in detecting arithmetic in lower order terms in the 1-level density of families of $L$-functions, which help explain the behavior of zeros at or near the central point. Recently, it was observed that the second moments of the Satake parameters exhibit a bias in many families of of elliptic curve $L$-functions, and it was conjectured that similar biases exist in other families. We resolve the conjecture for various families of Dirichlet $L$-functions and families of symmetric lifts of holomorphic newforms on GL(2) by exploiting powerful trace formulas and orthogonality relations. These projects are joint with Megumi Asada, Owen Barrett, Eva Fourakis, Gwyn Moreland and Blaine Talbut.

Dan Nichols, University of Massachusetts
Analog of the Collatz Conjecture in Polynomial Rings of Characteristic 2
The Collatz conjecture (also known as the $3n+1$ problem) concerns the behavior of integer sequences $\left\{ T^k(n) \right\}_{k=0}^\infty$, where $T$ is a particular transformation map on the positive integers. In this talk we will discuss an analog of the Collatz conjecture in polynomial rings of characteristic 2 which exhibits some interesting properties. We will outline some theoretical results, including a theorem concerning distribution of stopping times analogous to one proved by Terras and Everett for the original $3n+1$ problem. We will also present experimental data on stopping times and cycle lengths.

Carl Pomerance, Dartmouth College
The ranges of some familiar arithmetic functions
We consider $4$ functions from elementary number theory: $\sigma$ (the sum-of-divisors function), $\varphi$ (Euler's function), $\lambda$ (Carmichael's universal exponent function), and $s$ (the sum-of-proper-divisors function. In particular we discuss the distribution of the values of these functions, and coincidences of values. Most of the problems considered have a fairly long history, some over 80 years. We report on recent progress. (Various parts of this work are joint with Kevin Ford, Tristan Freiberg, Florian Luca, and Paul Pollack.)

Gautier Ponsinet, Université Laval
Functional equation for multi-signed Selmer groups
A. Lei and K. Büyükboduk have recently defined a signed Selmer group for abelian variety at supersingular prime. We will motivate and show a functional equation for this Selmer group, generalizing a result of B.D. Kim for elliptic curves.

David Rohrlich, Boston University
Almost abelian Artin representations
The purpose of this talk is to point out a connection between two seemingly unrelated achievements: Anderson's paper (Duke Math. J. 114, 2002) extending the Kronecker-Weber theorem to the case of "almost abelian" extensions of $\mathbf{Q}$ and Shintani's paper (J. Math. Soc. Japan 30, 1978) proving certain cases of Stark's conjecture for real quadratic fields.

Multiple zeta values: A combinatorial approach to structure.
Multiple zeta functions are a multivariate version of the Riemann zeta function. There are many open problems concerning these values, for example, it's not even known if these numbers are rational or even algebraic (although it is strongly suspected that they are transcendental). However, these values satisfy many interesting algebraic relations between them. A new approach to understanding multiple zetas is to study purely their algebraic structure. I will talk about a few spaces (which turn out to have the nice structure of a Lie algebra) that are essentially equivalent to a formal version of these zetas, and where all the interesting questions turn into combinatorial questions.

Jonathan Sands, University of Vermont
Derivatives of L-functions and annihilation of ideal class groups.
We consider certain abelian Galois extensions K/F of relative degree 6 with F real quadratic. Letting G denote the Galois group and S the set of ramified primes, we first computationally verify a version of Stark's conjecture that the derivative of the S-imprimitive equivariant L-function, when multiplied by an equivariant regulator, yields an element of the rational group ring of G. Subsequent multiplication by an annihilator of roots of unity then produces an element of the integral group ring, and further computation confirms that this element annihilates the ideal class group of K. This provides evidence for an analog of the Brumer-Stark conjecture that uses the leading coefficients of L-functions at the origin and not just the values. This is joint work with Brett Tangedal.

Andrew Schultz, Wellesley College
Deformations of the Weyl character formula via ice models.
By assigning polynomial weights to certain symmetry classes of alternating sign matrices, Okada gave deformations of the Weyl denominator formula in types $B$, $C$ and $D$. In this talk we generalize these results by replacing Okada's families of alternating sign matrices with more general families of ice models from statistical mechanics. This allows us to use some local conservation rules --- particularly the Yang-Baxter equation --- to evaluate the corresponding partition functions. We will see that a certain specialization of these polynomials returns the Weyl character formula.

Ari Shnidman, Boston College
Counting cusps via isogeny volcanoes
I will present solutions to various counting problems in number theory, topology, and algebraic geometry. The key tool is a refinement of an unpublished result of Lichtenbaum describing Ext groups of CM elliptic curves. The general formula involves taking walks along certain graphs called isogeny volcanoes. This is joint work with Julian Rosen.

Naomi Tanabe, Dartmouth College
Determining Hilbert modular forms by central values of Rankin-Selberg convolutions.
Identifying an automorphic forms by studying the special values of L-functions of its twists has been discussed in various setting. In joint work with Alia Hamieh, we generalize these results to Hilbert modular forms by analyzing a twisted first moment of Rankin-Selberg convolutions.

Karen Taylor, Bronx Community College
Hyperbolic Fourier Expansions of Modular Forms on the Full Modular Group
Classically, modular forms are described by their (parabolic) Fourier expansions. In this talk, we discuss hyperbolic Fourier expansions of modular forms. We give an explicit exact formula for the $n$-th hyperbolic Fourier coefficient of a modular form on the full modular group.

Bianca Thompson, Smith College
A very elementary proof of a conjecture of B. and M. Shapiro for cubic rational functions
Using essentially only algebra, we give a proof that a cubic rational function over $\mathbb{C}$ with only real critical points is equivalent to a real rational function. We also determine all fields $\mathbb{Q}_p$ over which a reasonable generalization holds.

Christelle Vincent, University of Vermont
Computing equations of hyperelliptic curves whose Jacobian has CM.
It is known that given a totally imaginary sextic field with totally real cubic subfield (a so-called CM sextic field) there exists a non-empty finite set of abelian varieties of dimension 3 that have CM by this field. In this talk we present an algorithm that takes as input such a field, and outputs a period matrix for such an abelian variety. We then check computationally if the abelian variety is the Jacobian of a hyperelliptic curve, and compute an equation for the curve if this is the case. This is joint work with J. Balakrishnan, S. Ionica and K. Lauter.

Jan Vonk, McGill University
Stable models of Hecke operators
Many linear operators in number theory arise as linearisations of correspondences between varieties. We will discuss how to postpone this linearisation as long as possible, and study correspondences between curves as geometric objects. After presenting a potential-semistability theorem for correspondences analogous to the Deligne--Mumford theorem for curves, we indicate how to extract spectral information from the combinatorics of the special fibre of a semi-stable correspondence, and revisit some classical methods known for Hecke operators.

Shou-Wu Zhang
Faltings heights and Zariski density of CM abelian varieties
The moduli of CM abelian varieties are the simplest objects in the category of Shimura varieties, and have been intensively studied related to Hilbert's 12th problem and the BSD conjecture. In this talk, I will discuss some recent progress on two different kinds of problems: 1) Colmez' conjecture on Faltings' heights of CM abelian varieties in terms of Artin L-functions, and 2) André-Oort's conjecture on Zariski density of CM points on Shimura varieties.

Michael Zieve, University of Michigan
Arithmetic of prime-degree functions
I will present several arithmetic results about prime-degree functions. For instance, I will describe the possible sizes of the image $f(\mathbb{F}_q)$ where $f(x)$ is a prime-degree rational function over $\mathbb{F}_q$. I will also describe all irreducible polynomials $f(x,y)\in \mathbb{Q}[x,y]$ with prime y-degree for which there are infinitely many rational numbers c such that f(x,c) is reducible. Finally, I will describe all prime-degree rational functions $f(x) \in \mathbb{Q}(x)$ for which the induced function on $P^1(\mathbb{Q})$ is noninjective over infinitely many values, and explain how this result suggests a vast generalization of Mazur's theorem on uniform boundedness of rational torsion on elliptic curves. The proofs rely on studying the possibilities for the fundamental invariants (i.e., the monodromy group and ramification type) of morphisms of complex curves.

## List of Participants

Harry Altman, University of Michigan
Jonathan Bayless, Husson University
Daniel Buck, University of Maine (grad)
Kenneth Bundy, University of Maine (grad)
Hugo Chapdelaine, Université Laval
Liubomir Chiriac, University of Massachusetts
Michael Chou, University of Connecticut (grad)
Harris Daniels, Amherst College
Henri Darmon, McGill University
Chantal David, Concordia University
Danielle David, University of Maine (grad)
Daniel Disegni, McGill University
David Dummit, University of Vermont
Taylor Dupuy, Hebrew University / University of Vermont
Zhenguang Gao, Framingham State University
Eva Goedhart, Smith College
Leo Goldmakher, Williams College
Fernando Gouvêa, Colby College
Theodore Halnon, Pennsylvania State University (undergrad)
Jeffrey Hatley, Union College
Jeffrey Hoffstein, Brown University
Thomas Hulse, Colby College
Dubi Kelmer, Boston College
Ju-Lee Kim, Massachusetts Institute of Technology
Paul Kinlaw, Husson University
Hershy Kisilevsky, Concordia University
Andrew Knightly, University of Maine
Dimitris Koukoulopoulos, University of Montréal
David Krumm, Colby College
Chan Ieong Kuan, University of Maine
Prateek Kunwar, University of Maine (grad)
John Larson, University of Maine (undergrad)
Hao (Billy) Lee, McGill University
Antonio Lei, Université Laval
Claude Levesque, Université Laval
Li-Mei Lim, Bard College at Simon's Rock
Álvaro Lozano-Robledo, University of Connecticut
Sergey Lvin, University of Maine
Christian Maire, Besançon / CRM / University of Montréal
Ayesha Maliwal, University of Maine (grad)
Kenneth McMurdy, Ramapo College of New Jersey
Steven Miller, Williams College
Daniel Nichols, University of Massachusetts (grad)
Robert Niemeyer, University of Maine
Nigel Pitt, University of Maine
Carl Pomerance, Dartmouth College
Yannan Qiu, University of Maine
Caroline Reno, University of Maine (grad)
David Rohrlich, Boston University
Jonathan Sands, University of Vermont
Andrew Schultz, Wellesley College
Ariel Shnidman, Boston College
Chip Snyder, University of Maine
Blaine Talbut, University of Chicago (undergrad)
Naomi Tanabe, Dartmouth College
Karen Taylor, Bronx Community Collge