# 2017 Maine/Québec Number Theory Conference

The University of Maine, Orono
October 14-15, 2017

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
Elsevier

Full List of Participants

## Saturday, October 14, 2017 - Bennett Hall

Click a name to view the abstract for the talk
 8:30 - 8:55 Coffee/Tea/Snacks, 137 Bennett Time 137 Bennett 140 Bennett 141 Bennett 8:55-9:00 Welcoming Remarks Nigel Pitt, Chair of Mathematics and Statistics 9:00-9:50 Henryk Iwaniec Families of orthogonal arithmetic functions 10:00-10:20 Jan Vonk Fundamental groups and rational points on curves Claude Levesque On Thue equations slides Naomi Tanabe Sign of Fourier Coefficients for Hilbert Modular Forms 10:30-10:50 Edgar Costa Computing zeta functions of nondegenerate toric hypersurfaces Michael Chou Growth of torsion of elliptic curves from $\mathbb{Q}$ to the maximal abelian extension. Fan Zhou Voronoi formula and its generalizations Break 11:10-11:30 Michael Mossinghoff The Lind-Lehmer constant for certain $p$-groups Farshid Hajir On the invariant factors of class groups in unramified towers of number fields Jared Lichtman, Steven J. Miller, Eric Winsor, and Jianing Yang Lower order biases in Fourier coefficients of elliptic curve and cuspidal newform families 11:40-12:00 Dijana Kreso Decomposable recursively defined polynomials Christian Maire Analytic Lie extensions of number fields with cyclic fixed points and tame ramification Dimitris Koukoulopoulos Pretentious methods for L-functions Lunch: Sandwiches at the University Club, Fogler Library. Go upstairs and to the right. Seating is limited. (Memorial Union is also open for lunch.) Time 137 Bennett 140 Bennett 141 Bennett 1:30-2:20 John Friedlander Exceptional characters and their consequences 2:30-2:50 Corentin Perret-Gentil Integral monodromy groups and applications in number theory David Dummit Classes of order 4 in the strict class group of number fields and unramified quadratic extensions of unit type Yujin Kim, Steven J. Miller, and Shannon Sweitzer Variance of Gaussian Primes Across Sectors and The Hecke L-Function Ratios Conjecture 3:00-3:20 Niko Laaksonen Prime Geodesic Theorem in $\mathbb{H}^{3}$ Anna Haensch Almost universal ternary sums of polygonal numbers Erik Wallace Bounds of the rank of the Mordell--Weil group of Jacobians of Hyperelliptic Curves 3:20-3:50 Tea/Refreshments 3:50-4:05(students) Brandon Alberts Nonabelian Cohen-Lenstra Moments for the Quaternion Group [slides] Sara Chari Metacommutation of Primes in Quaternion Orders of Class Number One Robert McDonald Torsion Subgroups of Elliptic Curves over Function Fields of Genus 0 4:15-4:35 Liang Xiao Some remarks on the ghost conjecture of Bergdall and Pollack Daniel Smertnig Arithmetical invariants of local quaternion orders Hershy Kisilevsky The Non-Square Part of Analytic Sha 4:45-5:00(students) Sam Schiavone Computing A Database of Belyi Maps: A Progress Report Michael Musty 2-solvable Belyi maps Wanlin Li Vanishing of hyperelliptic L-functions at the central point 5:10-5:30 Dubi Kelmer Shrinking target problems, homogenous dynamics, and Diophantine approximations Hugo Chapdelaine Real analytic Eisenstein series with non-trivial multiplier systems Matilde Lalin A geometric generalization of the square sieve and applications to cyclic covers Dinner: High Tide Restaurant, 5 South Main Street, Brewer. Light hors d'oeuvres at 6:30; Seating at 6:45

## Sunday, October 15, 2017 - Donald P. Corbett Business Building (DPC)

 8:30 - 9:00 am Coffee/Tea/Snacks, DPC Lobby Time DPC 105 DPC 107 DPC 115 9:00-9:20 Ari Shnidman Intersections of Heegner-Drinfeld cycles Carl Pomerance New results on an ancient function 9:30-9:50 Álvaro Lozano-Robledo A probabilistic model for the distribution of ranks of elliptic curves over $\mathbb Q$ [paper] Guhan Venkat Plus/Minus Beilinson - Flach Euler system Kim Klinger-Logan Differential equations in automorphic forms 10:00-10:15(students) Angelica Babei Tiled orders and the building for $SL_n(\mathbb{Q}_p)$ Gautier Ponsinet On the stucture of signed Selmer groups for abelian varities at supersingular primes Kyle Pratt A lower bound for the least prime in an arithmetic progression 10:25-10:45 Harris Daniels Torsion subgroups of rational elliptic curves over infinite extensions of $\mathbf{Q}$. Robert Lemke-Oliver Three-isogeny descent for quadratic twists of abelian varieties Paul Kinlaw An Explicit Lower Bound for the Counting Function of Squarefree Products of Three Primes. 10:55-11:15 Caleb Shor A characterization of the complement of a free numerical semigroup Antonio Lei Second Chern Classes for Supersingular Elliptic Curves Xianchang Meng Chebyshev's bias for total number of prime factors of arithmetic progressions 11:25-11:40(students) Shucheng Yu Logarithm laws for hitting time functions on homogeneous spaces Benjamin Breen The 2-Selmer group for number fields of even degree Vincent Ouellet On the middle prime factors of integers 11:50-12:10 Lior Bary-Soroker Twin primes going function field Henri Darmon A tamely ramified Perrin-Riou conjecture

## Abstracts (by last name)

Brandon Alberts, University of Wisconsin (grad)
Nonabelian Cohen-Lenstra Moments for the Quaternion Group
In this talk I will briefly review a conjecture of Melanie Wood generalizing Cohen-Lenstra heuristics for counting the expected number of unramified G-extensions of quadratic fields, for G any fixed finite group. I verify this conjecture for the case when G is the quaternion group of order 8. Moreover, in joint work with Jack Klys we asymptotically count the number of unramified G-extensions of quadratic fields of discriminant smaller than X as X goes to infinity.

Angelica Babei, Dartmouth College (grad)
Tiled orders and the building for $SL_n(\mathbb{Q}_p)$
In finding the formula for traces of Hecke operators, Hijikata used the fact that the normalizer up to units of an Eichler order of $M_2(\mathbb{Q}_p)$ has size 2. We can obtain the latter result by studying the Bruhat-Tits tree for $\operatorname{SL}_2(\mathbb{Q}_p)$ where Eichler orders correspond to pairs of vertices and the path between them. The normalizer of a given Eichler order then acts on the tree by swapping the two vertices. Analogously, the building for $\operatorname{SL}_n(\mathbb{Q}_p)$ provides a way to find normalizers of a generalization of Eichler orders called tiled orders, by studying the symmetries of convex polytopes associated to these orders.

Lior Bary-Soroker, Tel-Aviv University
Twin primes going function field
The study of function field versions of number theoretic problems may be used to get new insights and to provide new evidence for difficult conjectures. In this talk I will discuss some of the recent progress on problems on correlations of arithmetic functions, including the Hardy-Littlewood prime tuple conjecture on the correlation of the prime indicator function which is the quantative version of the twin prime conjecture. I will also survey some of the main challenges we are facing in light of those recent results.

Benjamin Breen, Dartmouth College (grad)
The 2-Selmer group for number fields of even degree
This talk focuses on the study of the 2-Selmer group for number fields with even degree. The 2-Selmer group plays a crucial role in the relationship between the class group and the narrow class group. We look at modeling the 2-Selmer group in order to derive heuristics on the 2-rank of the narrow class group.

Hugo Chapdelaine, Université Laval
Real analytic Eisenstein series with non-trivial multiplier systems
We shall begin by recalling some known facts about classical modular forms with non-trivial multiplier systems; the main example being the unary theta function of weight $1/2$ considered by Riemann in his famous proof of the meromorphic continuation and functional equation of the Riemann zeta function. We shall then proceed to define two types of real analytic Eisenstein series with non-trivial multiplier systems; in particular, this approach allows the construction of families of real analytic Eisenstein of half-integral weights. Finally, if time permits, we shall present some possible applications of these real analytic Eisenstein series via the so-called theta correspondance (or some of its more recent refinements which lead for example to the so-called "Borcherds products").

Sara Chari, Dartmouth College (grad)
Metacommutation of Primes in Quaternion Orders of Class Number One
In a quaternion order of class number one, factorization of elements is unique only up to units and metacommutation, or rearrangement of the prime factors. The fact that multiplication is not commutative causes an element to induce a permutation on the set of primes of a given norm. We discuss this permutation and previously known results about the cycle structure, sign, and number of fixed points in the case of the Hurwitz order in the Hamiltonians. We generalize these results to quaternion orders of class number one.

Michael Chou, Tufts University
Growth of torsion of elliptic curves from $\mathbb{Q}$ to the maximal abelian extension.
Torsion an elliptic curve over a number field is finite due to the Mordell-Weil theorem. However, even in certain infinite extensions of $\mathbb{Q}$ we have that torsion is finite. Ribet proved that, when base extended to the maximal abelian extension of $\mathbb{Q}$, the torsion of an elliptic curve over $\mathbb{Q}$ is finite. In this talk, we give a classification of all possible torsion structures appearing in this way.

Edgar Costa, Dartmouth College
Computing zeta functions of nondegenerate toric hypersurfaces
We report on an ongoing joint project with Kiran Kedlaya and David Harvey on the computation of zeta functions of nondegenerate toric hypersurfaces over finite fields using p-adic cohomology

Harris Daniels, Amherst College
Torsion subgroups of rational elliptic curves over infinite extensions of $\mathbf{Q}$.
Let $E/\mathbf{Q}$ be an elliptic curve. In this talk we consider the question of what torsion subgroups can occur when we base extend $E$ to some infinite field extensions $K/\mathbf{Q}$. For example, if $K$ is the compositum of all cubic extensions of $\mathbf{Q}$, we show that the torsion subgroup of $E(K)$ is finite and determine 20 possibilities for its structure. This work is partially joint with Álvaro Lozano-Robledo, Filip Najman, and Andrew Sutherland.

Henri Darmon, McGill University
A tamely ramified Perrin-Riou conjecture
I will describe a generalisation of a conjecture of Perrin Riou which I studied recently with Alan Lauder, and a "tamely ramified" refinement of it, in the spirit of Mazur and Tate's tame refinements of the p-adic Birch and Swinnerton Dyer conjecture. The latter is the fruit of ongoing exchanges with Michael Harris, Victor Rotger and Akshay Venkatesh.

David Dummit, University of Vermont
Classes of order 4 in the strict class group of number fields and unramified quadratic extensions of unit type
This talk will explain why, for example, the strict class group of a totally real field K with a totally positive system of fundamental units contains at least (n-1)/2 (n odd) or n/2 -1 (n even) independent elements of order 4 and make some remarks on unramified extensions generated by square roots of units.

John Friedlander, University of Toronto
Exceptional characters and their consequences

Anna Haensch, Duquesne University
Almost universal ternary sums of polygonal numbers
In 1796 Gauss showed that every natural number can be written as the sum of three triangular numbers. In 2009, Chan and Oh determined when a weighted sum of triangular numbers (i.e. triangular numbers with coefficients) represents all but finitely many natural numbers. We say such a sum is almost universal. In this talk we will determine when a sum of three generalized m-gonal numbers is almost universal. We will approach this question first from an algebraic, and then from analytic point of view, exploiting the capabilities of each method, and realizing new connections between the machinery.

Farshid Hajir, University of Massachusetts
On the invariant factors of class groups in unramified towers of number fields
In this talk, I will report on joint work with Christian Maire in which we study how the exponents of class groups vary in towers contained in L/K where L is the maximal unramified p-extension of K. Note that according to a conjecture of Fontaine and Mazur, such towers do not have p-adic analytic Galois group. Our main result is that in contrast to the p-adic analytic case, the average exponent of p-class groups (a concept we define) in unramified p-towers can be bounded. The proof uses a combination of group theory and a key arithmetic result of Tsfasman and Vladut generalizing a classic theorem of Brauer and Siegel.

Henryk Iwaniec, Rutgers University
Families of orthogonal arithmetic functions

Dubi Kelmer, Boston College
Shrinking target problems, homogenous dynamics, and Diophantine approximations
The shrinking target problem for a dynamical system tries to answer the question of how fast can a sequence of targets shrink, so that a typical orbit will keep hitting them indefinitely. In this talk I will describe some new and old results on this problem for homogenous dynamics, that is, the dynamical systems given by group actions on a homogenous space. I will then describe some applications of such results to problems in Diophantine approximations.

Yujin Kim (Columbia, undergrad), Steven J. Miller (Williams), and Shannon Sweitzer (UC Riverside, undergrad)
Variance of Gaussian Primes Across Sectors and The Hecke L-Function Ratios Conjecture
A Gaussian prime is a prime element in the ring of Gaussian integers $\mathbb{Z}[i]$. While many statistical properties of the ordinary primes carry over to this new setting, as the Gaussian integers lie in the plane interesting questions about their geometric properties can be asked, which have no classical analogue among the ordinary primes. Specifically, to each Gaussian prime $a+bi$ we associate an angle whose tangent is the ratio $b/a$. Hecke showed that these angles are uniformly distributed as $p$ varies, and Kubilius proved uniform distribution in somewhat short arcs. Motivated by a random matrix theory (RMT) model and a function field analogue, Rudnick and Waxman gave a conjecture for the variance of such angles across short arcs. While many number theoretic results show agreement between the main term of a calculation and RMT, far fewer results exist in which the secondary terms are known. We report on progress in applying the $L$-function Ratios Conjecture to a family of Hecke $L$-functions to derive a formula which computes the variance of Gaussian primes across short arcs. Our result breaks the universality of behavior picked up by RMT by capturing the subtle arithmetic properties found in the correction factor. Though previous conjectures deviated from empirical data, we manage to accurately match the theory to numerical computations by including these lower order terms. Our work extends the ratio conjecture's machinery to a new family of Hecke $L$-functions, and demonstrates the application of this machinery to a question of longstanding arithmetic interest.

Paul Kinlaw, Husson University
An Explicit Lower Bound for the Counting Function of Squarefree Products of Three Primes.
We will look at recent joint work with Jonathan Bayless on the problem of determining a tight explicit lower bound for the counting function of squarefree products of three prime factors. Time permitting, we will discuss related problems.

Hershy Kisilevsky, Concordia University
The Non-Square Part of Analytic Sha
Let $L(E/\mathbb{Q},s)$ be the $L$-function of an elliptic curve $E$ defined over the rational field $\mathbb{Q}$. We examine the central value $L(E,1,\chi)$ of non-vanishing twists of $L(E/\mathbb{Q},s)$ by Dirichlet characters $\chi$. In particular we show that the square-free part of the algebraic part of $L(E/F,1)$ is non-trivial for infinitely many extensions $F/\mathbb{Q}$.

Kim Klinger-Logan, University of Minnesota (grad)
Differential equations in automorphic forms
Physicists such as Greene, Vanhove, et al. show that differential equations involving automorphic forms govern the behavior of gravitons. One particular point of interest is solutions to $(\Delta-\lambda)u=E_{\alpha} E_{\beta}$ on the exceptional group $E_8$. We use spectral theory solve $(\Delta-\lambda)u=E_{\alpha}E_{\beta}$ on the more natural space $\operatorname{SL}_2(\mathbb{Z})\backslash \operatorname{SL}_2(\mathbb{R})$. The construction of such a solution uses Arthur truncation, the Maass-Selberg formula, and automorphic Sobolev spaces.

Dimitris Koukoulopoulos, Université de Montréal
Pretentious methods for L-functions
I will discuss ongoing work with K. Soundararajan on proving general prime number theorems that extend classical results for L-functions. Our main theorem gives a classification of those multiplicative functions whose partial sums present significant cancellation. As a byproduct of our methods, a new proof of the classical zero-free regions for L-functions is obtained via sieve theory.

Dijana Kreso, University of British Columbia
Decomposable recursively defined polynomials
In this talk I will present results that come from a joint work with Clemens Fuchs and Christina Karolus from University of Salzburg (Austria). We studied complex polynomials defined by a recurrence relation, which are decomposable, i.e. representable as a composition of lower degree polynomials. Let $(G_n(x))_{n=0}^{\infty}$ be a sequence of complex polynomials defined by a second order recurrence relation. Assuming $G_n(x)=g(h(x))$ where $\deg g, \deg h\geq 2$, we showed that under certain assumptions, and provided that $h(x)$ is not of particular type, $\deg g$ may be bounded by a constant independent of $n$. Our result is motivated by and similar in flavor to a result of Zannier from 2007 about lacunary polynomials. The possible ways of writing a polynomial as a composition of lower degree polynomials were studied by several authors. Results in this area of mathematics have applications to various other areas, e.g. number theory, complex analysis, etc. In my talk I will mention some Diophantine applications.

Niko Laaksonen, McGill University
Prime Geodesic Theorem in $\mathbb{H}^{3}$
On hyperbolic manifolds the lengths of primitive closed geodesics (prime geodesics) have many similarities with the usual prime numbers. In particular, they obey an asymptotic distribution analogous to the Prime Number Theorem. In this talk we present improvements on estimates for the error term in the Prime Geodesic Theorem for some hyperbolic 3-manifolds.

Matilde Lalin, Université de Montréal
A geometric generalization of the square sieve and applications to cyclic covers
We study a generalization of the quadratic sieve to a geometric setting. We apply this to counting points of bounded height on an l-cyclic cover over the rational function field and we consider a question of Serre. In addition to the geometric quadratic sieve, we use Fourier analysis over function fields, deep results of Deligne and Katz about cancelation of mixed character sums over finite fields, and a bound on the number of points of bounded height due to Browning and Vishe.
This is joint work with A. Bucur, A. C. Cojocaru and L. B. Pierce.

Antonio Lei, Université Laval
Second Chern Classes for Supersingular Elliptic Curves
Classical Iwasawa Theory often studies Galois modules up to pseudo-null. For example, the main conjecture relates the characteristic ideal of a Galois module to a p-adic L-function, which does not contain any information about pseudo-null submodules contained inside the Galois module.
A recent work of Bleher-Chinburg-Greenberg-Kakde-Pappas-Sharifi-Taylor describes some pseudo-null modules in terms of Katz' p-adic L-functions. In this talk, we will explain how to obtain a similar result for plus and minus p-adic L-functions coming from elliptic curves with supersingular reduction. This is joint work with Bharathwaj Palvannan.

Robert Lemke-Oliver, Tufts University
Three-isogeny descent for quadratic twists of abelian varieties
We determine the average size of the $\phi$-Selmer group in any quadratic twist family of abelian varieties having an isogeny $\phi$ of degree 3 over any number field. This has several applications towards the rank statistics in such families of quadratic twists. For example, it yields the first known quadratic twist families of absolutely simple abelian varieties over $\mathbf{Q}$, of dimension greater than one, for which the average rank is bounded; in fact, we obtain such twist families in arbitrarily large dimension. In the case that $E/F$ is an elliptic curve admitting a 3-isogeny, we prove that the average rank of its quadratic twists is bounded; if $F$ is totally real, we moreover show that a positive proportion of these twists have rank 0 and a positive proportion have $3$-Selmer rank 1. We also obtain consequences for torsion in Tate-Shafarevich groups of elliptic curves. This is joint work with Manjul Bhargava, Zev Klagsbrun, and Ari Shnidman.

Claude Levesque, Université Laval
On Thue equations.
This will be a succinct report on the results of Michel Waldschmidt and myself on the Thue equations $F(X,Y)=m$.

Wanlin Li, University of Wisconsin (grad)
Vanishing of hyperelliptic L-functions at the central point
We obtain a lower bound on the number of quadratic Dirichlet L-functions over the rational function field which vanish at the central point $s = 1/2$. This is in contract with the situation over the rational numbers, where a conjecture of Chowla predicts there should be no such L-functions. The approach is based on the observation that vanishing at the central point can be interpreted geometrically, as the existence of a map to a fixed abelian variety from the hyperelliptic curve associated to the character.

Jared Lichtman (Dartmouth, undergrad), Steven J. Miller (Williams), Eric Winsor (Michigan, undergrad), and Jianing Yang (Colby, undergrad)
Lower order biases in Fourier coefficients of elliptic curve and cuspidal newform families
Random Matrix Theory, originally developed to model energy levels of heavy nuclei, has had remarkable success in predicting the main term of the behavior of zeros of $L$-functions. Unfortunately these models cannot see the lower order terms, where the arithmetic of the family lives; similar to the Central Limit Theorem, the behavior of these terms control the rate of convergence. We prove results on lower order terms for many families of $L$-functions. In particular, we look at one-parameter families of elliptic curves, where these terms are related to understanding the observed excess rank. In all such families previously studied, the second moment of the Fourier coefficients of the $L$-functions had a negative bias (i.e., converged to the main term from below). We extend these techniques to more general families of elliptic curves and prove the bias is present there as well. Building on the Four (Gauss) and Six (Euler) Order Theorem, we also compute arbitrary moments of special families of elliptic curves with $j(T) = \{0, 1728\}$ and see biases in higher moments. As the modularity theorem connects elliptic curves to weight $2$ modular forms, it is natural to consider biases in other families of $L$-functions. Via the Petersson formula, we compute all higher moments of Fourier coefficients from Dirichlet $L$-functions and cuspidal newforms of level $N$ and weight $k$.

Álvaro Lozano-Robledo, University of Connecticut
A probabilistic model for the distribution of ranks of elliptic curves over $\mathbb Q$
In this talk, we propose a new probabilistic model for the distribution of ranks of elliptic curves in families of fixed Selmer rank, and compare the predictions with previous results, and with the databases of curves over the rationals that we have at our disposal. In addition, we document a phenomenon we refer to as Selmer bias that seems to play an important role in the data and in our models.

Christian Maire, Besançon / Cornell University
Analytic Lie extensions of number fields with cyclic fixed points and tame ramification
(Joint work with Farshid Hajir.) In this lecture, I will present an extension of the strategy of Boston from the 90's concerning the tame version of the Fontaine-Mazur conjecture. More precisely, we are interested in using group-theoretical information to derive consequences for tamely ramified Galois representations, specially for the groups Sl(n,Z_p).

Robert McDonald, University of Connecticut (grad)
Torsion Subgroups of Elliptic Curves over Function Fields of Genus 0
Let $K=\mathbb F_q(T)$ be a function field over a finite field of characteristic $p$, and $E/K$ be an elliptic curve. It is known that $E(K)$ is a finitely generated abelian group, and that for a given $p$, there is a finite, effectively calculable, list of possible torsion subgroups which can appear. In this talk, we show the complete list of possible full torsion subgroups which can appear, and appear infinitely often, for a given $p$.

Xianchang Meng, CRM, Université de Montréal
Chebyshev's bias for total number of prime factors of arithmetic progressions
We consider the total number of prime factors for integers up to x among different arithmetic progressions. Numerical experiments suggest that some arithmetic progressions have more number of prime factors than others. Greg Martin conjectured that this strong bias should be true for all x large enough. Under the Generalized Riemann Hypothesis and Linear Independence Conjecture, we prove that this phenomenon is true for a set of x with logarithmic density 1.

Michael Mossinghoff, Davidson College
The Lind-Lehmer constant for certain $p$-groups
In 2005, Lind formulated an analogue of Lehmer's well-known problem regarding the Mahler measure for general compact abelian groups, and defined a Lehmer constant for each group. Since then, this Lind-Lehmer constant has been determined for many finite abelian groups, including all but a thin set of finite cyclic groups. We establish some new congruences satisfied by the Lind Mahler measure for finite $p$-groups, and use them to determine the Lind-Lehmer constant for many groups of this form. We also develop an algorithm that determines a small set of possible values for a given $p$-group of a particular form. This method is remarkably effective, producing just one permissible value in all but a handful of trials. This is joint work with Dilum De Silva, Vincent Pigno, and Chris Pinner.

Michael Musty, Dartmouth College (grad)
2-solvable Belyi maps
In this talk we consider Galois Belyi maps that have a 2-solvable Galois group. By a theorem of Beckmann, the associated curve has good reduction away from 2. We discuss a method to explicitly compute such Belyi maps and provide examples.

Vincent Ouellet, Université Laval (grad)
On the middle prime factors of integers
We obtain an asymptotic formula for the sum of the reciprocals of the so-called $\beta$-positioned prime factors of integers $n\le x$ counting multiplicities, thus improving and generalizing an earlier result of De Koninck and Luca about the middle prime factor of an integer. This is joint work with Jean-Marie De Koninck and Nicolas Doyon (Laval University).

Corentin Perret-Gentil, CRM, Montréal
Integral monodromy groups and applications in number theory
I will talk about the determination of integral monodromy groups associated to Kloosterman sums over finite fields, and applications to the study of the distribution of the latter.

Carl Pomerance, Dartmouth College
New results on an ancient function
I refer to the function of Pythagoras that sends $n$ to $s(n)$, the sum of the divisors of $n$ that are less than $n$. A still-open conjecture of Catalan & Dickson: each orbit in the $s$-dynamical system (i.e., $n,s(n),s(s(n)),\dots$) is bounded. Modeling such a sequence as a random geometric progression, Bosma & Kane showed that the average of $\log(s(n)/n)$ for $n$ even is a constant $\lambda<0$ (and for $n$ odd, it's $-\infty$). A new result: the average of $\log(s(s(n))/s(n))$ for $n$ even is also $\lambda$. Pythagoras noted 2-cycles in the $s$-dynamical system, the so-called amicable numbers. It's been known since 1981 that the reciprocal sum of the amicable numbers is finite, and in 2011 Bayless & Klyve showed this sum is $<656{,}000{,}000$. Recently with Nguyen we improved the bound to $222$. I also report on some new results with Pollack & Thompson on fibers of $s$.

Gautier Ponsinet, Université Laval (grad)
On the stucture of signed Selmer groups for abelian varities at supersingular primes
Let $p$ be an odd prime and $A$ an abelian varietiy defined over $Q$. When studying Iwasawa theory for $A$, one looks at the structure of the $p$-Selmer groups associated to $A$ over the $\mathbb{Z}_p$ cyclotomique extension. Unfortunately, when $A$ has good supersingular reduction at $p$, this Selmer group is not a cotorsion module over the Iwasawa algebra. (All these terms will be explained in the talk). Following ideas of S. Kobayashi and F. Sprung, K. Büyükboduk and A. Lei define signed Selmer groups for $A$ which are conjectured to be cotorsion module. In this talk, assuming this conjecture, I will explain some results about the structure of these signed Selmer groups similar to classical results about the $p$-Selmer groups in the ordinary case.

Kyle Pratt, University of Illinois Urbana Champaign (grad)
A lower bound for the least prime in an arithmetic progression
Fix $k$ a positive integer, and let $\ell$ be coprime to $k$. Let $p(k,\ell)$ denote the smallest prime in the residue class $\ell \pmod{k}$, and set $P(k)$ to be the maximum of $p(k,\ell)$ over residue classes $\ell$. In this talk I will describe joint work with Junxian Li and George Shakan, in which we showed that for almost every $k$ one has
$P(k) \gg \phi(k) \log k \log_2 k \log_4 k/\log_3 k$.
This improves an earlier bound of Pomerance, and answers a question of Ford, Green, Konyagin, Maynard, and Tao. I will discuss some ideas used in the proof, which has roots in recent work on large gaps between primes. I will also discuss some heuristics about the size of $P(k)$.

Sam Schiavone, Dartmouth College (grad)
Computing A Database of Belyi Maps: A Progress Report
In the paper Numerical calculation of three-point branched covers of the projective line, we presented a method for computing equations of Belyi maps based on the correspondence described by Grothendieck in his celebrated work Esquisse d'un Programme. In this talk, we discuss the progress we have made in exhaustively computing all Belyi maps of low degree using this method. Joint work with Michael Musty, Jeroen Sijsling, and John Voight.

Ari Shnidman, Boston College
Intersections of Heegner-Drinfeld cycles
I'll present a formula which relates the intersection of two Heegner-Drinfeld cycles in the moduli stack of shtukas to derivatives of certain toric period integrals. The formula is a "higher order" generalization of the Gross-Kohnen-Zagier formula, in the function field setting, and is inspired by recent work of Yun and W. Zhang. This is joint work with Ben Howard.

Caleb Shor, Western New England University
A characterization of the complement of a free numerical semigroup
Let $\mathbb{N}_0$ denote the set of non-negative integers. A numerical semigroup $S$ is a subset of $\mathbb{N}_0$ that is closed under addition, contains 0, and has finite complement in $\mathbb{N}_0$. Two interesting quantities are the Frobenius number and genus of $S$ which are, respectively, the maximum and cardinality of its complement. Every numerical semigroup can be written as the set of non-negative linear combinations of elements of some finite set $G=\{g_1,\dots,g_k\}$ with $\gcd(G)=1$. If $k\le 2$, then formulas for the Frobenius number and genus of $S$ are well-known, dating to the work of Sylvester in the 19th century. If $k\ge 3$, far less is known.
In this talk, we will consider free numerical semigroups, which are generated by sets of integers that can be ordered to satisfy some certain smoothness conditions. (Examples of such sets include geometric sequences, compound sequences, supersymmetric sequences, and certain arithmetic sequences of length 3.) The smoothness conditions lead to an explicit description of the Apéry set of a numerical semigroup, which we can use along with a generalization of an identity of Tuenter. This gives us a complete characterization of the complement of a free numerical semigroup, leading to results for the Frobenius number, the genus, power sums of complement elements, and other interesting quantities. As an application we will see how to quickly sum all of the so-called "non-McNugget numbers."

Daniel Smertnig, Dartmouth College
Arithmetical invariants of local quaternion orders
In a Noetherian ring every non-zero-divisor can be written as a product of atoms (irreducible elements). Usually, these factorizations are not unique. Arithmetical invariants quantify the failure of this uniqueness.
In the noncommutative setting, recently arithmetical invariants of hereditary orders in central simple algebras over global fields have been studied. The non-hereditary case has so far not been considered. As a first step towards this, we consider quaternion orders over discrete valuation rings. We characterize finiteness of the elasticity, a basic invariant. The proof, which splits into several cases, relies on a classification of quaternion orders in terms of ternary quadratic forms.
Joint work with Nicholas R. Baeth.

Naomi Tanabe, Bowdoin College
Sign of Fourier Coefficients for Hilbert Modular Forms
Signs of Fourier coefficients have been studied extensively in various setting. In this talk, I will survey some results on sign changes for Hilbert modular forms and explain our resent result concerning half integral weight forms. It is a joint project with Surjeet Kaushik and Narasimha Kumar.

Guhan Venkat, Université Laval
Plus/Minus Beilinson - Flach Euler system
In this talk, we will see how one can ontain an integral Plus/Minus Euler system associated to the Rankin Selberg product of a pair of supersingular modular forms starting from an Euler system if rank 2. The results have Iwasawa theoretic applications and is joint work with Kazim Buyukboduk, Antonio Lei and David Loeffler.

Jan Vonk, McGill University
Fundamental groups and rational points on curves
It has long been known that the number of rational points on a curve of genus at least 2 has finitely many rational points. I will discuss a theory of Minhyong Kim which often provides alternative proofs, that have the advantage they can be made effective. This will be applied to the "cursed modular curve" $X_{ns}(13)$, resulting in a complete determination of rational points where traditional methods fail. This is joint work with Balakrishnan, Dogra, Müller, and Tuitman.

Erik Wallace, University of Connecticut (grad)
Bounds of the rank of the Mordell--Weil group of Jacobians of Hyperelliptic Curves
In a recent article written in collaboration with Álvaro Lozano-Robledo and Harris Daniels we extend work of Lehmer, Shanks, and Washington on cyclic extensions, and elliptic curves associated to the simplest cubic fields. In particular, we give families of examples of hyperelliptic curves $C: y^2=f(x)$ defined over $\mathbb{Q}$, with $f(x)$ of degree $p$, where $p$ is a Sophie Germain prime, such that the rank of the Mordell--Weil group of the jacobian $J/\mathbb{Q}$ of $C$ is bounded by the genus of $C$ and the $2$-rank of the class group of the (cyclic) field defined by $f(x)$, and exhibit examples where this bound is sharp.

Liang Xiao, University of Connecticut
Some remarks on the ghost conjecture of Bergdall and Pollack
The goal of this talk is to understand the $p$-adic valuations of the $U_p$-operators acting on the space of modular forms. In 1990s, Gouvea and Mazur computed many numerical data and made interesting conjectures, such as the Gouvea-Mazur conjecture, and Gouvea's conjecture on distribution of slopes. This topic was later developed by Buzzard and Calegari, and their students, by providing finer computational data and more precise conjectures. Recently, Bergdall and Pollack introduced the ghost conjecture that unifies these previous conjectures, and give the precise information of slopes of modular forms, at least in the "Buzzard-regular" case. In this talk, I will reformulate Bergdall and Pollack's conjecture in a purely abstract setting, and explain where it fits in the p-adic local Langlands conjecture for $GL_2(\mathbf{Q}_p)$. If time permits, I will discuss some theoretical evidence of this conjecture. This is a joint work with Ruochuan Liu and Bin Zhao.

Shucheng Yu, Boston College (grad)
Logarithm laws for hitting time functions on homogeneous spaces
In a probability space with an ergodic action, fix a measurable set of positive measure, the first hitting time function measures the time needed for the orbit of a point to enter this set. Intuitively, the smaller the set is, the larger the hitting time is. We establish a logarithm law for hitting time functions on a finite volume homogeneous space with the flow generated by a group element. The main ingredient of our proof is a uniform rate of decay for matrix coefficients along any unbounded flow. This is joint work with Dubi Kelmer.

Fan Zhou, University of Maine
Voronoi formula and its generalizations
The Voronoi formula is a Poisson-style summation formula connecting a sum of Fourier coefficients of an automorphic form twisted by additive characters and a contragredient sum. Firstly we present a proof of the Voronoi formula for coefficients of a large class of L-functions, in the style of the classical converse theorem of Weil. Our formula applies to full-level cusp forms, Rankin-Selberg convolutions, and certain isobaric sums. The key ingredient is the construction of a double Dirichlet series associated with these coefficients. This is joint work with Eren Mehmet Kiral. Secondly we introduce balanced Voronoi formulas, with hyper-Kloosterman sums of various dimensions on both sides. Those generalize a formula for GL(4) with ordinary Kloosterman sums on both sides by Miller-Li that was used by Blomer-Li-Miller to prove nonvanishing of GL(4) L-functions by GL(2)-twists. This is joint work with Stephen D. Miller.

## List of Participants

Brandon Alberts, University of Wisconsin (grad)
David Ayotte, Université Laval (grad)
Angelica Babei, Dartmouth College (grad)
Lior Bary-Soroker, Tel-Aviv University
David Bradley, University of Maine
Benjamin Breen, Dartmouth College (grad)
Crystel Bujold, Université de Montréal (grad)
Hugo Chapdelaine, Université Laval
Sara Chari, Dartmouth College (grad)
Jaeho Choi, University of Maine (grad)
Michael Chou, Tufts University
Edgar Costa, Dartmouth College
Harris Daniels, Amherst College
Henri Darmon, McGill University
David Dummit, University of Vermont
Michele Fornea, McGill University (grad)
John Friedlander, University of Toronto
Zhenguang Gao, Framingham State University
Ramesh Gupta, University of Maine
Pushpa Gupta, University of Maine
Anna Haensch, Duquesne University
Farshid Hajir, University of Massachusetts
Henryk Iwaniec, Rutgers University
Daniel Keliher, Tufts University (grad)
Dubi Kelmer, Boston College
Yujin Kim, Columbia University (undergrad)
Paul Kinlaw, Husson University
Hershy Kisilevsky, Concordia University
Kim Klinger-Logan, University of Minnesota (grad)
Andrew Knightly, University of Maine
Dimitris Koukoulopoulos, Université de Montréal
Dijana Kreso, University of British Columbia
Niko Laaksonen, McGill University
Matilde Lalin, Université de Montréal
Antonio Lei, Université Laval
Robert Lemke-Oliver, Tufts University
Claude Levesque, Université Laval
Wanlin Li, University of Wisconsin (grad)
Jared Lichtman, Dartmouth College (undergrad)
David Lilienfeldt, McGill University (grad)
Abel Lourenco, University of Maine (grad)
Álvaro Lozano-Robledo, University of Connecticut
Mostafa Mache, Université Laval (grad)
Christian Maire, Besançon / Cornell University
Robert McDonald, University of Connecticut (grad)
Xianchang Meng, CRM, Université de Montréal
Steven J. Miller, Williams College
Michael Mossinghoff, Davidson College
Michael Musty, Dartmouth College (grad)
Jungbae Nam, Concordia University (grad)
Kunjakanan Nath, Université de Montréal (grad)
Isabella Negrini, McGill University (grad)
Vincent Ouellet, Université Laval (grad)
Solly Parenti, University of Wisconsin (grad),
Zachary Parker, University of Vermont (grad),
Corentin Perret-Gentil, Université de Montréal
Nigel Pitt, University of Maine
Carl Pomerance, Dartmouth College
Gautier Ponsinet, Université Laval (grad)
Kyle Pratt, University of Illinois Urbana-Champaign (grad)
James Rickards, McGill University (grad)
Alicia Rossi, University of Vermont (undergrad)
Giovanni Rosso, Concordia University
Sam Schiavone, Dartmouth College (grad)
Ari Shnidman, Boston College
Caleb Shor, Western New England University
Daniel Smertning, Dartmouth College
Jerrod Smith, University of Maine
Chip Snyder, University of Maine
Shannon Sweitzer, UC Riverside (undergrad)
Naomi Tanabe, Bowdoin College
Guhan Venkat, Université Laval
Christelle Vincent, University of Vermont
Jan Vonk, McGill University
Erik Wallace, University of Connecticut
Zheng Wei, University of Maine
Benjamin Weiss, University of Maine
Eric Winsor, University of Michigan (undergrad)
Alic Wise, University of Maine (grad)
Liang Xiao, University of Connecticut
Peter Xu, McGill University (grad)
Jianing Yang, Colby College (undergrad)
Shucheng Yu, Boston College (grad)
Peter Zenz, McGill University (grad)
Fan Zhou, University of Maine