# 2019 Maine/Québec Number Theory Conference

The University of Maine
October 5-6, 2019

Financial Support
The National Science Foundation
The Journal of Number Theory (Elsevier)
At the University of Maine:
The College of Liberal Arts and Sciences
The Office of the Vice President for Research
The Department of Mathematics and Statistics

Full List of Participants

## Saturday, October 5, 2019 - Donald P. Corbett Business Building (DPC)

Click a name to view the abstract for the talk
 8:30 - 8:55 Coffee/Tea/Snacks, DPC Lobby Time 100 DPC 8:55-9:00 Welcoming Remarks Emily Haddad, Dean of the College of Liberal Arts and Sciences 9:00-9:50 Andrew Granville Distributions in number theory; inspiration from mathematical physics 117 DPC 115 DPC 107 DPC 105 DPC 10:00-10:20 Adam Logan Integral points on continued fraction varieties slides Allysa Lumley Distribution of Values of L-functions over Function Fields Jonathan Sands A Stickelberger theorem for L-functions of graph coverings. David Krumm Algebraic preperiodic points of entire transcendental functions 10:30-10:50 Brandon Alberts Counting Towers of Number Fields Sacha Mangerel Multiplicative Functions in Short Arithmetic Progressions and Applications Amy DeCelles The automorphic heat kernel, spectral and geometric points of view slides Julian Rosen Iterates of a derivation in positive characteristic Break 11:10-11:30 Asher Auel Brauer classes split by genus one curves Aled Walker Linear inequalities in primes Robert Hough Equidistribution of the lattice shape of number fields Cathy Swaenepoel Prime numbers with preassigned digits 11:40-12:00 Jennifer Park Everywhere local solubility for hypersurfaces in products of projective spaces Eran Assaf Computing Modular Forms for Arbitrary Congruence Subgroups David Lowry-Duda Consequences of spectral poles in arithmetic problems Dylan King and Catherine Wahlenmayer Crescent Configurations In Non-Euclidean Norms Lunch: A light deli lunch is provided in Wells Conference Center, Room 2 (map). Go upstairs. (The Bear's Den Pub in the Memorial Union is also open for lunch.) Time 117 DPC 115 DPC 107 DPC 105 DPC 1:30-1:45(students) Tomer Reiter Isogenies of Elliptic Curves over $\mathbb{Q}(2^{\infty})$ David Lilienfeldt Generalised Heegner cycles and Griffiths goups of infinite rank slides Andrei Shubin Small gaps between primes from subsets Crystel Bujold Long large character sums 1:55-2:10 (students) Garen Chiloyan A classification of isogeny-torsion graphs. slides Debanjana Kundu Iwasawa Theory of Fine Selmer Groups Bin Guan Averages of central values of triple product L-functions William Verreault Bounds for the counting function of the Jordan-Pólya numbers 2:20-2:40 Céline Maistret Arithmetic invariants and parity of ranks of abelian surfaces. Steven J. Miller Optimal Test Functions for $n$-Level Densities and Applications to Central Point Vanishing video   slides Josh Zelinsky On the total number of prime factors of an odd perfect number slides 2:50-3:10 Antonio Lei Congruences of anticyclotomic p-adic L-functions Naomi Tanabe Additive Twists of Fourier Coefficients Carl Pomerance Primitive sets slides Tea/Refreshments 3:40-4:00 Ciaran Schembri Examples of genuine QM abelian surfaces which are modular Yeongseong Jo Rankin-Selberg L-functions via Good Sections Maria Nastasescu Mean values and subconvexity results for a degree 8 Euler product 4:10-4:30 John Voight Counting elliptic curves with an isogeny of degree three slides Fatma Cicek A Discrete Analogue of Selberg's Central Limit Theorem Thomas Hulse The Impossible Vanishing Spectrum slides Paul Kinlaw A Variation of Mertens' Theorem for Almost Primes slides 4:40-4:55(students) Caleb McWhorter Torsion of Rational Elliptic Curves over Nonic Galois Fields Anthony Doyon Explicit anticyclotomic extensions Isabella Negrini On the classification of rigid meromorphic cocycles slides Kunjakanan Nath Primes of the form $a^2+b^2+h$ for some $h \in \{1, \dotsc, H\}$ in tuples 5:05-5:20(students) Soumya Sankar Proportion of ordinary curves in some families Grant Molnar The Arithmetic of Modular Grids slides Nha Truong An example of explicit computation of the Hecke operator. slides Antoine Poulin Probabilistic properties of p-adic polynomials Dinner: High Tide Restaurant, 5 South Main Street, Brewer. Light hors d'oeuvres at 6:30; Seating at 6:45

## Sunday, October 6, 2019 - Donald P. Corbett Business Building (DPC)

 8:30 - 9:00 am Coffee/Tea/Snacks, DPC Lobby Time 117 DPC 115 DPC 107 DPC 105 DPC 9:00-9:15(students) Hao Li Congruence relation of GSpin Shimura varieties Benjamin Breen Heuristics for abelian fields: totally positive units and narrow class groups. Daniel Keliher Comparing the number of $D_4$ and $S_4$ quartic extensions of function fields Cédric Dion On the $\pm$ factorization of $p$-adic $L$-functions associated to an elliptic curve with supersingular reduction 9:25-9:40(students) Mostafa Mache GL(2) real analytic Eisenstein series twisted by quasi-characters Hugues Bellemare Finding explicit uniformizers of $\mathbb{Q}_p(\zeta_{p^2},\sqrt[p]{p})$ Matthew Friedrichsen A Bias Towards $D_4$ Extensions of Quadratic Number Fields slides James Rickards Intersection Numbers of Modular Geodesics slides 9:50-10:10 Hugo Chapdelaine Introduction to Green functions and theta lifting identities slides Elliot Benjamin The narrow 2-class field tower of some real quadratic number fields with 2-class group isomorphic to $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$ slides Claire Burrin Windings of prime geodesics 10:20-10:40 Keshav Aggarwal A new subconvex bound for GL(3) in t-aspect Sara Chari Metacommutation of primes in locally Eichler orders Scott Ahlgren Maass forms and the mock theta function f(q) Break 11:00-11:20 Li-Mei Lim Zeros of L-Functions of Half-Integral Weight Automorphic Forms David Dummit Signature Ranks of Units in Real Biquadratic and Multiquadratic Extensions (joint with H. Kisilevsky) slides Matthew Welsh Roots of cubic congruences 11:30-lunchtime Roman Holowinsky Simple delta methods and subconvexity problems

## Abstracts (by last name)

Keshav Aggarwal, University of Maine
A new subconvex bound for GL(3) in t-aspect
We revisit Munshi's proof of the t-aspect subconvex bound for GL(3) L-functions, and we are able to remove the 'conductor lowering' trick. This simplification along with a more careful stationary phase analysis allows us to improve Munshi's bound to $L(1/2+it,\pi) \ll_{\pi, \epsilon} (1+|t|)^{3/4-3/40+\epsilon}$.

Scott Ahlgren, University of Illinois at Urbana-Champaign
Maass forms and the mock theta function f(q)
Let f(q) be the "third order" mock theta function of Ramanujan. In 1964, George Andrews proved an asymptotic formula for its Fourier coefficients, and made two conjectures about the asymptotic series (these coefficients have an important combinatorial interpretation). The first of these conjectures was proved in 2009 by Bringmann and Ono. Here we prove the second conjecture, and we obtain a power savings bound in Andrews' original asymptotic formula. The proofs rely on uniform bounds for sums of Kloosterman sums which follow from the spectral theory of Maass forms of half integral weight and in particular from a new estimate which we derive for the Fourier coefficients of such forms. This is joint work with Alexander Dunn.

Brandon Alberts, University of Connecticut
Counting Towers of Number Fields
Fix a number field $K$ and a finite transitive subgroup $G \le S_n$. Malle's conjecture proposes asymptotics for counting the number of G-extensions of number fields $F/K$ with discriminant bounded above by X. A recent and fruitful approach to this problem introduced by Lemke Oliver, Wang, and Wood is to count inductively. Fix a normal subgroup $T\lhd G$. Step one: for each $G/T$-extension $L/K$, first count the number of towers of fields $F/L/K$ with ${\rm Gal}(F/L) \cong T$ and ${\rm Gal}(F/K)\cong G$ with discriminant bounded above by $X$. Step two: sum over all choices for the $G/T$-extension $L/K$. In this talk we discuss the close relationship between step one of this method and the first Galois cohomology group. This approach suggests a refinement of Malle's conjecture which gives new insight into the problem. We give the solution to step one when $T$ is an abelian normal subgroup of $G$, and convert this into nontrivial lower bounds for Malle's conjecture whenever $G$ has an abelian normal subgroup.

Eran Assaf, Dartmouth College
Computing Modular Forms for Arbitrary Congruence Subgroups
Let $\Gamma$ be a congruence subgroup of $\operatorname{PSL}_2(\mathbb{Z})$. Current methods for explicitly computing the space of modular forms $M_k(\Gamma)$ reduce to the cases of the standard groups $\Gamma_0$ and $\Gamma_1$. However, this reduction entails a significant increase of the level. In our current work, we compute those spaces directly, thus reducing computational costs.

Asher Auel, Dartmouth College
Brauer classes split by genus one curves
It is an open problem, even over the rational numbers, to decide whether every Brauer class is split by the function field of a genus one curve. The problem has been solved for Brauer classes of index at most 5 over any field. In this talk, I'll report on how new cases can be settled by an analysis of certain torsors over the moduli space of pointed genus one curves that arise from classical geometric constructions. This is joint work with Danny Krashen.

Finding explicit uniformizers of $\mathbb{Q}_p(\zeta_{p^2},\sqrt[p]{p})$
Let $p$ be an odd prime. While studying extensions of $\mathbb{Q}_p$, the field of $p$-adic numbers, one encounters the totally ramified extensions $\mathbb{Q}_p(\zeta_{p^m})$ and $\mathbb{Q}_p(\sqrt[p^n]{p})$. Both of these extensions have natural uniformizers: $\zeta_{p^m}-1$ is a uniformizer for $\mathbb{Q}_p(\zeta_{p^m})$ while $\sqrt[p^n]{p}$ is a uniformizer for $\mathbb{Q}_p(\sqrt[p^n]{p})$. It is thus natural to seek for a uniformizer of $K_{m,n} = \mathbb{Q}_p(\zeta_{p^m},\sqrt[p^n]{p})$, the compositum of these two extensions. There is an easy answer for $K_{1,n}$, but as soon as $m \geq 2$, things get more complicated. The goal of this talk is to present a method that gives a uniformizer for the extension $K_{2,1} = \mathbb{Q}_p(\zeta_{p^2},\sqrt[p]{p})$. If time permits, we will discuss difficulties encountered when trying to adapt the method to more general cases.

Elliot Benjamin, Capella University
The narrow 2-class field tower of some real quadratic number fields with 2-class group isomorphic to $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$
Let $k$ be a real quadratic number field with 2-class group isomorphic to $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$, norm of fundamental unit equal to 1, and discriminant $d_k$ divisible by only positive prime discriminants. We distinguish between k having narrow 2-class field tower length 2 and $\ge 3$ based upon the biquadratic residue symbols of two of the primes dividing $d_k$, or equivalently the rank of the 2-class group of the narrow 2-class field of k. This is joint work with Chip Snyder.

Heuristics for abelian fields: totally positive units and narrow class groups.
We describe heuristics in the style of Cohen-Lenstra for narrow class groups and units in abelian extensions of odd degree. These results stem from a model for the 2-Selmer group of a number field. We conclude with computational evidence for cyclic extensions of degree n = 3,5,7.

Crystel Bujold, University of Montréal (grad)
Long large character sums
In this talk, I will discuss a lower bound for the maximal value achieved by a character sum of length $q/(\log q)^B$, as we go through the characters modulo $q$. As an aside, I will also mention an interesting result on lattices that arises as a by-product of the proof.

Claire Burrin, Rutgers University
Windings of prime geodesics
Closely related to the study of the multiplier system of Dedekind's eta-function is the Rademacher function, which measures the winding of any closed geodesic around the cusp of the modular orbifold. Ten years ago, Sarnak computed the statistics of these windings using Selberg's trace formula, and in joint work with Flemming von Essen, we extend his results and the setting to counting windings of closed geodesics on any cusped hyperbolic orbifold. Moreover, we obtain as an application of our results the following analogue of Dirichlet's prime number theorem: there exist infinitely many closed primitive geodesics whose winding number lies in a prescribed arithmetic progression.

Hugo Chapdelaine, Université Laval
Introduction to Green functions and theta lifting identities
In this talk we shall give an introduction to Green functions and explain how they relate to classical automorphic objects. By the end of the talk we shall discuss a theta lifting identity which involves Green functions on different quatrnion algebras. Such identities have been discovered in the 70s by John Fay and they arise precisely because Green functions satisfy a "reproducing identity". If time permits, we shall also explain how such identities, when combined simultaneously with "Borcherds unfolding trick" and the Selberg point-pair invariants, can potentially be used to obtain non-trivial relations between automorphic data associated to the initial pair of quatenion algebras.

Sara Chari, Bates College
Metacommutation of primes in locally Eichler orders
We study the metacommutation problem in locally Eichler orders. From this arises a permutation of the set of locally principal left ideals of a given prime reduced norm. Previous results on the cycle structure were determined for locally maximal orders. As we extend these results, we present an alternative, combinatorial description of the metacommutation permutation as an action on the Bruhat-Tits tree.

Garen Chiloyan, University of Connecticut (grad)
A classification of isogeny-torsion graphs.
An isogeny graph is a nice visualization of the isogeny class of an elliptic curve. A theorem of Kenku shows sharp bounds on the number of distinct isogenies that a rational elliptic curve can have (in particular, every isogeny graph has at most 8 vertices). In this talk, we classify what torsion subgroups over $\mathbb{Q}$ can occur in each vertex of a given isogeny graph of elliptic curves defined over the rationals. This is joint work with Álvaro Lozano-Robledo.

Fatma Cicek, University of Rochester (grad)
A Discrete Analogue of Selberg's Central Limit Theorem
One of the most influential probabilistic results in analytic number theory is Selberg's central limit theorem. Roughly, it states that the logarithm of the Riemann zeta-function on the critical line has an approximate two-dimensional normal distribution. We assume RH and write $\rho=\frac12+i\gamma$ to consider the distribution of the following sequences
$\displaystyle\log{\zeta(\rho+z)} \qquad \hbox{and} \qquad \log{|\zeta^\prime(\rho)|}$
for $0 < \gamma \leq T$ where $z$ is a sufficiently small nonzero complex number and $T$ is large. Our results show that on the further assumption of a certain zero-spacing hypothesis, analogues of Selberg's central limit theorem hold for these sequences.

Amy DeCelles, University of St. Thomas, Minnesota
The automorphic heat kernel, spectral and geometric points of view
The automorphic heat kernel has many applications in number theory, e.g. estimates on the number of cuspidal eigenvalues of the automorphic Laplacian below a given bound, integral representations for zeta functions, sup-norm bounds on average for cusp forms. Using global automorphic Sobolev theory, we construct an automorphic heat kernel via an automorphic spectral expansion in terms of cusp forms, Eisenstein series, and residues of Eisenstein series. We prove the uniqueness of the automorphic heat kernel as an application of operator semigroup theory. Changing to a geometric perspective, we also construct the automorphic heat kernel by winding up a heat kernel on the corresponding symmetric space G/K. We use known growth estimates for heat kernels on symmetric spaces and the theory of gauges on groups to prove convergence, moderate growth, and square integrability on the quotient.

On the $\pm$ factorization of $p$-adic $L$-functions associated to an elliptic curve with supersingular reduction
Let $E/\mathbb{Q}$ be an elliptic curve over the rational numbers and $p$ be a prime number. A $p$-adic $L$-function $L_p(E,s)$ associated to $E$ is a $p$-adic analytic function that interpolates special values of the complex $L$-function $L(E,s)$. When the reduction type of $E$ is supersingular at $p$, $L_p(E,s)$ is no longer a bounded measure and so $L_p(E,s)$ is not amenable to be studied via classical Iwasawa theory. To solve this problem, Pollack defined plus and minus $p$-adic $L$-functions $L_p^{\pm}(E,s)$ which are bounded measures in the case when $E$ is supersingular at $p$ with $a_p=0$. In this talk, we give a review of the construction of Pollack's $\pm$ $p$-adic $L$-functions and derive some consequences of the functional equation in the same spirit as works of Bianchi and Wuthrich. Finally, we give explicit distributions for the power series $\log_p^{\pm}$ appearing in the factorization of $L_p(E,s)$ into $\pm$ $L$-functions. This is joint work with A. Lei and F. Sprung.

Explicit anticyclotomic extensions
Let p be a prime number and K an imaginary quadratic field. The anticyclotomic Zp-extension of K is the unique Zp-extension of K whose Galois group over Q is dihedral. Under certain circumstances, layers of this extension can be computed explicitly by a recent work of Bröker, Hubbard and Washington published in 2018. In particular, they explain how to compute the defining polynomial of each layer of the extension for the case p=3. In this talk, we will explore some generalizations of this work.

David Dummit, University of Vermont
Signature Ranks of Units in Real Biquadratic and Multiquadratic Extensions (joint with H. Kisilevsky)
We prove a number of results on the unit signature ranks of real biquadratic and multiquadratic fields. For example, we give explicit infinite families of real biquadratic fields K for each of the three possible unit signature ranks 1, 2, or 3, in the case when all three quadratic subfields of K have a totally positive fundamental unit. As one application we prove the rank of the totally positive units modulo squares in the totally real subfield of cyclotomic fields can be arbitrarily large.

A Bias Towards $D_4$ Extensions of Quadratic Number Fields
A result of Bhargava, following work of Cohen, Diaz y Diaz, and Olivier, shows about 83% of quartic fields over $\mathbb{Q}$ are $S_4$, while about 17% are $D_4$. Though we might expect most number fields to have more quartic $S_4$ extensions than $D_4$, this is not the case for quadratic number fields. We study the ratio of $D_4$ extensions to $S_4$ extensions for quadratic number fields and show that 100% of these number fields have arbitrarily more quartic $D_4$ than $S_4$ extensions. This work is joint with Daniel Keliher.

Andrew Granville, Université de Montréal
Distributions in number theory; inspiration from mathematical physics

Averages of central values of triple product L-functions
Feigon and Whitehouse studied central values of triple L-functions averaged over newforms of weight 2 and prime level. They proved some exact formulas applying the results of Gross and Kudla which link central values of triple L-functions to classical "periods". In this talk, I will show more results of this problem for more cases using Jacquet's relative trace formula, and some application of these average formulas to the non-vanishing problem.

Roman Holowinsky, Ohio State University
Simple delta methods and subconvexity problems
We will present several simple delta methods in application to classical subconvexity results in the GL(2) setting. We discuss how these methods are naturally connected to Munshi's Petersson delta method and suggest other applications.

Robert Hough, Stony Brook University
Equidistribution of the lattice shape of number fields
The ring of integers of a degree $n$ number field is a rank $n$ lattice in the canonical embedding, with a short vector in the direction of 1. Call the lattice shape of the number field the projection of this lattice in the direction orthogonal to 1, rescaled to have co-volume 1. I introduce a twisted class of Shintani zeta functions, twisted by an automorphic form, and explain how these objects can be used to give equidistribution statements for the distribution of the shape of low degree number fields ordered by discriminant.

Thomas Hulse, Boston College
The Impossible Vanishing Spectrum
We consider an apparently square-integrable object over its fundamental domain which is relevant to studying arithmetic progressions of squares. Curiously, the discrete and continuous parts of the spectrum of this object seem to vanish, but the object is nonzero. The resolution of this paradox is non-obvious and consequential to the final result. Joint work with Chan Ieong Kuan, David Lowry-Duda and Alexander Walker.

Yeongseong Jo, University of Iowa
Rankin-Selberg L-functions via Good Sections
In 1990's Bump and Ginzburg establish the integral representation yielding symmetric square L-functions for GL(n) and the twisted version is subsequently constructed by Takeda. Unfortunately the local functional equation involves intertwining operator opposed to Fourier transform appearing in the well-known Rankin-Selberg integrals by Jacquet, Piatetski-Shapiro, and Shalika. In this talk, we present the notion of "Good Sections" introduced by Piatetski-Shapiro and Rallis. We show that in Asai, Rankin-Selberg and exterior square cases, non-archimedean local factors via Rankin-Selberg integrals attached to either Schwartz-Bruhat functions or good sections agree. We propose a concept of exceptional poles in the context of good section which can be carried out on the analysis of symmetric square or cubic Asai L-functions.

Comparing the number of $D_4$ and $S_4$ quartic extensions of function fields
In this talk we will describe an asymptotic formula for enumerating quartic dihedral extensions of a function field which is analogous to the one of Cohen, Diaz y Diaz, and Olivier for number fields. We'll use this and other tools to study when a function field has more $D_4$ than $S_4$ quartic extensions and show that by means of base extension we can introduce a bias in favor of $D_4$.

Dylan King, Wake Forest University (undergrad)

Crescent Configurations In Non-Euclidean Norms
We investigate a variant of the Erdös distinct distances problem. We say that $n$ points in the plane lie in general position if no three points lie on a line and no four points lie on a circle. A crescent configuration is a set of $n$ points in the plane in general position such that for all $1 \leq i \leq n - 1$, there is a distance $d_i$ which appears as the distance between two points exactly $i$ times. Erdös, I. Pàlàsti, A. Liu, and C. Pomerance have found crescent configurations for $n \leq 8$, but Erdös conjectured that there exists $N$ such that there do not exist crescent configurations on $N$ or more points. Little further progress has been made on this problem in the past three decades. One can study crescent configurations in the more general setting of a normed metric space $(X, d)$. The original question of Erdös asks for $(\mathbb{R}^2, L^2)$ crescent configurations, while a 2015 paper by D. Burt et al. provides a construction for $(\mathbb{R}^n, L^2)$ configurations for $n\geq 3$ by taking advantage of the freedom of higher dimensional spaces. Since the geometry of a space is heavily dependent upon the norm with which it is equipped, it is natural to nature of crescent configurations to vary with space and norm. We explore the properties of crescent configurations in generic normed spaces, and as a case study examine $(\mathbb{R}^2, L^p)$ for $1 \leq p \leq \infty$. In these non-Euclidean settings, we provide explicit constructions of some crescent configurations and adjust the notion of general position so that the existence of such crescent configurations is non-trivial. We show the existence of crescent configurations on $4$ points in any normed space, and carefully demonstrate a dichotomy in behavior between those norms whose unit balls contain a line segment, such as $L^1$, and those that do not, such as $L^p$ for $p \in (1,\infty)$. Using computational search, we identify crescent configurations on $7$ and $8$ points in $L^1$, $L^\infty$ respectively.

Paul Kinlaw, Husson University
A Variation of Mertens' Theorem for Almost Primes
A recent result of G. Tenenbaum establishes a version of Mertens' second theorem for the sum of $1/(p_1...p_k)$ over all ordered k-tuples of prime numbers with product not exceeding x. Here, terms can occur in the sum more than once, if the order of appearance of different primes in the product is changed. For instance, the case k=3 includes the term 1/30 exactly six times, the term 1/12 exactly three times, and the term 1/8 exactly once. The result was originally established by D. Popa for k=2 and k=3 using the hyperbola method for primes, and then extended by Tenenbaum to arbitrary k using the Selberg-Delange method from complex analysis. The sum is equal to a degree k polynomial in loglog x with coefficients given by an explicit formula, plus an error term which tends to zero as x tends to infinity. We will discuss a variation of this result where each term is counted exactly once. In other words, we consider the reciprocal sum of k-almost primes not exceeding x. Time permitting, we will discuss progress with related problems, such as determining explicit bounds for the error terms and addressing the question of sign changes in the error terms.

David Krumm, Reed College
Algebraic preperiodic points of entire transcendental functions
Motivated by questions in transcendental number theory, K. Mahler asked in 1976 whether there exists an entire transcendental function $f:\mathbb {C}\to\mathbb{C}$ with the property that $f(\overline{\mathbb{Q}})\subseteq\overline{\mathbb{Q}}$ and $f^{-1}(\overline{\mathbb{Q}})\subseteq\overline{\mathbb{Q}}$. Mahler's question was answered in the affirmative by Marques and Moreira in 2016. In this talk we will discuss some dynamical properties of this type of function $f$, in particular the structure of the directed graph of algebraic preperiodic points of $f$.

Debanjana Kundu, University of Toronto (grad)
Iwasawa Theory of Fine Selmer Groups
In this talk we will talk about the Classical Iwasawa $\mu=0$ Conjecture and relate it to Coates-Sujatha Conjecture A for fine Selmer Groups. We will provide new evidence for Conjecture A and if time permits talk about why fine Selmer groups are interesting outside of Iwasawa theory.

Antonio Lei, Université Laval
Let f and g be two modular forms that are congruent modulo a prime number p. Furthermore, suppose that f and g are both ordinary at p. Vatsal proved that the cyclotomic p-adic L-functions attached to f and g are also congruent mod p after choosing appropriate periods. This result has led to several proofs of special cases of the Iwasawa main conjecture (thanks to works of Greenberg-Vatsal and Emerton-Pollack-Weston). Recently, Kriz and Li have obtained results towards Goldfeld's conjecture on quadratic twists of elliptic curves by comparing Heegner points of congruent elliptic curves. Based on the method of Kriz and Li, we show that anticyclotomic p-adic L-functions attached to congruent Hilbert modular forms satisfy a congruent relation, generalizing the result of Vatsal on cyclotomic p-adic L-functions. This is joint work with Daniel Delbourgo.

Congruence relation of GSpin Shimura varieties
I will introduce the background of Eichler-Shimura congruence relation, its modern generalization and my recent work on this topic for GSpin Shimura variety.

Generalised Heegner cycles and Griffiths goups of infinite rank
Algebraic cycles occupy a central role in modern algebraic geometry and number theory. Several deep conjectures are concerned with their behaviour, the most famous of which are the Hodge conjecture, the Tate conjecture and the Beilinson-Bloch conjecture. The latter relates cycles to special values of L-functions. Using a distinguished collection of cycles, known as generalised Heegner cycles, we prove that a certain cycle group has infinite rank. The group in question is called the Griffiths group and measures the discrepancy between homological and algebraic equivalence. This is joint work with Massimo Bertolini, Henri Darmon and Kartik Prasanna.

Li-Mei Lim, Boston University
Zeros of L-Functions of Half-Integral Weight Automorphic Forms
While automorphic L-functions are expected to satisfy a Riemann hypothesis, L-functions associated to half-integral weight cusp forms in fact do not. In this talk, we'll discuss ongoing work exploring the distribution and behavior of the zeros of these half-integral weight L-functions. (Joint with Thomas Hulse and David Lowry-Duda.)

Adam Logan, TIMC and Carleton University
Integral points on continued fraction varieties
It is well-known that the continued fraction expansion of a quadratic irrational is eventually periodic. In this talk we will discuss continued fractions of quadratic irrationals from a geometric point of view, describing affine varieties $V_{k,n}(\alpha)$ that parametrize continued fractions associated to $\sqrt{\alpha}$ in which an initial segment of length $k$ is followed by a periodic part of length $n$.
We are particularly interested in the integral points on these varieties. When $n$ is small, we can show that the integral points on $V_{k,n}$ are not potentially dense when $\alpha$ belongs to a number field. When k is small, the opposite appears to be true, and we can prove it in some cases. In small dimensions we study some of the intermediate cases by exhibiting compactifications by smooth normal crossings divisors and describing the log Iitaka fibrations explicitly. This is joint work with Bruce Jordan, Allan Keeton, and Genya Zaytman.

David Lowry-Duda, Brown University
Consequences of spectral poles in arithmetic problems
Many arithmetic lattice problems can be translated into questions concerning coefficients of modular forms. The behaviour of modular forms can sometimes be approached through spectral theory. Recently, I've encountered several examples where a line of spectral poles seems to correspond to an oscillating error term --- but our understanding of these poles is too poor to prove it. In this talk, we describe this phenomenon.

Allysa Lumley, University of Montréal
Distribution of Values of L-functions over Function Fields
Let $q\equiv 1 \pmod 4$ be a prime power. Consider $D$ to be a square-free monic polynomial over $\mathbb{F}_q[T]$ and $\chi_D$ the Kronecker symbol associated to $D$. In this talk we will discuss the distribution of large values for $L(\sigma,\chi_D)$ for $1/2< \sigma \le 1$. We will note the expected similarities to the situation over quadratic extensions of $\mathbb{Q}$ and the surprising differences.

GL(2) real analytic Eisenstein series twisted by quasi-characters
In this talk we shall present a proof of the functional equation, in the usual $s$ variable, satisfied by a wide class of GL(2) real analytic Eisenstein series. The proof relies on classical Fourier analysis applied to a suitable "decorated Gaussian function" evaluated at a positive definite quadratic form.

Céline Maistret, Boston University
Arithmetic invariants and parity of ranks of abelian surfaces.
Let K be a number field and A/K an abelian surface. By the Mordell-Weil theorem, the group of K-rational points on A is finitely generated and as for elliptic curves, its rank is predicted by the Birch and Swinnerton-Dyer conjecture. A basic consequence of this conjecture is the parity conjecture: the sign of the functional equation of the L-series determines the parity of the rank of A/K. Under suitable local constraints and finiteness of the Shafarevich-Tate group, we prove the parity conjecture for principally polarized abelian surfaces. We also prove analogous unconditional results for Selmer groups.

Sacha Mangerel, Université de Montréal
Multiplicative Functions in Short Arithmetic Progressions and Applications
There are many results in the literature (notably the Barban-Davenport-Halberstam theorem and refinements thereof by Hooley and Montgomery) that establish cancellation for the variance
$\displaystyle\sum_{a \pmod{q}, (a,q) = 1} \left|\sum_{n \leq x} f(n) 1_{n \equiv a \pmod{q}} - \frac{\chi_1(a)}{\phi(q)} \sum_{n \leq x} f(n)\bar{\chi}_1(n)\right|^2,$
of a bounded multiplicative function $f$, where $\chi_1$ is a character modulo $q$ that correlates with $f$ (often, $\chi_1$ is principal in these investigations, when $f$ is a suitably well-behaved function such as the Mobius function). Typically, these estimates work except for a possible exceptional set of moduli $q \leq x$ of size $\ll_A x/(\log x)^A$ that depend on $f$ in an unknown way, and potentially in a restricted regime of $q$ (e.g., $q \geq x/(\log x)^{A})$. I will discuss some new results concerning estimates for this variance that apply whenever $x/q$ tending to infinity arbitrarily slowly without particular constraints on $f$, aside from a set of exceptional $q$ (independent of $f$) with a nearly power savings in the exceptional set (and for certain special families of moduli, including prime of smooth moduli, with much fewer or no exceptions). Some applications of the method to Linnik-type problems and to function field variants will also be discussed. (Based on joint work with O. Klurman and J. Teräväinen.)

Torsion of Rational Elliptic Curves over Nonic Galois Fields
The Mordell-Weil Theorem states that, for an elliptic curve $E/K$, the group of $K$-rational points is a finitely generated abelian group. Over the past two decades, much progress has been made in understanding the possible torsion subgroups of elliptic curves defined over number fields. This talk will briefly discuss some of this progress and then discuss the classification of the possible torsion subgroups for $E(K)$, where $E$ is a rational elliptic curve and $K$ is a nonic Galois field.

Steven J. Miller, Williams College
Optimal Test Functions for $n$-Level Densities and Applications to Central Point Vanishing
Spacings between zeros of $L$-functions occur throughout modern number theory, such as in Chebyshev's bias and the class number problem. Montgomery and Dyson discovered in the 1970's that random matrix theory (RMT) seems to model these spacings away from the central point $s=1/2$. While we have an incomplete understanding as to why a correlation exists between RMT and number theory, this interplay has proved useful for conjecturing answers to classical problems. These RMT models are insensitive to finitely many zeros, and thus miss the behavior near the central point. This is the most arithmetically interesting place; for example, the Birch and Swinnerton Dyer conjecture states that the rank of the Mordell-Weil group equals the order of vanishing of the associated $L$-function there. To investigate the zeros near the central point, Katz and Sarnak developed a new statistic, the $n$-level density; one application is to bound the average order of vanishing at the central point for a given family of $L$-functions by an integral of a weight against some test function $\phi$. It is therefore of interest to choose $\phi$ optimally to minimize the integral and obtain the best bound possible. While the $1$-level density has been studied in prior work, larger $n$ yield better bounds, but new technical problems emerge in the higher level densities. By restricting to a smaller class of test functions which split into products of test functions of a single variable, we are able to reduce the problem to an analogue of the one-dimensional case through a careful choice of $n-1$ of the $n$ test functions. This allows us to overcome the issues posed by the greater complexity of the higher level densities, as an appropriate choice yields a similar $1$-level problem but with a different weighting function. We explicitly prove, for a given set of $n-1$ admissible factors, the optimal choice of the final factor. Furthermore we estimate the corresponding integral to give a formula for the value as a function of the factors in the decomposition of $\phi$, obtaining strong estimates on both the average and excess ranks. Joint with Charles Devlin VI (Michigan).

The Arithmetic of Modular Grids
Zagier duality between sequences of modular forms has been discovered and proven in various contexts. In this talk, we review this history of Zagier duality and demonstrate how it arises from the Bruinier-Funke pairing. This perspective allows us to prove Zagier duality holds for certain canonical bases of quite general spaces.

Maria Nastasescu, Northwestern University
Mean values and subconvexity results for a degree 8 Euler product (joint work with Jeff Hoffstein and Min Lee)
We investigate some consequences of a result of Michel and Ramakrishnan that gives an explicit evaluation of Gross/Zagier type formulae. We introduce an additional parameter in their expression and sum over it to obtain a sum of certain products of L-functions. In particular, we study a weighted average over weight 2, level N holomorphic newforms of a degree 8 Euler product of three central L-values that depend on form f. We give an expression of this average in terms of double shifted Dirichlet series and analyze these series to express the average in terms of a main term plus an error term. We also estimate the growth of an individual term in the sum to get a subconvexity result.

Kunjakanan Nath, Université de Montréal (grad)
Primes of the form $a^2+b^2+h$ for some $h \in \{1, \dotsc, H\}$ in tuples
Let $\mathbb{P}_H$ be the set of primes of the form $a^2+b^2+h$ for some $h \in \{1, \dotsc, H \}$ and some $a, b \in \mathbb{Z}$ with $(a, b)=1$. In this talk, we will show that for each $m\geq 2$, there exists a constant $H_m$ such that the interval $[n, n+H_m]$ contains at least $m$ primes from the set $\mathbb{P}_{H_m}$ for inifintely many values of $n$. Our proof is based on a combination of the Maynard-Tao sieve and the semi-linear sieve.

On the classification of rigid meromorphic cocycles
In their paper Singular moduli for real quadratic fields: a rigid analytic approach, Darmon and Vonk classified rigid meromorphic cocycles, i.e. elements of $\text{H}^1(\text{SL}_2(\mathbb{Z}[1/p]), \mathcal{M}^\times)$ where $\mathcal{M}^\times$ is the multiplicative group of rigid meromorphic functions on the $p$-adic upper-half plane. They also evaluated these cocycles at RM points and observed analogies with the CM values of modular functions on $\text{SL}_2(\mathbb{Z})\setminus \mathcal{H}$. In this mainly expository talk, we recall some known facts and present an ongoing project aimed at generalising the work of Darmon and Vonk to congruence subgroups of $\text{SL}_2(\mathbb{Z}[1/p])$.

Jennifer Park, Ohio State University
Everywhere local solubility for hypersurfaces in products of projective spaces
Poonen and Voloch proved that the Hasse principle holds for either 100% or 0% of most families of hypersurfaces (specified by degrees and the number of variables). In this joint work with Tom Fisher and Wei Ho, we study one of the special families of hypersurfaces not accounted for by Poonen and Voloch, and we show that the explicit proportion of everywhere locally soluble (2,2)-curves in $P^1 \times P^1$ is about 87.4%.

Carl Pomerance, Dartmouth College
Primitive sets
A set of integers larger than 1 is primitive if no member divides another. Erdös has proved that the sum of $1/(n \log n)$ over a primitive set is bounded, and conjectured that the set of primes gives the largest such sum. We make progress on this conjecture and show a connection to the literature on prime number races. This is joint work with Jared Lichtman and Greg Martin.

In this talk, we explore a link between the ideal generated by polynomials over the ring of $p$-adic integers and the ideal generated by their projections mod $p^k$. This allows us to extend certain probabilistic results on polynomials over finite rings to polynomials over the ring of $p$-adic integers.

Isogenies of Elliptic Curves over $\mathbb{Q}(2^{\infty})$
Let $\mathbb{Q}(2^{\infty})$ be the compositum of all quadratic extensions of $\mathbb{Q}$. Torsion subgroups of rational elliptic curves base changed to $\mathbb{Q}(2^{\infty})$ were classified by Laska, Lorenz and Fujita. Recently Daniels, Lozano-Robledo, Najman, and Sutherland classified torsion subgroups of rational elliptic curves base changed to $\mathbb{Q}(3^{\infty})$, the compositum of all cubic extensions of $\mathbb{Q}$. We classify all cyclic isogenies of prime power degree of rational elliptic curves base changed to $\mathbb{Q}(2^{\infty})$, and give a list of possible degrees of cyclic isogenies for all but finitely many elliptic curves over $\mathbb{Q}(2^{\infty})$.

Intersection Numbers of Modular Geodesics
Let $\Gamma$ be a discrete subgroup of $\text{PSL}(2,\mathbb{R})$, and consider closed geodesics on $\Gamma\backslash\mathbb{H}$. In this talk, we will restrict to $\Gamma\backslash\mathbb{H}$ being a Shimura curve, and we will study the intersections of pairs of these closed geodesics. We will generate a formula for the "total intersection" of a pair of real quadratic discriminants, and show how to produce weight two modular forms on $\Gamma_0(N)$. We finish by connecting intersection numbers to the work of Gross and Zagier on factorizing $j(\tau_1)-j(\tau_2)$, the work of Duke, Imamoglu, and Töth on linking numbers in $\text{SL}(2,\mathbb Z)\backslash\text{SL}(2,\mathbb R)$, and the work of Darmon and Vonk on a real quadratic analogue of the $j-$function.

Julian Rosen, University of Maine
Iterates of a derivation in positive characteristic
Given two derivations of a ring, their composition is typically not a derivation. However, in characteristic $p$, the $p$-fold composition of a derivation with itself is again a derivation. In this talk, I will explain some results about the family of derivations $D^p\mod p$, when $D$ is a fixed derivation in characteristic $0$ and $p$ varies. This is joint work with Hunter Brooks.

Jonathan Sands, University of Vermont
A Stickelberger theorem for L-functions of graph coverings.
We consider an unramified abelian Galois covering of a graph X by a graph Y, and denote the group of automorphisms of Y over X by G. For the graph Y, the Jacobian J(Y) is a group with a variety of other names whose order is the tree-number of Y. In our situation, J(Y) becomes a module over the group ring Z[G]. Using L-functions, we define an element in this group ring and show that it annihilates the group J(Y). This is an analog of the classical Stickelberger theorem for cyclotomic fields. This is joint work with Kyle Hammer, Thomas Mattman, and Daniel Vallieres at CSU Chicco.

Soumya Sankar, University of Wisconsin (grad)
Proportion of ordinary curves in some families
A curve over a field of characteristic $p$ is called ordinary if the $p$-torsion of its Jacobian is geometrically as large as is possible. We ask the question: what is the probability that a curve is ordinary? We answer this for curves in some special families, namely Artin-Schreier and superelliptic families.

Ciaran Schembri, Dartmouth College
Examples of genuine QM abelian surfaces which are modular
Let K be an imaginary quadratic field. Modular forms for GL(2) over K are known as Bianchi modular forms. Standard modularity conjectures assert that every weight 2 rational Bianchi newform has either an associated elliptic curve over K or an associated abelian surface with quaternionic multiplication over K. We give explicit evidence in the way of examples to support this conjecture in the latter case. Furthermore, the quaternionic surfaces given correspond to genuine Bianchi newforms, which answers a question posed by J. Cremona in 1992 as to whether this phenomenon can happen.

Small gaps between primes from subsets
In the talk we will consider the distribution of primes with the property $\{ p^{\alpha} \} < 1/2$ where $\alpha$ is positive and non-integer. We will also describe the idea of the proof for the analogue of Bombieri-Vinogradov Theorem and corresponding results on bounded gaps between such primes.

Cathy Swaenepoel, Université de Montréal
Prime numbers with preassigned digits
Bourgain (2015) estimated the number of prime numbers with a proportion $c>0$ of preassigned digits in base 2 ($c$ is an absolute constant not specified). We present a generalization of this result in any base $g\geq2$ and we provide explicit admissible values for the proportion $c$ depending on $g$. Our proof, which adapts, develops and refines Bourgain's strategy, is based on the circle method and combines techniques from harmonic analysis together with results on zeros of Dirichlet $L$-functions, notably a very sharp zero-free region due to Iwaniec.

Naomi Tanabe, Bowdoin College
In this talk, we study the sum of additively twisted Fourier coefficients of Hilbert modular forms. We also survey some results pertaining to cancellation in the twists. This is a joint ongoing project with Alia Hamieh.

Nha Truong, University of Connecticut (grad)
An example of explicit computation of the Hecke operator.
This is an example in the ongoing project to prove the Ghost Conjecture of Bergdall and Pollack, which predicts the slope of p-adic automorphic forms. Using Jacquet-Langland correspondence, we can study the overconvergent p-adic automorphic from for definite quaternion algebra instead. The advantage of this approach is that we can work combinatorically. For example, we were able to compute the matrix of the Hecke operator $U_5$ on the space 5-adic automorphic form and deduce some interesting facts.

Bounds for the counting function of the Jordan-Pólya numbers
A positive integer $n$ is said to be a Jordan-Pólya number if it can be written as a product of factorials. Let ${\cal J}$ be the set of Jordan-Pólya numbers and let ${\cal J}(x)$ be its counting function. I will go over the proof that, given any $\varepsilon>0$, there exists a number $x_0=x_0(\varepsilon)$ such that, for all $x\ge x_0$,
$\displaystyle \exp\left\{ (2-\varepsilon) \frac{\sqrt{\log x}}{\log \log x} \right\} < {\cal J}(x) < \exp\left\{ (2+\varepsilon)\sqrt{\log x\log\log x} \right\}.$
This is joint work with Jean-Marie De Koninck, Nicolas Doyon, and Arthur A. Bonkli Razafindrasoanaivolala.

John Voight, Dartmouth College
Counting elliptic curves with an isogeny of degree three
We count by height the number of elliptic curves over the rationals that possess an isogeny of degree 3. This is joint work with Carl Pomerance and Maggie Pizzo.

Aled Walker, Université de Montréal
Linear inequalities in primes
The circle method provides a powerful tool for estimating the number of solutions to m simultaneous linear equations in prime variables, provided that the number of such variables is at least 2m+1. In a programme of work completed in 2010, Green-Tao-Ziegler created the theory of 'higher order fourier analysis' and, using this, they reduced the number of prime variables that are required to m+2. In this talk we will discuss a programme of work on the related topic of diophantine inequalities in primes, in which we have recently shown that higher order fourier analysis may also be used to estimate the number of solutions, provided that there are at least m+2 prime variables.

Matthew Welsh, Rutgers University
Roots of cubic congruences
We discuss a correspondence between pairs of "disjoint" roots of a cubic congruence and ideals in the associated cubic order, focusing on cube roots of 2 for concreteness. We then see how this correspondence leads to a parameterization and then an approximation of the roots. The parameterization also reveals a surprising connection between the roots and the arithmetic of binary cubic forms.

Joshua Zelinsky, Hopkins School, New Haven, CT
On the total number of prime factors of an odd perfect number
Let $N$ be an odd perfect number with $\omega(N)$ . Let $\omega$ be the number of distinct prime factors of $N$ and let $\Omega$ be the number of total prime factors of $N$. Ochem and Rao showed $\Omega \geq \frac{18\omega -31}{7}$. We discuss improvements on this inequality and related open problems involving cyclotomic polynomials.

## List of Participants

Keshav Aggarwal, University of Maine
Scott Ahlgren, University of Illinois Urbana-Champaign
Brandon Alberts, University of Connecticut
Eran Assaf, Dartmouth College
Jonathan Bayless, Husson University
Elliot Benjamin, Capella University
James Brody, Bates College
Crystel Bujold, Université de Montréal (grad)
Claire Burrin, Rutgers University
Jack Buttcane, University of Maine
Hugo Chapdelaine, Université Laval
Sara Chari, Bates College
Garen Chiloyan, University of Connecticut (grad)
Fatma Cicek, University of Rochester (grad)
Tyrone Crisp, University of Maine
Amy DeCelles, University of St. Thomas, Minnesota
David Dummit, University of Vermont
Andrew Granville, Université de Montréal
Bin Guan, City College of New York (grad)
Roman Holowinsky, Ohio State University
Robert Hough, Stony Brook University
Thomas Hulse, Boston College
Henryk Iwaniec, Rutgers University
Yeongseong Jo, University of Iowa
Dylan King, Wake Forest University (undergrad)
Paul Kinlaw, Husson University
Hershy Kisilevsky, Concordia University
Andrew Knightly, University of Maine
Mitsuo Kobayashi, Dartmouth College
David Krumm, Reed College
Debanjana Kundu, University of Toronto (grad)
Antonio Lei, Université Laval
Li-Mei Lim, Boston University
Shenhui Liu, University of Maine
Adam Logan, TIMC and Carleton University
David Lowry-Duda, Brown University
Allysa Lumley, Université de Montréal (grad)
Céline Maistret, Boston University
Sacha Mangerel, Université de Montréal (grad)
Steven J. Miller, Williams College
Maria Nastasescu, Northwester University
Kunjakanan Nath, Université de Montréal (grad)
Jennifer Park, Ohio State University
Nigel Pitt, University of Maine
Carl Pomerance, Dartmouth College
Julian Rosen, University of Maine
Stelios Sachpazis, Université de Montréal (grad)
Jonathan Sands, University of Vermont
Soumya Sankar, University of Wisconsin (grad)
Ciaran Schembri, Dartmouth College
Andrei Shubin, California Institute of Technology (grad)
Lloyd Simons, St. Michael's College
Chip Snyder, University of Maine
Cathy Swaenepoel, Université de Montréal
Naomi Tanabe, Bowdoin College
Nha Truong, University of Connecticut (grad)