Maine-Québec
Number Theory Conference
University of Maine
October 2nd and 3rd , 2021

Organizers
Jack Buttcane
Andrew Knightly

Saturday
9:00-9:10
9:10-10:00

Virtual Room A
Welcome

Virtual Room B

Valentin Blomer(Universität Bonn)
Density theorems for GL(n)10:00-10:10

Break
Robert Hough(SUNY Stony Brook)
Subconvexity of the
Shintani zeta function

Antonio Lei(Université Laval)
Anticyclotomic Selmer groups for
CM elliptic curves at inert primes

Eun Hye Lee(Stony Brook University)
Multiple Dirichlet series associatedto binary cubic forms

Wanlin Li(CRM)
Abelian varieties of prescribedorder over finite fields

David Lowry-Duda(ICERM)
Zeros of Half Integral Weight
Dirichlet Series

Jaitra Chattopadhyay(Indian I.T. Guwahati)
On Some Families of
Non-Pólya Bi-quadratic Fields

Kunjakanan Nath(Université de Montréal)Distribution of primes with a missing
digit in arithmetic progressions

Katharina Müller(Université Laval)
Iwasawa µ invariantsof fine Selmer groups

Saurabh Singh(Indian Institute of Technology)
Delta symbol and its applications

Subramani Muthukrishnan(IIIT D&M, Chennai)
Euclidean Quartic number fields

2:20-2:30

Break

2:30-2:50

Joshua Zelinsky(Hopkins School)
Mertens’ Theorem and perfect numbers

Anwesh Ray(University of British Columbia)
Constructing Galois representations
ramified at one prime

John Voight(Dartmouth College)
Stickelberger’s discriminant theorem
for algebras

Debanjana Kundu(UBC Vancouver/ PIMS)
Arithmetic Statistics andIwasawa Theory

Krystian Gajdzica(Jagiellonian University)
Arithmetic properties of the restricted
partition function pA (n, k)

Pavel Čoupek Video(Purdue University)
Ramification bounds for mod p étale
cohomology via prismatic cohomology

4:05-4:15

Break

4:15-4:30(students)

Nihar Gargava(EPFL)
Lattice packings through division algebras

Elad Zelingher(Yale University)
On regularization of integrals of
matrix coefficients associatedwith spherical models

Peter Zenz
(McGill University)
Quantum Variance for holomorphic
Hecke cusp forms on the vertical geodesic

Siva Nair(Université de Montréal)
A Family of Polynomials with28
Mahler Measure 5π
2 ζ(3)

4:55-5:05

Break

5:05-5:20(students)

Alexandre de Faveri(Caltech)
Simple zeros of GL(2) L-functions

Isabella Negrini(McGill University)
A Shimura-Shintani correspondence
for rigid analytic cocycles

10:10-10:30
10:30-10:40
10:40-11:00

Break

11:00-11:30
11:30-11:50

Coffee11:50-12:00
12:00-12:20

Break

12:20-2:00
2:00-2:20

Lunch

2:50-3:00
3:00-3:20

Break

3:20-3:50
3:50-4:05(students)

Coffee4:30-4:40
4:40-4:55(students)

Virtual Room C

Tariq Osman(Queen’s University)
A New Bound for a Particular Familyof Quadratic Weyl Sums

Stelios Sachpazis(Université de Montréal)
On multiplicative functions
with small averages

Break
Luochen Zhao
(Johns Hopkins University)
Sum expressions for
Kubota-Leopoldt p-adic L-functions
Chung-Hang Kwan Video(Columbia University)
Moment formulae for automorphic
L-functions of higher-rank groups

Sunday
Virtual Room A
9:00-9:20
9:20-9:30
9:30-9:50

Abbas Maarefparvar Video(Inst. for Research in Fund. Sciences)
Polya-Ostrowski groups
in algebraic number fields

Yeongseong Jo(The University of Maine)
The weak test vector problems
and local periods

Matilde Lalin(Université de Montréal)Sums of certain arithmetic functions
over Fq [T ] and symplectic distributions

Break

10:20-10:50
10:50-11:10

CoffeeEdmund Karasiewicz(University of Utah)
The twisted Satake transform andthe Casselman-Shalika formula

Jonathan Love
(McGill University (CRM-ISM))
Products of Pythagorean slopes

Kim Klinger-Logan(Kansas State University)
Graviton Scattering and Differential
Equations in Automorphic Forms

Sudhir Pujahari(University of Warsaw)
Distribution of moments of traces of
Frobenius in arithmetic progressions

Joshua Males(University of Manitoba)
Mock modular forms andEichler-Selberg relations

Manami Roy(Fordham University)
Local data for rational elliptic curves
with non-trivial torsion

Eric Stubley(McGill University)
Classical forms of low weight in p-adicfamilies of ordinary modular forms

Andrew Schultz(Wellesley College)
Realizing module structures of square
power classes over biquadratic extensions

11:10-11:20
11:20-11:40

Break

11:40-11:50
11:50-12:10

Break

12:10-12:20
12:20-12:40

Break

12:40-2:00
2:00-2:50

Lunch
Theresa Anderson(Purdue University)
Two meetings of
number theory and analysis

2:50-3:00
3:00-3:15(students)

BreakHari Iyer(Harvard College)
One-level density of a family of
L-functions over function fields

3:15-3:25

Subham Roy(Université de Montréal)
Mahler measure of a family ofpolynomials for arbitrary tori

George Hauser(Rutgers University)
Eisenstein Series on
Higher Covers of SL(2,R)

Doyon Kim
(Rutgers University)
Automorphic distributions andZeros of L-functions

Yusheng Lei(Boston College)
Theta liftings on double coversof orthogonal groups

4:05-4:15

Break

4:15-4:30(students)

Grant Molnar(Dartmouth College)
The LCM product and
Gronwall’s theorem

Zhining Wei(Ohio State University)
Linear Relations of Siegel Poincaré
Series and Non-vanishing of the Central
Values of Spinor L-functions

3:40-3:50
3:50-4:05(students)Félix Baril Boudreau(University of Western Ontario)
Computing an L-function
modulo a prime

Break
Simran Khunger(Carnegie Mellon University)
Extending support in calculating the nth
moment of the 1-level density of
low lying zeroes of orthogonal families

3:25-3:40(students)

Virtual Room C

BreakJonathan Bayless(Univ. of Maine at Augusta)
Higher Mertens constants
for almost primes

9:50-10:00
10:00-10:20

Virtual Room B

Min Lee(University of Bristol)
Non-vanishing ofsymmetric cube L-functions

Cédric Dion
(Université Laval)
Functional equation for
chromatic Selmer groups

Break
Anthony Doyon(Université Laval)
Congruences between additive elliptic
curves and Ramanujan’s tau function

Tyler Genao(University of Georgia)
Typically bounding torsion on
elliptic curves: j(E) ∈ F and beyond1

Abstracts
Theresa Anderson, Purdue University
Two meetings of number theory and analysis
In many recent works, number theory and analysis go beyond working side by side and teamup in an interconnected back and forth interplay to become a powerful force. Here I describe
two distinct meetings of the pair, which result in sharp counts for equilateral triangles in
Euclidean space and statistics for how often a random polynomial has Galois group not isomorphic to the full Sn .

Félix Baril Boudreau, University of Western Ontario (student)
Computing an L-function modulo a prime
Let E be an elliptic curve with non-constant j-invariant over a function field K with constant field of size an odd prime power q. Its L-function L(T, E/K) belongs to 1 + T Z[T ].
Inspired by the algorithms of Schoof and Pila for computing zeta functions of curves overfinite fields, we propose an approach to compute L(T, E/K). The idea is to compute, forsufficiently many primes ` invertible in K, the reduction L(T, E/K) mod `. The L-functionis then recovered via the Chinese remainder theorem. When E(K) has a subgroup of orderN ≥ 2 coprime with q, Chris Hall showed how to explicitly calculate L(T, E/K) mod N .
We present novel theorems going beyond Hall’s.

Jonathan Bayless, University of Maine at Augusta
Higher Mertens constants for almost primes
For k ≥ 1, a k-almost prime is a positive integer with exactly k prime factors, counted withmultiplicity. In this article we give elementary proofs of precise asymptotics for the reciprocal sum of k-almost primes. Our results match the strength of those of classical analytic
methods. We also study the limiting behavior of the constants appearing in these estimates,
which may be viewed as higher analogues of the Mertens constant β = 0.2614 . . . This is
joint work with Paul Kinlaw and Jared Lichtman.

Valentin Blomer, Universität Bonn
Density theorems for GL(n)The generalized Ramanujan conjecture predicts that all cuspidal automorphic representations for GL(n) are tempered. A density theorem is a quantitative version of the statementthat non-tempered representations become rarer the further their Langlands parameters at
a given place are away from the unitary axis. In many cases this is a good substitute for the2

Ramanujan conjecture, in the same way as the Bombieri-Vinogradov theorem can be used
a substitute for the Riemann hypothesis for Dirichlet L-functions. In this talk I show how
a relative trace formula together with new bounds for GL(n) Kloosterman sums can obtainstrong density theorems, and I present some applications.

Jaitra Chattopadhyay, Indian Institute of Technology Guwahati
On Some Families of Non-Pólya Bi-quadratic Fields
An algebraic number field K is called a Pólya field if the ring of integer-valued polynomials has a regular basis. This behavior is governed by the Pólya group P o(K), a particularsubgroup of the ideal class group ClK of K. The study of Pólya groups for various number
fields is of considerable interest in recent times. In this talk, we shall compute the Pólyagroups of three possibly infinite families of bi-quadratic fields, two of which are known tohave non-principal Euclidean ideal class under mild conditions. This is a joint work with
Anupam Saikia.

Pavel Čoupek, Purdue University (student)
Ramification bounds for mod p étale cohomology via prismatic cohomology
Given a smooth proper formal scheme X over OK where K is a p–adic field and p is an odd
iprime, we give an upper bound for ramification of the Fp –representations Hét
(XCK , Z/pZ)
in terms of p, i, and e, the absolute ramification index of K, without any restriction on thesize of i and e. In order to achieve this, a crucial input is the recently developed prismatici
i(XAinf /Ainf ), resp., and(X/S) and H∆
cohomology in its Breuil–Kisin and Ainf –instances, H∆
a series of conditions (Crs )s≥0 that control the Galois action on the elements of the Breuil–Kisin cohomology groups inside the Ainf –cohomology groups.
VIDEO OF LECTURE

Alexandre de Faveri, Caltech (student)
Simple zeros of GL(2) L-functions
Let f ∈ Sk (Γ1 (N )) be a primitive holomorphic form of arbitrary weight k and level N . Weshow that the completed L-function of f has Ω(T δ ) simple zeros with imaginary part in
2[−T, T ], for any δ < 27
. This is the first power bound in this problem for f of non-trivial
level, where previously the best results were Ω(log log log T ) for N odd, due to Booker, Milinovich, and Ng, and infinitely many simple zeros for N even, due to Booker. In addition,
for f of trivial level (N = 1), we also improve an old result of Conrey and Ghosh on thenumber of simple zeros.

3

Cédric Dion, Université Laval (student)
Functional equation for chromatic Selmer groups
Let p be a fixed odd prime and let E be an elliptic curve over the rational numbers. Supposethat E has good ordinary reduction at p. It is well known that the p-adic L-function of E
over the cyclotomic extension satisfies a functional equation. It also follows from Mazur’s
work that the p-primary Selmer group of E over the same extension satisfies a so-called
algebraic functional equation. In this talk, we investigate the setting where E has good supersingular reduction at p and the Selmer groups considered are Sprung’s chromatic Selmer
groups. We show that these Selmer groups also satisfy a functional equation over the cyclotomic extension.

Anthony Doyon, Université Laval (student)
Congruences between additive elliptic curves and Ramanujan’s tau function
Let E denote an elliptic curve
P defined over Q. By modularity, one can associate a modular
form fE to E, say fE (z) = n≥1 an (E)q n where q = e2πiz . Let ∆ be the unique normalized
cuspidal modular form of weight 12 and level 1 given by the following q-expansion∆(z) = qY(1 − q n )24 =n≥1Xτ (n)q nn≥1

where τ : N → Z is known as Ramanujan’s tau function. Let p be a prime. This talk focuseson the following question: For which elliptic curves E does an (E) ≡ τ (n) (mod p) holds for
all n ≥ 1?
For p = 2 and 3, we present families of elliptic curves satisfying this condition. We also study
the link between p-adic Mazur-Tate elements attached to ∆ and those attached to certain
elliptic curves E having additive reduction at p. Under certain hypotheses, we can prove an
equality of p-adic Iwasawa µ and λ invariants of Mazur-Tate elements.

Krystian Gajdzica, Jagiellonian University (student)
Arithmetic properties of the restricted partition function pA (n, k)
Let A = (an )n∈N+ be a sequence of positive integers. The function pA (n, k) counts the number of multi-color partitions of n into parts in {a1 , . . . , ak }. We present several arithmetic
properties of the sequence (pA (n, k) (mod m))n∈N for an arbitrary fixed integer m > 2, andapply them to the special cases of A. In particular, for a fixed parameter k, we investigate
both lower and upper bounds for the odd density of pA (n, k). Moreover, we perform somenew results related to restricted m-ary partitions and state a few open questions at the endof the talk.

4

Nihar Gargava, EPFL, Switzerland (student)
Lattice packings through division algebras
We will show the existence of lattice packings in a sparse family of dimensions. This construction will be a generalization of Venkatesh’s lattice packing result of 2013. In our construction,
we replace the appearance of the cyclotomic number field with a division algebra over therationals. This improves the best known lower bounds on lattice packing problem in many
dimensions. The talk will cover previously known bounds, an overview of the new bounds
and a live numerical simulation of Siegel’s mean value theorem.

Tyler Genao, University of Georgia (student)
Typically bounding torsion on elliptic curves: j(E) ∈ F and beyond
In 1996, Merel proved the “strong uniform boundedness conjecture” for torsion subgroupsof elliptic curves over number fields of a fixed degree, suggested by previous work of Mazur,
Kamienny, Kenku and Momose. Unfortunately, explicit upper bounds on torsion subgroups
over degree d > 3 number fields are greater than exponential in d. Despite this, special families of elliptic curves, such as the family of elliptic curves with complex multiplication, havetorsion subgroups which satisfy a “typical order”: there exists a bound on torsion subgroupsof this family which works over all degrees, provided one ignores those torsion subgroups
whose number field degrees lie in a subset of Z+ of arbitrarily small upper density. Suchfamilies of elliptic curves are said to be typically bounded in torsion. In this talk, we will
explore some of the main ideas involved in proving that the family EF of elliptic curves with
j-invariant lying in a fixed number field F is typically bounded in torsion. We will also show
why this implies that a certain larger family of elliptic curves is typically bounded in torsion,and see how such a family fits into a general study of Q-curves.

George Hauser, Rutgers University (student)
Eisenstein Series on Higher Covers of SL(2,R)
Eisenstein series on covers of SL(2) are the subject of much interest in the theory of higher
theta functions, although it is necessary to extend the base field to contain roots of unity.
In this talk, I will explore a construction of some Eisenstein series on high covers of SL(2,R)
without extending the base field. The main result is the computation of the poles andresidues of this series, which are automorphic forms on the universal cover of SL(2,R). For
technical convenience we will use language of automorphic distributions.

Robert Hough, SUNY Stony Brook
Subconvexity of the Shintani zeta function
We prove a subconvex estimate for Shintani’s zeta function enumerating binary cubic forms,
and for its Maass form twisted version. The estimate uses methods developed by Bhargava

5for enumerating orbits in representation spaces, and is expected to obtain subconvex estimates for other zeta functions of prehomogeneous vector spaces. The estimates are novel as
the degree is high. Joint work with Eun Hye Lee.

Hari Iyer, Harvard College (student)
One-level density of a family of L-functions over function fields
Katz and Sarnak conjectured a correspondence between n-level density statistics of zeroesof families of L-functions and the eigenvalues of random matrix ensembles. The particularensemble depends on the symmetry type of the family, which is a classical compact group(unitary, symplectic, or orthogonal). The latter are often studied by random matrix theory(RMT). We build upon previous work by Waxman, which has shown that L-functions associated with Hecke characters on the Gaussian integers Z[i] have zeroes which are modeledby the eigenvalues of symplectic matrices. We consider analogous L-functions associatedwith “super-even” characters in the function field setting. Though these characters havebeen studied from an RMT perspective as q → ∞ (for Fq [t]), we instead consider the limit
where the degree K of the modulus of the Dirichlet character is large; also note that this isequivalent to the large conductor limit, since K log q is proportional to the average logarithmic conductor of the family of super-even L-functions evaluated at s = 1/2. We computethe limiting one-level density for this family of L-functions and we show that it matches a
symplectic distribution for a class of test functions f whose Fourier transform fˆ is compactly
supported in (−1, 1). We directly calculate the main term and a lower order term for the
one-level density. In addition, we apply the L-functions Ratios Conjecture to compute theone-level density, and we show agreement with the unconditional result for restricted support
to arbitrarily lower order O(K −a ) for all a > 1.

Yeongseong Jo, The University of Maine
The weak test vector problems and local periods
Let F be a non-archimedean local field of characteristic zero and π an irreducible admissible
generic representation of GLm (F ). By definition, the local L-function associated to π is apriori given by a finite sum of Rankin-Selberg type integrals. The purpose of this talk is
to finds a pair of Whittaker functions and Schwartz-Bruhant functions so that so-called theformal L-function can be expressed as a single integral. This is referred to the weak test
problem and in general the formal L-function divides the original one. Time permitting, we
explain how one can extend it to archimedean local fields. The last part is joint work withPeter Humphries.

Edmund Karasiewicz, University of Utah
The twisted Satake transform and the Casselman-Shalika formula6

The Fourier coefficients of automorphic forms are an important object of study due to theirconnection to L-functions. In the adelic framework, constructions of L-functions involving
Fourier coefficients (e.g. Langland-Shahidi and Rankin-Selberg methods) naturally lead tospherical Whittaker functions on p-adic groups. Thus we would like to understand these
spherical Whittaker functions to better understand L-functions. Casselman-Shalika determined a formula for the spherical Whittaker functions, and basic algebraic manipulations
reveal that their formula can be more succinctly expressed in terms of characters of theLanglands dual group. We will describe a new proof of the Cassllman-Shalika formula thatprovides a conceptual explanation of the appearance of characters.

Simran Khunger, Carnegie Mellon University (student)
Extending support in calculating the nth moment of the 1-level density of low lying zeroes oforthogonal families of L-functions
We study low-lying zeroes of L-functions and their n-level density, which relies on a smooth
test function φ whose Fourier transform has compact support. Assuming GRH, we establish tools to compute the nth centered moments of the 1-level density of low-lying zeroes of
L-functions associated with weight k, prime level N cuspidal newforms as N → ∞, where1
1
φ̂ has support in n−a, n−a
for n ≥ 2a. The Katz-Sarnak Density conjecture claims thatthe n-level density of certain families of L-functions is the same as the distribution of eigenvalues of a family of orthogonal random matrices. Our work helps establish new evidence
for the Katz-Sarnak Density conjecture in larger support and aids in bounding the order ofvanishing of L-functions at the central point.

Doyon Kim, Rutgers University (student)
Automorphic distributions and Zeros of L-functions
The automorphic distribution attached to a holomorphic modular form f (z) is defined to be
the boundary values τ (x) = limy→0+ f (x + iy). We can define the Mellin transform of the
distribution τ (x) and derive the functional equation of the L-function of f (z). In this talk, I
will define the automorphic distribution, present some of their properties and describe how
we can prove that there are infinitely many zeros of the L-function on the critical line using
automorphic distributions.

Kim Klinger-Logan, Kansas State University
Graviton Scattering and Differential Equations in Automorphic Forms
Green, Russo, and Vanhove have shown that the scattering amplitude for gravitons (hypothetical particles of gravity represented by massless string states) is closely related toautomorphic forms through differential equations. Green, Miller, Russo, Vanhove, Pioline,

7

and K-L have used a variety of methods to solve eigenvalue problems for the invariant Laplacian on different moduli spaces to compute the coefficients of the scattering amplitude offour gravitons. We will discuss recent work with S. Miller and D. Radchenko to solve themost complicated of these differential equations (∆ − λs )u = Ea2 on SL2 (Z)\h.

Debanjana Kundu, UBC Vancouver/ PIMS
Arithmetic Statistics and Iwasawa Theory
We will report on recent work (with collaborators) where we study the average behaviour of
the Iwasawa invariants for the Selmer groups of elliptic curves, setting out new directions inarithmetic statistics and Iwasawa theory.

Chung-Hang Kwan, Columbia University (student)
Moment formulae for automorphic L-functions of higher-rank groups
The central values of L-functions are of great arithmetic interests. One of the ways to accesstheir statistics is via moment calculations and moment problems of L-functions have been
central to analytic number theory. There have been many spectacular advances and applications in the past decades. In this talk, we will focus on the higher-rank situation, discussing
the techniques, hurdles, and some of my on-going work in this direction.
VIDEO OF LECTURE

Matilde Lalin, Université de MontréalSums of certain arithmetic functions over Fq [T ] and symplectic distributions
In 2018 Keating, Rodgers, Roditty-Gershon and Rudnick established relationships of themean-square of sums of the divisor function dk (f ) over short intervals and over arithmetic
progressions for the function field Fq [T ] to certain integrals over the ensemble of unitary matrices. We consider similar problems leading to distributions over the ensemble of symplectic
matrices. We also consider analogous questions involving convolutions of the von Mangoldt
function. This is joint work with Vivian Kuperberg.

Eun Hye Lee, Stony Brook University
Multiple Dirichlet series associated to binary cubic forms
In this talk, I will construct a double Dirichlet series associated to a space of certain binary
cubic forms and show the domain of meromorphy continues to the whole of C2 .

8

Min Lee, University of Bristol
Non-vanishing of symmetric cube L-functions
The non-vanishing of L-series at the central of the critical strip has long been a subjectof great interest. The main purpose of this talk is to prove that there are infinitely many
Maass–Hecke cuspforms over the imaginary quadratic field of discriminant -3 such that thecentral values of their symmetric cube L-functions do not vanish. This is done by usinga result of Ginzburg, Jiang and Rallis which shows that the non-vanishing at the central
point of the critical strip of the symmetric cube L-series of any GL(2) automorphic form isequivalent to the non-vanishing of a certain triple product integral. We use spectral theory
and the properties of the cubic theta function to show that the non-vanishing of this triple
product occurs for infinitely many cusp forms. This is a joint work with Jeff Hoffstein andJunehyuk Jung.

Antonio Lei, Université Laval
Anticyclotomic Selmer groups for CM elliptic curves at inert primes
Let E/Q be a CM elliptic curve and K an imaginary quadratic field where p is inert. The
goal of this talk is to shed light on the structure of the fine Selmer group of E over the
anticyclotomic extension K∞ of K, which was initially systematically studied by Coates andSujatha in early 2000’s. Burungale, Kobayashi and Ota recently proved a longstanding conjecture of Rubin on the structure of the local points on E over K∞ from the 1980’s. We will
disucss how this allows us to link the fine Selmer group to the plus and minus Selmer groupsof E over K∞ . This generalizes (and simplifies) a recent result of myself and Sujatha in thecyclotomic setting.

Yusheng Lei, Boston College (student)
Theta liftings on double covers of orthogonal groups
In 2003, Bump-Friedberg-Ginzburg constructed the generalized theta representations on the
double cover of odd special orthogonal groups. The same authors later use such theta representations to construct the generalized theta lifting of a cuspidal automorphic representationon the double cover of an odd special orthogonal group to the double cover of an even special
orthogonal group. For a fixed automorphic representation, one may ask what the smallest
rank for an even special orthogonal group is so that the theta lifting to its double cover
is non-zero. The case of generic representations was treated by Bump-Friedberg-Ginzburg.
In this talk, I will talk about the general framework of the generalized theta liftings andsome progress that predicts the first non-zero occurrence of the liftings of a fixed non-genericcuspidal automorphic representation.

9

Wanlin Li, CRM
Abelian varieties of prescribed order over finite fields
Given a prime power q and a positive integer n, one can ask whether every integer in theinterval given by the Hasse–Weil bound is the order of some abelian variety of dimension
n over Fq . In this talk, I will discuss results on this topic from joint work with Raymond
van Bommel, Edgar Costa, Bjorn Poonen and Alexander Smith. Given a prime power q andn  1, we prove that every integer in a large subinterval of the Hasse–Weil interval is theorder of a geometrically simple ordinary principally polarized abelian variety of dimension
n over Fq . As a consequence, we generalize a result of Howe and Kedlaya for F2 to showthat for each q, every sufficiently large positive integer is realizable as the order of an abelianvariety over Fq . Our result also improves upon the best known constructions of sequences ofsimple abelian varieties with point counts towards the extremes of the Hasse–Weil interval.
We also determine, for fixed n, the largest subinterval of the Hasse–Weil interval consisting of realizable integers, asymptotically as q grows; this gives an asymptotically optimal
improvement of a 1998 theorem of DiPippo and Howe. Our methods are effective and inparticular we prove that if q ≤ 5, then every positive integer is realizable.

Jonathan Love, McGill University (CRM-ISM)
Products of Pythagorean slopes
A “Pythagorean slope” is a ratio of the lengths of the legs of a rational right triangle; by a2tclassical parametrization, this is equivalent to being of the form 1−t
2 for some t ∈ Q ∩ (0, 1).
Products of Pythagorean slopes naturally arise when studying certain configurations of points
in Rn with pairwise rational distances. In this talk, we will classify the set of products of
two Pythagorean slopes using a family of elliptic curves, and use this classification to provethat every positive rational number can be written as a product of three Pythagorean slopes(in infinitely many ways).

David Lowry-Duda, ICERM
Zeros of Half Integral Weight Dirichlet Series
Automorphic L-functions are well-studied and are believed to satisfy a number of major
conjectures, such as the Riemann hypothesis. Dirichlet series associated to half-integral
weight modular forms are in many ways similar to their better-studied cousins, but are not
L-functions and in various ways behave much worse. In this talk, I’ll describe an ongoing investigation into half-integral weight modular forms and their Dirichlet series. We’ll see thatthese series don’t satisfy a Riemann hypothesis, but are still in many ways understandable.
This includes joint work with Thomas Hulse, Li-Mei Lim, and Mehmet Kiral.

10

Abbas Maarefparvar, Institute for Research in Fundamental Sciences (IPM)
Polya-Ostrowski groups in algebraic number fields
Pólya fields are number fields whose Bhargava generalized factorials are principal ideals (Thisis a modern definition, classically they are defined in terms of regular bases, or back at thetime of Hilbert in terms of the action of the Galois group on the class group of a number
field). As a “measure of the obstruction” for a number field to be a Pólya field, the notionof Pólya group was introduced later in 1997 by Cahen-Chabert, and a number field is Pólya
iff its Pólya group is trivial. Pólya fields encompass all PID number fields, however they
are more tractable than the PID ones. Whereas Gauss’ class number one conjecture is stillwide open for the real quadratic fields and it took a while to prove that Gauss’ list of imaginary class number one fields were complete (Baker-Stark-Heegner), all Pólya quadratic fields
are completely classified, essentially thanks to Hilbert’s computation of ambiguous ideals as
demonstrated by Pólya (more generally, Pólya groups in Galois extensions will be “controllable” throughout Galois cohomology and ramifications). In this talk, I would like to give a
brief history of Pólya fields and Pólya groups along with many various examples.
VIDEO OF LECTURE

Joshua Males, University of Manitoba
Mock modular forms and Eichler-Selberg relations
Eichler-Selberg type relations for scalar-valued mock modular forms were proven by Mertensin a celebrated paper in Adv. Math (2016), thereby proving an older conjecture of Cohen. In
two recent papers, the speaker and A. Mono have extended these ideas to the vector-valued
world. The central idea is to use certain “higher” theta lifts and Serre duality, along withvarious properties of harmonic Maass forms. We will see that a wide class of vector-valued
mock modular forms satisfy Eichler-Selberg relations. In a slightly different direction, we
also offer a construction of locally harmonic Maass forms on grassmanians of even latticesof any indefinite signature.

Grant Molnar, Dartmouth College (student)
The LCM product and Gronwall’s theorem
In this talk, we define the LCM product, due originally to R. D. von Sterneck. We relate theLCM product to Dirichlet convolution, and then prove an analogue to Grönwall’s Theorem
for kth LCM powers of the identity function on the natural numbers.

Katharina Müller, Université Laval
Iwasawa µ invariants of fine Selmer groups
We compare Iwasawa Invariants of fine Selmer groups of congruent abelian varieties oversuitable p-adic Lie-extensions and also give a relation to ideal class groups. This is joint
work with Sören Kleine.

11

Subramani Muthukrishnan, Indian Institute of Information Technology D&M,
Chennai
Euclidean Quartic number fields
We prove that all abelian quartic number fields of class number one are Euclidean.

Siva Nair, Université de Montréal (student)
28
A Family of Polynomials with Mahler Measure 5π
2 ζ(3)The Mahler measure of a Laurent polynomial P (x1 , . . . , xn ) is the integral of log |P | over theunit n-torus defined by |xi | = 1 for all i. Interest in this quantity arose from the fact that
the Mahler measure of certain polynomials is quite remarkable and not just any random
real number. If P is univariate, then the measure is given by Jensen’s formula in termsof its roots, and in the multivariable case, it has been observed to yield special values of
L-functions. Here, we will discuss ongoing work (with Prof. Matilde Lalı́n) that evaluates
the Mahler measure of a family of polynomials in terms of ζ(3).

Kunjakanan Nath, Université de MontréalDistribution of primes with a missing digit in arithmetic progressions
One of the fundamental questions in number theory is to find primes in any subset of thenatural numbers. In general, it’s a difficult question and leads to open problems like thetwin prime conjecture, Landau’s problem and many more. Recently, Maynard considered
the set of natural numbers with a missing digit and showed that it contains infinitely many
primes whenever the base b ≥ 10. In fact, he has established the right order of the upper
and the lower bounds when the base b = 10 and an asymptotic formula whenever b is large.
In this talk, we will consider the distribution of primes with a missing digit in arithmetic
progressions for base b large enough. In particular, we will show an analogue of the BombieriVinogradov type theorems for primes with a missing digit for large base b. The proof relieson the circle method, which in turn is based on the Fourier structure of the digital set andthe Fourier transform of primes over arithmetic progressions on an average. Finally, we will
give its application to count the primes of the form p = 1 + m2 + n2 with a missing digit fora large odd base.

Isabella Negrini, McGill University (student)
A Shimura-Shintani correspondence for rigid analytic cocycles
In their paper Singular moduli for real quadratic fields: a rigid analytic approach, Darmon
and Vonk introduced rigid meromorphic cocycles, i.e. elements of H 1 (SL2 (Z[1/p]), M× )where M× is the multiplicative group of rigid meromorphic functions on the p-adic upperhalf plane. Their values at RM points belong to narrow ring class fields of real quadratic

12

fiends and behave analogously to CM values of modular functions on SL2 (Z) \ H. In this talkwe will present some of our results on rigid meromorphic and analytic cocycles of weight k, aswell as some of the progress we made towards developing a Shimura Shintani correspondence
for rigid analytic cocycles.

Tariq Osman, Queen’s University (student)
A New Bound for a Particular Family of Quadratic WeylP
Sums


1 2
Consider quadratic Weyl sums of the form SN (x; α, β) := N
n
+
βn
x
+
nα
where
n=1 e
2
b
a
, β = 2mwith a, b and m odd, and gcd(a, b, m) = 1. We show that there exists a conα = 2m√
stant C = C(m) such that |SN (x; α, β)| ≤ C N for all x. This is joint work with Francesco
Cellarosi.

Sudhir Pujahari, University of Warsaw
Distribution of moments of traces of Frobenius in arithmetic progressions
In this talk we will study moments of the trace of Frobenius of elliptic curves if the trace isrestricted to a fixed arithmetic progression. We will see an asymptotic formulas for moments
for cases for which the prime p goes to infinity for fixed r and cases where the power r goesto infinity with fixed p. As a special case we recover a result of Birch proving Sato-Tatedistribution for certain family of elliptic curves. Moreover, we will see that these results follow from similar asymptotic formulas relating sums and moments of Hurwitz class numbers
where the sums are restricted to certain arithmetic progressions. This is a joint work with
Kathrin Bringmann and Ben Kane.

Anwesh Ray, University of British Columbia
Constructing Galois representations ramified at one prime
A prime p is said to be regular if p does not divide the class number of Q(µp ). Given an integer n > 1 and a prime p greater than or equal to 4bn/2c + 1, Ralph Greenberg constructeda Galois representation ρ : Gal(Q̄/Q) → GLn (Zp ) with large image with minimal ramification. In fact, ρ is constructed so as to unramified at all primes ` 6= p. We prove that such
Galois representations ramified at one prime can be constructed for irregular primes as well.
These representations are constructed by lifting suitably chosen residual representations ρ̄with image in the diagonal torus in GLn (Fp ) for which the associated deformation problem isunobstructed. The results are of interest from the perspective of the Inverse Galois problem.

Manami Roy, Fordham University
Local data for rational elliptic curves with non-trivial torsion

13

We consider a parameterized family ET of elliptic curves with the property that they parameterize all elliptic curves E/Q which contain T in their torsion subgroup, T is one offourteen possible non-trivial torsion subgroups allowed by Mazur’s Torsion Theorem. Using
these parameterized families and Tate’s Algorithm, we explicitly classify the Néron types,
the conductor exponents fp , and the local Tamagawa numbers cp of E/Q at the primes p
where it has additive reduction. Consequently, we find all rational elliptic curves with a2-torsion and 3-torsion point which have global Tamagawa number equal to 1. This is a
joint work with Alexander J. Barrios.

Subham Roy, Université de Montréal (student)
Mahler measure of a family of polynomials for arbitrary tori
The (logarithmic) Mahler measure of a polynomial P in n variables is defined as the meanof log |P | restricted to the standard n-torus (Tn = {(x1 , . . . , xn ) ∈ (C∗ )n : |xi | = 1, ∀ 1 ≤ i ≤n}). The Mahler measure has been related to special values of L-functions, and this has been
explained in terms of regulators. Bertin proved several formulas relating the Mahler measure of K3-surfaces to the L-function of its transcendental lattice. The generalized Mahler
measure is defined as the integral over an arbitrary torus. In this talk, I will discuss few
results we obtained involving the relation between the generalized Mahler measure and thestandard (logarithmic) Mahler measure of one of Bertin’s K3-surfaces.

Stelios Sachpazis, Université de Montréal (student)
On multiplicative functions with small averages
There are results of analytic number theory which use information about the prime valuesof a multiplicative function, in order to extract information about its averages. Examples of such results include Wirsing’s theorem and the Landau-Selberg-Delange method. In
this talk, we are interested in the opposite direction. In particular, we are going to seethat if f is a suitable divisor-bounded multiplicative function with small partial sums, thenf (p) ≈ −piγ1 − . . . − piγm on average, where the γj ’s are the imaginary parts of the zeros of
the Dirichet series corresponding to f . This extends an existing result of Koukoulopoulos
and Soundararajan and it is built upon ideas coming from previous work of Koukoulopoulos
for the case where |f | ≤ 1.

Andrew Schultz, Wellesley College
Realizing module structures of square power classes over biquadratic extensions
Recently the Galois module structure for square power classes has been computed in the casewhere the underlying Galois group is the Klein 4 group. Despite the fact that the modular
representation theory allows for an infinite number of non-isomorphic indecomposable types,
it turns out that this particular module features at most 9 classes of indecomposable summands. We discuss the realizability of these summand types by drawing ideas from Galois

14embedding problems and the Brauer group.

Saurabh Singh, Indian Institute of Technology
Delta symbol and its applications
The circle method is a method employed by Hardy, Ramanujan, and Littlewood to solve
many asymptotic problems in additive number theory, particularly in deriving an asymptotic
formula for the partition function p(n). In this talk we shallfirst recall the formula developed by Kloosterman and Duke-Friedlander-Iwaniec and thendiscuss some applications of the delta symbol.

Eric Stubley, McGill University
Classical forms of low weight in p-adic families of ordinary modular forms
I’ll discuss a new strategy for proving theorems of the flavour “a p-adic family of ordinary
modular forms contains infinitely many classical weight one forms if and only if it has complex multiplication.” The original version of this theorem is due to Ghate–Vatsal, whose
proof makes heavy use of the fact that the Galois representation attached to a classicalweight one eigenform has finite image. I’ll discuss a new proof strategy which avoids using
this finite image property, and sketch how this strategy can be applied to new settings suchas partial weight one Hilbert modular forms.

John Voight, Dartmouth College
Stickelberger’s discriminant theorem for algebras
Stickelberger proved that the discriminant of a number field is congruent to 0 or 1 modulo
4. We generalize this to an arbitrary (not necessarily commutative) ring of finite rank overZ using techniques from linear algebra. Our proof, which only relies on elementary matrix
identities, is new even in the classical case. This is joint work with Asher Auel and Owen
Biesel.

Zhining Wei, Ohio State University (student)
Linear Relations of Siegel Poincaré Series and Non-vanishing of the Central Values of Spinor
L-functions
In this paper, we will first investigate the linear relations of a one parameter family of Siegel
Poincaré series. Then we give the applications to the non-vanishing of Fourier coefficients ofSiegel cusp eigenforms and the central values.

15

Elad Zelingher, Yale University (student)
On regularization of integrals of matrix coefficients associated with spherical models
The Ichino-Ikeda conjecture is a refinement of the Gan-Gross-Prasad conjecture. It roughly
states that the Gan-Gross-Prasad period can be expressed as an infinite product of local
integrals of matrix coefficients. However, in order to define these local integrals, one must
assume that the representations involved are tempered everywhere. In this talk, I will explain how to extend the definition of the local integrals for non-tempered representations forthe case of principal series.

Joshua Zelinsky, Hopkins School
Mertens’ Theorem and perfect numbers
Given a perfect number N , we are interested in inequalities relating N , p(N ), and ω, where
p is the smallest prime factor of N , and where ω(N ) is the number of distinct prime factorsof N . We will discuss recent results in this direction, and their connection to Mertens’ theorem. We will also discuss open question about two functions closely connected to Mertens’
theorem.

Peter Zenz, McGill University (student)
Quantum Variance for holomorphic Hecke cusp forms on the vertical geodesic
In this talk we explore a distribution result for holomorphic Hecke cusp forms on the verticalgeodesic. More precisely, we show how to evaluate the quantum variance of holomorphic
Hecke cusp forms on the vertical geodesic for smooth, compactly supported test functions.
The variance is related to an averaged shifted-convolution problem that we evaluate asymptotically. We encounter an off-diagonal term that matches exactly with a certain diagonalterm, a feature reminiscent of moments of L-functions.

Luochen Zhao, Johns Hopkins University (student)
Sum expressions for Kubota-Leopoldt p-adic L-functions
An observation due to Delbourgo shows that Kubota-Leopoldt p-adic L-functions can be
written as convergent infinite sums, assuming p is odd. We will discuss how these sum expressions hold in general without the assumption, by using integral representations of suchp-adic L-functions. Interesting applications, such as a direct proof of the Ferrero-Greenberg
formula, will also be presented.