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.
Hugues Bellemare, McGill University (grad)
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.
Benjamin Breen, Dartmouth College (grad)
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.
Cédric Dion, Université Laval (grad)
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.
Anthony Doyon, Université Laval (undergrad)
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.
Matthew Friedrichsen, Tufts University (grad)
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
Bin Guan, CUNY (grad)
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.
Daniel Keliher, Tufts University (grad)
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)
Catherine Wahlenmayer, Gannon 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
Congruences of anticyclotomic p-adic L-functions
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.
Hao Li, Boston College (grad)
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.
David Lilienfeldt, McGill University (grad)
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.
Mostafa Mache, Université Laval (grad)
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.)
Caleb McWhorter, Syracuse University (grad)
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).
Grant Molnar, Dartmouth College (grad)
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.
Isabella Negrini, McGill University (grad)
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.
Antoine Poulin, Université Laval (undergrad)
Probabilitstic properties of p-adic polynomials
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.
Tomer Reiter, Emory University (grad)
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 $\mathbb{Q}(2^{\infty})$.
James Rickards, McGill University (grad)
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.
Andrei Shubin, Cal Tech (grad)
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
Additive Twists of Fourier Coefficients
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.
William Verreault, Université Laval (undergrad)
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.
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List of Participants
Keshav Aggarwal, University of Maine
Scott Ahlgren, University of Illinois Urbana-Champaign
Brandon Alberts, University of Connecticut
Eran Assaf, Dartmouth College
David Ayotte, Concordia University (grad)
Jonathan Bayless, Husson University
Hugues Bellemare, McGill University (grad)
Elliot Benjamin, Capella University
David Bradley, University of Maine
Benjamin Breen, Dartmouth College (grad)
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)
Jake Chinis, McGill University (grad)
Fatma Cicek, University of Rochester (grad)
Tyrone Crisp, University of Maine
Amy DeCelles, University of St. Thomas, Minnesota
Cédric Dion, Université Laval (grad)
Anthony Doyon, Université Laval (undergrad)
David Dummit, University of Vermont
Wilson Fotsing, Université Laval (grad)
Matthew Friedrichsen, Tufts University (grad)
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
Daniel Keliher, Tufts University (grad)
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
Hao Li, Boston College (grad)
David Lilienfeldt, McGill University (grad)
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)
Mostafa Mache, Université Laval (grad)
Céline Maistret, Boston University
Sacha Mangerel, Université de Montréal (grad)
Caleb McWhorter, Syracuse University (grad)
Steven J. Miller, Williams College
Grant Molnar, Dartmouth College (grad)
Maria Nastasescu, Northwester University
Kunjakanan Nath, Université de Montréal (grad)
Isabella Negrini, McGill University (grad)
Jennifer Park, Ohio State University
Nigel Pitt, University of Maine
Carl Pomerance, Dartmouth College
Antoine Poulin, Université Laval (undergrad)
Tomer Reiter, Emory University (grad)
James Rickards, McGill University (grad)
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)
William Verreault, Université Laval (undergrad)
John Voight, Dartmouth College
Catherine Wahlenmayer, Gannon University (undergrad)
Aled Walker, Université de Montréal
Matthew Welsh, Rutgers University
Joshua Zelinsky, Hopkins School, New Haven
Peter Zenz, McGill University (grad)
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