Brandon Alberts, University of Wisconsin (grad)
Nonabelian Cohen-Lenstra Moments for the Quaternion Group
In this talk I will briefly review a conjecture of Melanie Wood generalizing
Cohen-Lenstra heuristics for counting the expected number of unramified G-extensions of
quadratic fields, for G any fixed finite group. I verify this conjecture for the case when
G is the quaternion group of order 8. Moreover, in joint work with Jack Klys we
asymptotically count the number of unramified G-extensions of quadratic fields of
discriminant smaller than X as X goes to infinity.
Angelica Babei, Dartmouth College (grad)
Tiled orders and the building for $SL_n(\mathbb{Q}_p)$
In finding the formula for traces of Hecke operators, Hijikata used the fact that the
normalizer up to units of an Eichler order of $M_2(\mathbb{Q}_p)$ has size 2. We can obtain
the latter result by studying the Bruhat-Tits tree for $\operatorname{SL}_2(\mathbb{Q}_p)$ where Eichler
orders correspond to pairs of vertices and the path between them. The normalizer of a given
Eichler order then acts on the tree by swapping the two vertices. Analogously, the
building for $\operatorname{SL}_n(\mathbb{Q}_p)$ provides a way to find normalizers of a generalization of
Eichler orders called tiled orders, by studying the symmetries of convex polytopes
associated to these orders.
Lior Bary-Soroker, Tel-Aviv University
Twin primes going function field
The study of function field versions of number theoretic problems may
be used to get new insights and to provide new evidence for difficult
conjectures. In this talk I will discuss some of the recent progress on problems
on correlations of arithmetic functions, including the Hardy-Littlewood prime
tuple conjecture on the correlation of the prime indicator function which is the
quantative version of the twin prime conjecture.
I will also survey some of the main challenges we are facing in light of those
recent results.
Benjamin Breen, Dartmouth College (grad)
The 2-Selmer group for number fields of even degree
This talk focuses on the study of the 2-Selmer group for number fields with even
degree. The 2-Selmer group plays a crucial role in the relationship between the
class group and the narrow class group. We look at modeling the 2-Selmer group in
order to derive heuristics on the 2-rank of the narrow class group.
Hugo Chapdelaine, Université Laval
Real analytic Eisenstein series with non-trivial multiplier systems
We shall begin by recalling some known facts about classical modular forms with
non-trivial multiplier systems; the main example
being the unary theta function of weight $1/2$ considered by Riemann in his
famous proof of the meromorphic continuation and functional equation of
the Riemann zeta function. We shall then proceed to define two types of
real analytic Eisenstein series with non-trivial multiplier systems; in particular,
this approach allows the construction of families of real analytic
Eisenstein of half-integral weights. Finally, if time permits, we shall
present some possible applications of these real analytic Eisenstein series
via the so-called theta correspondance (or some of its more recent refinements
which lead for example to the so-called "Borcherds products").
Sara Chari, Dartmouth College (grad)
Metacommutation of Primes in Quaternion Orders of Class Number One
In a quaternion order of class number one, factorization of elements
is unique only up to units and metacommutation, or rearrangement of the prime
factors. The fact that multiplication is not commutative causes an element to
induce a permutation on the set of primes of a given norm. We discuss this
permutation and previously known results about the cycle structure, sign, and
number of fixed points in the case of the Hurwitz order in the Hamiltonians. We
generalize these results to quaternion orders of class number one.
Michael Chou, Tufts University
Growth of torsion of elliptic curves from $\mathbb{Q}$ to the maximal abelian
extension.
Torsion an elliptic curve over a number field is finite due to the
Mordell-Weil theorem. However, even in certain infinite extensions of $\mathbb{Q}$ we have
that torsion is finite. Ribet proved that, when base extended to the maximal abelian
extension of $\mathbb{Q}$, the torsion of an elliptic curve over $\mathbb{Q}$ is finite. In
this talk, we give a classification of all possible torsion structures appearing in this
way.
Edgar Costa, Dartmouth College
Computing zeta functions of nondegenerate toric hypersurfaces
We report on an ongoing joint project with Kiran Kedlaya and David Harvey on the
computation of zeta functions of nondegenerate toric hypersurfaces over finite
fields using p-adic cohomology
Harris Daniels, Amherst College
Torsion subgroups of rational elliptic curves over
infinite extensions of $\mathbf{Q}$.
Let $E/\mathbf{Q}$ be an elliptic curve. In this talk we consider the question of
what torsion subgroups can occur when we base extend $E$ to some infinite field extensions
$K/\mathbf{Q}$. For example, if $K$ is the compositum of all cubic extensions of $\mathbf{Q}$, we
show that the torsion subgroup of $E(K)$ is finite and determine 20 possibilities for its
structure. This work is partially joint with Álvaro Lozano-Robledo, Filip Najman, and
Andrew Sutherland.
Henri Darmon, McGill University
A tamely ramified Perrin-Riou conjecture
I will describe a generalisation of a conjecture of Perrin Riou which I
studied recently with Alan Lauder, and a "tamely ramified" refinement of it, in
the spirit of Mazur and Tate's tame refinements of the p-adic Birch and
Swinnerton Dyer conjecture. The latter is the fruit of ongoing exchanges with
Michael Harris, Victor Rotger and Akshay Venkatesh.
David Dummit, University of Vermont
Classes of order 4 in the strict class group of number fields and
unramified quadratic extensions of unit type
This talk will explain why, for example, the strict class group of a
totally real field K with a totally positive system of fundamental units contains
at least (n-1)/2 (n odd) or n/2 -1 (n even) independent elements of order 4
and make some remarks on unramified extensions generated by square roots
of units.
John Friedlander, University of Toronto
Exceptional characters and their consequences
Anna Haensch, Duquesne University
Almost universal ternary sums of polygonal numbers
In 1796 Gauss showed that every natural number can be written as the sum of three
triangular numbers. In 2009, Chan and Oh determined when a weighted sum of
triangular numbers (i.e. triangular numbers with coefficients) represents all but
finitely many natural numbers. We say such a sum is almost universal. In this
talk we will determine when a sum of three generalized m-gonal numbers is almost
universal. We will approach this question first from an algebraic, and then from
analytic point of view, exploiting the capabilities of each method, and realizing
new connections between the machinery.
Farshid Hajir, University of Massachusetts
On the invariant factors of class groups in unramified towers of number
fields
In this talk, I will report on joint work with Christian Maire in
which we study how the exponents of class groups vary in towers contained in L/K
where L is the maximal unramified p-extension of K. Note that according to a
conjecture of Fontaine and Mazur, such towers do not have p-adic analytic Galois
group. Our main result is that in contrast to the p-adic analytic case, the
average exponent of p-class groups (a concept we define) in unramified p-towers
can be bounded. The proof uses a combination of group theory and a key arithmetic
result of Tsfasman and Vladut generalizing a classic theorem of Brauer and
Siegel.
Henryk Iwaniec, Rutgers University
Families of orthogonal arithmetic functions
Dubi Kelmer, Boston College
Shrinking target problems, homogenous dynamics, and Diophantine
approximations
The shrinking target problem for a dynamical system tries to answer the question
of how fast can a sequence of targets shrink, so that a typical orbit will keep
hitting them indefinitely. In this talk I will describe some new and old results
on this problem for homogenous dynamics, that is, the dynamical systems given by
group actions on a homogenous space. I will then describe some applications of
such results to problems in Diophantine approximations.
Yujin Kim (Columbia, undergrad), Steven J. Miller (Williams),
and Shannon Sweitzer (UC Riverside, undergrad)
Variance of Gaussian Primes Across Sectors and The Hecke L-Function Ratios
Conjecture
A Gaussian prime is a prime element in the ring of Gaussian integers
$\mathbb{Z}[i]$. While many statistical properties of the ordinary primes carry
over to this new setting, as the Gaussian integers lie in the plane interesting
questions about their geometric properties can be asked, which have no classical
analogue among the ordinary primes. Specifically, to each Gaussian prime $a+bi$
we associate an angle whose tangent is the ratio $b/a$. Hecke showed that these
angles are uniformly distributed as $p$ varies, and Kubilius proved uniform
distribution in somewhat short arcs. Motivated by a random matrix theory (RMT)
model and a function field analogue, Rudnick and Waxman gave a conjecture for the
variance of such angles across short arcs. While many number theoretic results
show agreement between the main term of a calculation and RMT, far fewer results
exist in which the secondary terms are known. We report on progress in applying
the $L$-function Ratios Conjecture to a family of Hecke $L$-functions to derive a
formula which computes the variance of Gaussian primes across short arcs. Our
result breaks the universality of behavior picked up by RMT by capturing the
subtle arithmetic properties found in the correction factor. Though previous
conjectures deviated from empirical data, we manage to accurately match the
theory to numerical computations by including these lower order terms. Our work
extends the ratio conjecture's machinery to a new family of Hecke $L$-functions,
and demonstrates the application of this machinery to a question of longstanding
arithmetic interest.
Paul Kinlaw, Husson University
An Explicit Lower Bound for the Counting Function of Squarefree Products of Three
Primes.
We will look at recent joint work with Jonathan Bayless on the problem of
determining a tight explicit lower bound for the counting function of squarefree products
of three prime factors. Time permitting, we will discuss related problems.
Hershy Kisilevsky, Concordia University
The Non-Square Part of Analytic Sha
Let $L(E/\mathbb{Q},s)$ be the $L$-function of an elliptic curve $E$ defined over the rational
field $\mathbb{Q}$.
We examine the central value $L(E,1,\chi)$ of non-vanishing twists of $L(E/\mathbb{Q},s)$ by
Dirichlet characters $\chi$. In particular we show that the square-free part of the
algebraic part of $L(E/F,1)$ is non-trivial for infinitely many extensions $F/\mathbb{Q}$.
Kim Klinger-Logan, University of Minnesota (grad)
Differential equations in automorphic forms
Physicists such as Greene, Vanhove, et al. show that
differential equations involving automorphic forms govern the behavior
of gravitons. One particular point of interest is solutions to
$(\Delta-\lambda)u=E_{\alpha} E_{\beta}$ on the exceptional group $E_8$.
We use spectral theory solve $(\Delta-\lambda)u=E_{\alpha}E_{\beta}$ on
the more natural space $\operatorname{SL}_2(\mathbb{Z})\backslash
\operatorname{SL}_2(\mathbb{R})$.
The construction of such a solution uses Arthur truncation, the
Maass-Selberg formula, and automorphic Sobolev spaces.
Dimitris Koukoulopoulos, Université de Montréal
Pretentious methods for L-functions
I will discuss ongoing work with K. Soundararajan on proving general prime number
theorems that extend classical results for L-functions. Our main theorem gives a
classification of those multiplicative functions whose partial sums present
significant cancellation. As a byproduct of our methods, a new proof of the
classical zero-free regions for L-functions is obtained via sieve theory.
Dijana Kreso, University of British Columbia
Decomposable recursively defined polynomials
In this talk I will present results that come from a joint work with Clemens Fuchs
and Christina Karolus from University of Salzburg (Austria). We studied complex
polynomials defined by a recurrence relation, which are decomposable, i.e.
representable as a composition of lower degree polynomials. Let
$(G_n(x))_{n=0}^{\infty}$ be a sequence of complex polynomials defined by a second
order recurrence relation. Assuming $G_n(x)=g(h(x))$ where $\deg g, \deg h\geq 2$,
we showed that under certain assumptions, and provided that $h(x)$ is not of
particular type, $\deg g$ may be bounded by a constant independent of $n$. Our
result is motivated by and similar in flavor to a result of Zannier from 2007 about
lacunary polynomials. The possible ways of writing a polynomial as a composition of
lower degree polynomials were studied by several authors. Results in this area of
mathematics have applications to various other areas, e.g. number theory, complex
analysis, etc. In my talk I will mention some Diophantine applications.
Niko Laaksonen, McGill University
Prime Geodesic Theorem in $\mathbb{H}^{3}$
On hyperbolic manifolds the lengths of primitive closed geodesics (prime
geodesics)
have many similarities with the usual prime numbers. In particular, they obey
an asymptotic distribution analogous to the Prime Number Theorem. In this talk we
present improvements on estimates for the error term in the Prime Geodesic
Theorem
for some hyperbolic 3-manifolds.
Matilde Lalin, Université de Montréal
A geometric generalization of the square sieve and applications to cyclic
covers
We study a generalization of the quadratic sieve to a geometric
setting. We apply this to counting points of bounded height on an l-cyclic cover
over the rational function field and we consider a question of Serre. In addition
to the geometric quadratic sieve, we use Fourier analysis over function fields,
deep results of Deligne and Katz about cancelation of mixed character sums over
finite fields, and a bound on the number of points of bounded height due to
Browning and Vishe.
This is joint work with A. Bucur, A. C. Cojocaru and L. B. Pierce.
Antonio Lei, Université Laval
Second Chern Classes for Supersingular Elliptic Curves
Classical Iwasawa Theory often studies Galois modules up to pseudo-null. For example, the
main conjecture relates the characteristic ideal of a Galois module to a p-adic L-function,
which does not contain any information about pseudo-null submodules contained inside the
Galois module.
A recent work of Bleher-Chinburg-Greenberg-Kakde-Pappas-Sharifi-Taylor describes some
pseudo-null modules in terms of Katz' p-adic L-functions. In this talk, we will explain how
to obtain a similar result for plus and minus p-adic L-functions coming from elliptic
curves with supersingular reduction. This is joint work with Bharathwaj Palvannan.
Robert Lemke-Oliver, Tufts University
Three-isogeny descent for quadratic twists of abelian varieties
We determine the average size of the $\phi$-Selmer group in any quadratic twist
family of abelian varieties having an isogeny $\phi$ of degree 3 over any number field.
This has several applications towards the rank statistics in such families of quadratic
twists. For example, it yields the first known quadratic twist families of absolutely
simple abelian varieties over $\mathbf{Q}$, of dimension greater than one, for which the average
rank is bounded; in fact, we obtain such twist families in arbitrarily large dimension. In
the case that $E/F$ is an elliptic curve admitting a 3-isogeny, we prove that the average
rank of its quadratic twists is bounded; if $F$ is totally real, we moreover show that a
positive proportion of these twists have rank 0 and a positive proportion have $3$-Selmer
rank 1. We also obtain consequences for torsion in Tate-Shafarevich groups of elliptic
curves. This is joint work with Manjul Bhargava, Zev Klagsbrun, and Ari Shnidman.
Claude Levesque, Université Laval
On Thue equations.
This will be a succinct report on the results
of Michel Waldschmidt and myself
on the Thue equations $F(X,Y)=m$.
Wanlin Li, University of Wisconsin (grad)
Vanishing of hyperelliptic L-functions at the central point
We obtain a lower bound on the number of quadratic Dirichlet L-functions over the
rational function field which vanish at the central point $s = 1/2$. This is in
contract with the situation over the rational numbers, where a conjecture of
Chowla predicts there should be no such L-functions. The approach is based on the
observation that vanishing at the central point can be interpreted geometrically,
as the existence of a map to a fixed abelian variety from the hyperelliptic curve
associated to the character.
Jared Lichtman (Dartmouth, undergrad),
Steven J. Miller (Williams), Eric Winsor (Michigan, undergrad), and
Jianing Yang (Colby, undergrad)
Lower order biases in Fourier coefficients of elliptic curve and cuspidal
newform families
Random Matrix Theory, originally developed to model energy levels of heavy
nuclei, has had remarkable success in predicting the main term of the behavior of
zeros of $L$-functions. Unfortunately these models cannot see the lower order
terms, where the arithmetic of the family lives; similar to the Central Limit
Theorem, the behavior of these terms control the rate of convergence. We prove
results on lower order terms for many families of $L$-functions. In particular,
we look at one-parameter families of elliptic curves, where these terms are
related to understanding the observed excess rank. In all such families
previously studied, the second moment of the Fourier coefficients of the
$L$-functions had a negative bias (i.e., converged to the main term from below).
We extend these techniques to more general families of elliptic curves and prove
the bias is present there as well. Building on the Four (Gauss) and Six (Euler)
Order Theorem, we also compute arbitrary moments of special families of elliptic
curves with $j(T) = \{0, 1728\}$ and see biases in higher moments. As the
modularity theorem connects elliptic curves to weight $2$ modular forms, it is
natural to consider biases in other families of $L$-functions. Via the Petersson
formula, we compute all higher moments of Fourier coefficients from Dirichlet
$L$-functions and cuspidal newforms of level $N$ and weight $k$.
Álvaro Lozano-Robledo, University of
Connecticut
A probabilistic model for the distribution of ranks of elliptic curves
over $\mathbb Q$
In this talk, we propose a new probabilistic model for the distribution
of ranks of elliptic curves in families of fixed Selmer rank, and compare the
predictions with previous results, and with the databases of curves over the
rationals that we have at our disposal. In addition, we document a phenomenon we
refer to as Selmer bias that seems to play an important role in the data and in
our models.
Christian Maire, Besançon / Cornell University
Analytic Lie extensions of number fields with cyclic fixed points and tame
ramification
(Joint work with Farshid Hajir.)
In this lecture, I will present an extension of the strategy of Boston from the
90's concerning the tame version of the Fontaine-Mazur conjecture. More
precisely, we are interested in using group-theoretical information to derive
consequences for tamely ramified Galois representations, specially for the groups
Sl(n,Z_p).
Robert McDonald, University of Connecticut (grad)
Torsion Subgroups of Elliptic Curves over Function Fields of Genus 0
Let $K=\mathbb F_q(T)$ be a function field over a finite field of characteristic
$p$, and $E/K$ be an elliptic curve. It is known that $E(K)$ is a finitely
generated abelian group, and that for a given $p$, there is a finite, effectively
calculable, list of possible torsion subgroups which can appear. In this talk, we
show the complete list of possible full torsion subgroups which can appear, and
appear infinitely often, for a given $p$.
Xianchang Meng, CRM, Université de Montréal
Chebyshev's bias for total number of prime factors of arithmetic
progressions
We consider the total number of prime factors for integers up to x
among different arithmetic progressions. Numerical experiments suggest that some
arithmetic progressions have more number of prime factors than others. Greg
Martin conjectured that this strong bias should be true for all x large
enough. Under the Generalized Riemann Hypothesis and Linear Independence
Conjecture, we prove that this phenomenon is true for a set of x with logarithmic
density 1.
Michael Mossinghoff, Davidson College
The Lind-Lehmer constant for certain $p$-groups
In 2005, Lind formulated an analogue of Lehmer's well-known problem
regarding the Mahler measure for general compact abelian groups, and
defined a Lehmer constant for each group. Since then, this Lind-Lehmer
constant has been determined for many finite abelian groups, including all
but a thin set of finite cyclic groups. We establish some new congruences
satisfied by the Lind Mahler measure for finite $p$-groups, and use them to
determine the Lind-Lehmer constant for many groups of this form. We also
develop an algorithm that determines a small set of possible values for
a given $p$-group of a particular form. This method is remarkably
effective, producing just one permissible value in all but a handful of
trials. This is joint work with Dilum De Silva, Vincent Pigno, and Chris
Pinner.
Michael Musty, Dartmouth College (grad)
2-solvable Belyi maps
In this talk we consider Galois Belyi maps that have
a 2-solvable Galois group. By a theorem of Beckmann, the associated curve has good
reduction away from 2. We discuss a method to explicitly compute such Belyi maps and
provide examples.
Vincent Ouellet, Université Laval (grad)
On the middle prime factors of integers
We obtain an asymptotic formula for the sum of the reciprocals of the so-called
$\beta$-positioned prime factors of integers $n\le x$ counting multiplicities,
thus improving and generalizing an earlier result of De Koninck and Luca about
the middle prime factor of an integer.
This is joint work with Jean-Marie De Koninck and Nicolas Doyon (Laval
University).
Corentin Perret-Gentil, CRM, Montréal
Integral monodromy groups and applications in number theory
I will talk about the determination of integral monodromy groups associated to
Kloosterman sums over finite fields, and applications to the study of the
distribution of the latter.
Carl Pomerance, Dartmouth College
New results on an ancient function
I refer to the function of Pythagoras that sends $n$ to
$s(n)$, the sum of the divisors of $n$ that are less than $n$.
A still-open conjecture of Catalan & Dickson: each orbit
in the $s$-dynamical system (i.e., $n,s(n),s(s(n)),\dots$)
is bounded. Modeling such a sequence as a random geometric
progression, Bosma & Kane showed that the average of $\log(s(n)/n)$
for $n$ even is a constant $\lambda<0$ (and for $n$ odd, it's $-\infty$).
A new result: the average of $\log(s(s(n))/s(n))$ for $n$ even is
also $\lambda$. Pythagoras noted 2-cycles in the $s$-dynamical
system, the so-called amicable numbers. It's been known since
1981 that the reciprocal sum of the amicable numbers is finite,
and in 2011 Bayless & Klyve showed this sum is $<656{,}000{,}000$.
Recently with Nguyen we improved the bound to $222$. I also report
on some new results with Pollack & Thompson on fibers of $s$.
Gautier Ponsinet, Université Laval (grad)
On the stucture of signed Selmer groups for abelian varities at
supersingular primes
Let $p$ be an odd prime and $A$ an abelian varietiy defined over $Q$.
When studying Iwasawa theory for $A$, one looks at the structure of the
$p$-Selmer groups associated to $A$ over the $\mathbb{Z}_p$ cyclotomique
extension. Unfortunately, when $A$ has good supersingular reduction at $p$, this
Selmer group is not a cotorsion module over the Iwasawa algebra. (All these terms
will be explained in the talk). Following ideas of S. Kobayashi and F. Sprung, K.
Büyükboduk and A. Lei define signed Selmer groups for $A$ which are
conjectured to be cotorsion module. In this talk, assuming this conjecture, I
will explain some results about the structure of these signed Selmer groups
similar to classical results about the $p$-Selmer groups in the ordinary case.
Kyle Pratt, University of Illinois Urbana Champaign (grad)
A lower bound for the least prime in an arithmetic progression
Fix $k$ a positive integer, and let $\ell$ be coprime to $k$. Let $p(k,\ell)$ denote
the smallest prime in the residue class $\ell \pmod{k}$, and set $P(k)$ to be the
maximum of $p(k,\ell)$ over residue classes $\ell$.
In this talk I will describe joint work with Junxian Li and George Shakan,
in which we showed that for almost every $k$ one has
$P(k) \gg \phi(k) \log k \log_2 k \log_4 k/\log_3 k$.
This improves an earlier bound
of Pomerance, and answers a question of Ford, Green, Konyagin, Maynard,
and Tao. I will discuss some ideas used in the proof, which has roots in
recent work on large gaps between primes. I will also discuss some heuristics
about the size of $P(k)$.
Sam Schiavone, Dartmouth College (grad)
Computing A Database of Belyi Maps: A Progress Report
In the paper Numerical calculation of three-point branched covers of the
projective line, we presented a method for computing equations of Belyi maps based on the
correspondence described by Grothendieck in his celebrated work Esquisse d'un Programme.
In this talk, we discuss the progress we have made in exhaustively computing all Belyi maps
of low degree using this method. Joint work with Michael Musty, Jeroen Sijsling, and John
Voight.
Ari Shnidman, Boston College
Intersections of Heegner-Drinfeld cycles
I'll present a formula which relates the intersection of two
Heegner-Drinfeld cycles in the moduli stack of shtukas to derivatives
of certain toric period integrals. The formula is a "higher order"
generalization of the Gross-Kohnen-Zagier formula, in the function
field setting, and is inspired by recent work of Yun and W. Zhang.
This is joint work with Ben Howard.
Caleb Shor, Western New England University
A characterization of the complement of a free numerical semigroup
Let $\mathbb{N}_0$ denote the set of non-negative integers. A numerical semigroup $S$ is a
subset of $\mathbb{N}_0$ that is closed under addition, contains 0, and has finite
complement in $\mathbb{N}_0$. Two interesting quantities are the Frobenius number and genus
of $S$ which are, respectively, the maximum and cardinality of its complement. Every
numerical semigroup can be written as the set of non-negative linear combinations of
elements of some finite set $G=\{g_1,\dots,g_k\}$ with $\gcd(G)=1$. If $k\le 2$, then
formulas for the Frobenius number and genus of $S$ are well-known, dating to the work of
Sylvester in the 19th century. If $k\ge 3$, far less is known.
In this talk, we will consider free numerical semigroups, which are generated by sets of
integers that can be ordered to satisfy some certain smoothness conditions. (Examples of
such sets include geometric sequences, compound sequences, supersymmetric sequences, and
certain arithmetic sequences of length 3.) The smoothness conditions lead to an explicit
description of the Apéry set of a numerical semigroup, which we can use along with a
generalization of an identity of Tuenter. This gives us a complete characterization of the
complement of a free numerical semigroup, leading to results for the Frobenius number, the
genus, power sums of complement elements, and other interesting quantities. As an
application we will see how to quickly sum all of the so-called "non-McNugget numbers."
Daniel Smertnig, Dartmouth College
Arithmetical invariants of local quaternion orders
In a Noetherian ring every non-zero-divisor can be written as a product
of atoms (irreducible elements). Usually, these factorizations are not
unique. Arithmetical invariants quantify the failure of this uniqueness.
In the noncommutative setting, recently arithmetical invariants of
hereditary orders in central simple algebras over global fields have
been studied. The non-hereditary case has so far not been considered. As
a first step towards this, we consider quaternion orders over discrete
valuation rings. We characterize finiteness of the elasticity, a basic
invariant. The proof, which splits into several cases, relies on a
classification of quaternion orders in terms of ternary quadratic forms.
Joint work with Nicholas R. Baeth.
Naomi Tanabe, Bowdoin College
Sign of Fourier Coefficients for Hilbert Modular Forms
Signs of Fourier coefficients have been studied extensively in various
setting. In this talk, I will survey some results on sign changes for Hilbert
modular forms and explain our resent result concerning half integral weight
forms. It is a joint project with Surjeet Kaushik and Narasimha Kumar.
Guhan Venkat, Université Laval
Plus/Minus Beilinson - Flach Euler system
In this talk, we will see how one can ontain an integral Plus/Minus
Euler system associated to the Rankin Selberg product of a pair of supersingular
modular forms starting from an Euler system if rank 2. The results have Iwasawa
theoretic applications and is joint work with Kazim Buyukboduk, Antonio Lei and
David Loeffler.
Jan Vonk, McGill University
Fundamental groups and rational points on curves
It has long been known that the number of rational points on a curve of genus at
least 2 has finitely many rational points. I will discuss a theory of Minhyong Kim which
often provides alternative proofs, that have the advantage they can be made effective. This
will be applied to the "cursed modular curve" $X_{ns}(13)$, resulting in a complete
determination of rational points where traditional methods fail. This is joint work with
Balakrishnan, Dogra, Müller, and Tuitman.
Erik Wallace, University of Connecticut (grad)
Bounds of the rank of the Mordell--Weil group of Jacobians of
Hyperelliptic Curves
In a recent article written in collaboration with Álvaro Lozano-Robledo
and Harris Daniels
we extend work of Lehmer, Shanks, and Washington on cyclic extensions,
and elliptic curves associated to the simplest cubic fields. In particular, we give
families of examples of hyperelliptic curves $C: y^2=f(x)$ defined over $\mathbb{Q}$, with
$f(x)$ of degree $p$, where $p$ is a Sophie Germain prime, such that the rank of the
Mordell--Weil group of the jacobian $J/\mathbb{Q}$ of $C$ is bounded by the genus of $C$
and the $2$-rank of the class group of the (cyclic) field defined by $f(x)$,
and exhibit examples where this bound is sharp.
Liang Xiao, University of Connecticut
Some remarks on the ghost conjecture of Bergdall and Pollack
The goal of this talk is to understand the $p$-adic valuations of
the $U_p$-operators acting on the space of modular forms. In 1990s, Gouvea
and Mazur computed many numerical data and made interesting conjectures,
such as the Gouvea-Mazur conjecture, and Gouvea's conjecture on
distribution of slopes. This topic was later developed by Buzzard and
Calegari, and their students, by providing finer computational data and
more precise conjectures. Recently, Bergdall and Pollack introduced the
ghost conjecture that unifies these previous conjectures, and give the
precise information of slopes of modular forms, at least in the
"Buzzard-regular" case.
In this talk, I will reformulate Bergdall and Pollack's conjecture in a
purely abstract setting, and explain where it fits in the p-adic local
Langlands conjecture for $GL_2(\mathbf{Q}_p)$. If time permits, I will discuss
some theoretical evidence of this conjecture. This is a joint work with
Ruochuan Liu and Bin Zhao.
Shucheng Yu, Boston College (grad)
Logarithm laws for hitting time functions on homogeneous spaces
In a probability space with an ergodic action, fix a measurable set of positive
measure, the first hitting time function measures the time needed for the orbit of a point
to enter this set. Intuitively, the smaller the set is, the larger the hitting time is. We
establish a logarithm law for hitting time functions on a finite volume homogeneous space
with the flow generated by a group element. The main ingredient of our proof is a uniform
rate of decay for matrix coefficients along any unbounded flow. This is joint work with
Dubi Kelmer.
Fan Zhou, University of Maine
Voronoi formula and its generalizations
The Voronoi formula is a Poisson-style summation formula connecting a
sum of Fourier coefficients of an automorphic form twisted by additive characters
and a contragredient sum. Firstly we present a proof of the Voronoi formula for
coefficients of a large class of L-functions, in the style of the classical
converse theorem of Weil. Our formula applies to full-level cusp forms,
Rankin-Selberg convolutions, and certain isobaric sums. The key ingredient is the
construction of a double Dirichlet series associated with these coefficients.
This is joint work with Eren Mehmet Kiral. Secondly we introduce
balanced Voronoi formulas, with hyper-Kloosterman sums of various dimensions on
both sides. Those generalize a formula for GL(4) with ordinary Kloosterman sums
on both sides by Miller-Li that was used by Blomer-Li-Miller to prove
nonvanishing of GL(4) L-functions by GL(2)-twists. This is joint work with
Stephen D. Miller.
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List of Participants
Brandon Alberts, University of Wisconsin (grad)
David Ayotte, Université Laval (grad)
Angelica Babei, Dartmouth College (grad)
Lior Bary-Soroker, Tel-Aviv University
David Bradley, University of Maine
Benjamin Breen, Dartmouth College (grad)
Crystel Bujold, Université de Montréal (grad)
Hugo Chapdelaine, Université Laval
Sara Chari, Dartmouth College (grad)
Jaeho Choi, University of Maine (grad)
Michael Chou, Tufts University
Edgar Costa, Dartmouth College
Harris Daniels, Amherst College
Henri Darmon, McGill University
David Dummit, University of Vermont
Michele Fornea, McGill University (grad)
John Friedlander, University of Toronto
Zhenguang Gao, Framingham State University
Ramesh Gupta, University of Maine
Pushpa Gupta, University of Maine
Anna Haensch, Duquesne University
Farshid Hajir, University of Massachusetts
Henryk Iwaniec, Rutgers University
Daniel Keliher, Tufts University (grad)
Dubi Kelmer, Boston College
Yujin Kim, Columbia University (undergrad)
Paul Kinlaw, Husson University
Hershy Kisilevsky, Concordia University
Kim Klinger-Logan, University of Minnesota (grad)
Andrew Knightly, University of Maine
Dimitris Koukoulopoulos, Université de Montréal
Dijana Kreso, University of British Columbia
Niko Laaksonen, McGill University
Matilde Lalin, Université de Montréal
Antonio Lei, Université Laval
Robert Lemke-Oliver, Tufts University
Claude Levesque, Université Laval
Wanlin Li, University of Wisconsin (grad)
Jared Lichtman, Dartmouth College (undergrad)
David Lilienfeldt, McGill University (grad)
Abel Lourenco, University of Maine (grad)
Álvaro Lozano-Robledo, University of Connecticut
Mostafa Mache, Université Laval (grad)
Christian Maire, Besançon / Cornell University
Robert McDonald, University of Connecticut (grad)
Xianchang Meng, CRM, Université de Montréal
Steven J. Miller, Williams College
Michael Mossinghoff, Davidson College
Michael Musty, Dartmouth College (grad)
Jungbae Nam, Concordia University (grad)
Kunjakanan Nath, Université de Montréal (grad)
Isabella Negrini, McGill University (grad)
Vincent Ouellet, Université Laval (grad)
Solly Parenti, University of Wisconsin (grad),
Zachary Parker, University of Vermont (grad),
Corentin Perret-Gentil, Université de Montréal
Nigel Pitt, University of Maine
Carl Pomerance, Dartmouth College
Gautier Ponsinet, Université Laval (grad)
Kyle Pratt, University of Illinois Urbana-Champaign (grad)
James Rickards, McGill University (grad)
Alicia Rossi, University of Vermont (undergrad)
Giovanni Rosso, Concordia University
Sam Schiavone, Dartmouth College (grad)
Ari Shnidman, Boston College
Caleb Shor, Western New England University
Daniel Smertning, Dartmouth College
Jerrod Smith, University of Maine
Chip Snyder, University of Maine
Shannon Sweitzer, UC Riverside (undergrad)
Naomi Tanabe, Bowdoin College
Guhan Venkat, Université Laval
Christelle Vincent, University of Vermont
Jan Vonk, McGill University
Erik Wallace, University of Connecticut
Zheng Wei, University of Maine
Benjamin Weiss, University of Maine
Eric Winsor, University of Michigan (undergrad)
Alic Wise, University of Maine (grad)
Liang Xiao, University of Connecticut
Peter Xu, McGill University (grad)
Jianing Yang, Colby College (undergrad)
Shucheng Yu, Boston College (grad)
Peter Zenz, McGill University (grad)
Fan Zhou, University of Maine
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