Harry Altman, University of Michigan
Well-Ordering Phenomena in Computation of Natural Numbers
In this talk we will consider the complexity of computing natural
numbers in two very simple models of computation. One, known simply as
"integer complexity", is the smallest number of 1's needed to write a
number using any combination and multiplication; for a number $n$, we
denote this $\|n\|$. The other, the addition chain length of $n$, $l(n)$, is
the smallest number of additions needed to make $n$ starting from 1, where
numers may be reused freely once they are made. For each we define a
notion of "defect" by subtracting off a logarithmic lower bound
-- for integer complexity, we consider $\|n\|-3\log_3(n)$, and for addition
chain length, we consider $l(n)-\log_2(n)$. Surprisingly, if we consider
the set of all integer complexity defects, this turns out to be a
well-ordered set, of order type $\omega^\omega$; and if we consider the set
of all addition chain defects, this is also a well-ordered set of order
type $\omega^\omega$. In this talk we will discuss more detailed forms of
these results as well as some speculative extensions of them, including
to other models of computation.
David Burt (Williams College) and
Blaine Talbut (University of Chicago)
Large Gaps Between Zeros of $L$-Functions Associated to GL(2) Cusp Forms.
The spacing of gaps between zeros of $L$-functions encode arithmetic information.
The first nontrivial bound on the normalized spacing of large gaps between zeros
of $L$-functions of cuspidal newforms on GL(2) was made last year by Barrett et
al. We look to improve upon this bound using a method of Hall; similar
techniques have successfully improved lower bounds for gaps between zeros of the
Riemann Zeta function and Dirichlet $L$-functions. We report on progress on the
first step, deriving a shifted twisted second moment for such $L$-functions,
where the twist is by an arbitrary, sufficiently short Dirichlet polynomial. This
is joint work with Owen Barrett, Steven J. Miller and Caroline
Turnage-Butterbaugh.
Hugo Chapdelaine, Université Laval
A new proof of the functional equation of a special class of real analytic
Eisenstein series.
Let $E(z,s)$ be a real analytic Eisenstein with a unitary integral
weight.
Here $z$ varies over the Poincare upper half-plane (the symmetric space
associated to the number
field $\mathbb{Q}$) and $s$ varies over the right half-plane $Re(s)>1$. In this
talk we will explain how one may prove
the analytic continuation and the functional equation of the function $s \mapsto
E(z,s)$ using the analytic Fredholm
theorem for a certain family of compact operators associated to the hyperbolic
Laplacian.
This result automatically implies the analytic continuation and the functional
equation of the coefficients
of the Fourier series of $z\mapsto E(z,s)$.
In particular, one obtains a new proof of a functional equation of a class of
partial
zeta functions constructed by the author some years ago. Initially, the
functional equation was proved
using the Poisson summation formula. In this sense, it is fair to say the
classical role played
by the Poisson summation formula is being replaced by some spectral analysis and
the analytic Fredholm theorem. This set of ideas take its roots from the
pioneering work of Lax and
Philips who applied scattering theory to the study of automorphic forms. All
this work generalizes to
an arbitrary number field $K$.
Liubomir Chiriac, University of Massachusetts
Vanishing Traces of Frobenius in Artin-type Representations
In this talk we address Serre's lacunarity question of Frobenius
traces in Artin-type representations. Assuming the irreducibility of the
adjoint action we give a bound on the density of the vanishing traces,
which is independent of the dimension of the representation. In addition,
we construct an infinite family of representations of finite groups with an
irreducible adjoint action.
Michael Chou, University of Connecticut
Torsion of Rational Elliptic Curves over Quartic Galois Number Fields
The classification of the torsion subgroup of elliptic curves over $\mathbb{Q}$
was determined by Mazur. The classification over quadratic number fields was
completed due to work of Kamienny, Kenku, and Momose. However, over cubic fields
the classification is already incomplete. In this talk we discuss a refined
version of this problem: let $E$ be an elliptic curve defined over $\mathbb{Q}$
and $K$ be a number field of degree $d$; what groups appear as
$E(K)_{\text{tors}}$? In particular, we will present a classification over all
quartic galois number fields $K$ and show how the techniques used may be applied
to other fields.
Harris Daniels, Amherst College
Torsion Subgroups of Rational Elliptic Curves Over the Compositum of All
Cubic Fields
Let $E/\mathbf{Q}$ be an elliptic curve and let $\mathbf{Q}(3^\infty)$ be the
compositum of all cubic extensions of $\mathbf{Q}$.
In this talk we show that the torsion subgroup of $E(\mathbf{Q}(3^\infty))$ is
finite and determine 20 possibilities for its structure, along with a complete
description of the $\overline{\mathbf{Q}}$-isomorphism classes of elliptic
curves that fall into each case. We provide rational parameterizations for each
of the 16 torsion structures that occur for infinitely many
$\overline{\mathbf{Q}}$-isomorphism classes of elliptic curves, and a complete
list of $j$-invariants for each of the 4 that do not. This is joint work with
Álvaro Lozano-Robledo, Filip Najman, and Andrew Sutherland.
Henri Darmon, McGill University
Generalised Kato classes
I will discuss the conjectured properties of the eponymous objects of the title
and their connection with the p-adic Birch-Swinnerton-Dyer conjecture in the
spirit of Kato and Perrin-Riou.
Chantal David, Concordia University
One-level density in one-parameter families of elliptic curves with non-zero
average root number
This is joint work with Sandro Bettin and
Christophe Delaunay.
We present in this talk a (conjectural) formula for the one-level density of general
one-parameter families of elliptic curves, in term of $n$, the rank of $E$ over $\mathbb{Q}(t)$
and the average root number $W_E$ over the family. In the general case, $W_E$ is zero, and
the one-level density is given by orthogonal symmetries as predicted by the conjectures of Katz
and Sarnak. In the exceptional cases where $W_E \neq 0$, we find that the statistics are
given by a weighted sum of even orthogonal and odd orthogonal symmetries. The most dramatic
and counter-intuitive cases occur when $W_E = \pm 1$. In that case, the one-level
density exhibits even orthogonal symmetries when $(-1)^n W_E = 1$ and odd
orthogonal symmetries when $(-1)^n W_E = -1$, and there is a shift of the
symmetries (between orthogonal odd and orthogonal even) when $n$ is odd.
We also build several one-parameter families of elliptic curves with $W_E \neq 0$,
and which exhibit the shifts of the symmetries.
Taylor Dupuy, Hebrew University / University of Vermont
Examples of Varieties of General Type over Function Fields Whose Rational
Points are Not Dense
The Lang-Bombieri-Noguchi conjecture says that a variety of general type
over Q has nondense Q-points. In this talk we exhibit some examples of varieties
satisfying an analogous conjecture over function fields (in characteristic
zero). These examples cannot be embedded into abelian varieties and hence
provide examples of Lang-Bombieri-Noguchi outside Mordell-Lang (Falting's
Theorem). This is joint work with Daniel Litt.
Zhenguang (Jeff) Gao, Framingham State University
Triangular numbers in the Jacobsthal family
Using congruences, second-order Diophantine equations, and linear
algebra, we identify Jacobsthal and Jacobsthal-Lucas numbers that are also
triangular numbers. The conclusion is that there are 5 triangular Jacobsthal
numbers and there is only 1 triangular Jacobsthal-Lucas number.
Leo Goldmakher, Williams College
Mock characters
I will discuss recent work (joint with Jean-Paul Allouche) on a class
of functions which are, in some sense, as close as possible to being Dirichlet
characters without *actually* being Dirichlet characters. The talk will contain
results, conjectures, and manipulatives.
Jeffrey Hatley, Union College
Elliptic curves with maximally disjoint division fields
This talk will discuss a recent paper of the same title, written
jointly with Harris Daniels and James Ricci. We exhibit an infinite
two-parameter family of elliptic curves whose direct product Galois
representation has maximal image. This maximality is closely related to the
arithmetic of the number fields generated by the elliptic curves' torsion
points.
Jeffrey Hoffstein, Brown University
Shifted multiple Dirichlet series and second moments of GL(2) L-series
I'll explain what shifted multiple Dirichlet series are. I'll also
describe recent joint work with Min Lee in which we apply the meromorphic
properties of shifted multiple Dirichlet series to the problem of expressing the
mean square of $GL(2)$ $L$-series averaged over characters as a main term plus an
error term.
Thomas Hulse, Colby College
Averaging Average Orders with Shifted Series.
Here we consider Dirichlet series where the coefficients are related
to the average order of Fourier coefficients of holomorphic cusp forms. By
considering the spectral decomposition of certain shifted sums, we are able to
obtain a meromorphic continuation of these Dirichlet series which in turn yields
information about the cancellation of partial sums of these Fourier
coefficients. Joint work with Chan Ieong Kuan, David Lowry-Duda and Alexander
Walker.
Dubi Kelmer, Boston College
Counting lattice points in a random lattice
Consider the counting function counting the number of lattice points of a
bounded norm as a function on the space of unimodular lattices. By considering
local averages of this function, and in particular averages on all shears of a
lattice, it is possible to give optimal bounds for the remainder that hold on
average for any compact set in the space of lattices.
Julee Kim, MIT
Asymptotic behavior of supercuspidal characters and Sato-Tate
equidistribution for families
We establish properties of family of automorphic representations as we
vary prescribed supercuspidal representations at a given finite set of primes. We
prove the limit multiplicity property with error terms. Thereby we obtain a
Sato-Tate equidistribution for the Hecke eigenvalues. The main problem is to show
that the orbital integrals of matrix coefficients of supercuspidal
representations with increasing formal degree on a connected reductive $p$-adic
group tend to zero uniformly for every noncentral semisimple element. I will
start with a brief survey on Sato-Tate conjecture and many examples. This is a
joint work with Sug Woo Shin and Nicholas Templier.
Paul Kinlaw, Husson University
Repeated values of Euler's function
We consider solutions of $\varphi(n)=\varphi(n+1)$ and
$\sigma(n)=\sigma(n+1)$. Both equations are conjectured to have infinitely many
solutions. Work of Erdös, Pomerance and Sárközy shows that the sum of
reciprocals of solutions is convergent.
We will discuss recent joint work with Jonathan Bayless, including explicit
bounds on the counting functions of smooth numbers as well as numbers with $k$
distinct prime factors. We use these results as tools to put explicit numerical
bounds on the sum of reciprocals of solutions of $\varphi(n)=\varphi(n+1)$ and
$\sigma(n)=\sigma(n+1)$. We show that there are infinitely many $n$ such that
$\varphi(n)=\varphi(n+k)$ for some $k<\sqrt[3]{n}$, and the same for $\sigma$.
Dimitris Koukoulopoulos, University of Montréal
Sums of Euler products and statistics of elliptic curves
I will present a new approach to several statistical questions about elliptic
curves over finite fields, such as the average Lang-Trotter conjecture and the
vertical Sato-Tate conjecture. The starting point is a theorem of Gekeler that
provides a probabilistic reinterpretation of Deuring's theorem about the number
of elliptic curves in a given isogeny class. In the heart of our approach lies a
general technical theorem about averages of Euler products. As a corollary of
this general result, we obtain new proofs of various results, some already known
and some of which are new. One of the new results is the vertical Sato-Tate
conjecture for very short intervals. This is joint work with Chantal David and
Ethan Smith.
David Krumm, Colby College
A local-global principle in the dynamics of quadratic polynomials
Let $K$ be a number field and let $f\in K[x]$ be a polynomial. For any
nonnegative integer $n$, let $f^n$ denote the $n$-fold composition of $f$ with
itself. We say that an element $\alpha\in K$ is periodic for $f$ if
there exists a positive integer $n$ such that $f^n(\alpha)=\alpha$. In that
case, the least such $n$ is called the period of $\alpha$. It is clear
that if $f$ has a point of period $n$ in $K$, then it has a point of period $n$
in every extension of $K$; in particular, for every place $v$ of $K$, $\,f$ has a
point of period $n$ in the completion $K_v$. In this talk we will discuss
whether the converse holds: if $f$ has a point of period $n$ in every completion
of $K$, must it then have a point of period $n$ in $K$?
Antonio Lei, Université Laval
Asymptotic behaviour of the Shafarevich-Tate groups of modular forms
Let $E$ be an elliptic curve with good ordinary reduction at a prime
$p$. Mazur has studied the $p$-primary part of the the Shafarevich-Tate group
of $E$ over the $\mathbb{Z}_p$-cyclotomic extensions of a number field. In
particular, he showed that there is an asymptotic formula for the size of these
groups in terms of the Iwasawa invariants of the Selmer group of $E$. This has
been generalized to supersingular primes by Kobayashi and Sprung using plus and
minus Selmer groups. In this talk, I shall discuss a generalization of these
results to modular forms using the machinery of Wach modules developped by
Loeffler, Zerbes and myself.
Patrick Letendre, Université Laval
New upper bounds for the number of divisors function.
Let $\tau(n)$ stand for the number of divisors of the positive integer
$n$. We obtain new upper bounds for $\tau(n)$ in terms of $n$ and the number of
distinct prime factors of $n$. This is joint work with Jean-Marie De Koninck
(Laval University).
Claude Levesque, Université Laval
On Thue equations.
We will survey some joint results with Michel Waldschmidt
on Thue equations.
Li-Mei Lim, Bard College at Simon's Rock
Counting Square Discriminants
Hee Oh and Nimish Shah prove that the number of integral binary quadratic forms
whose coefficients are bounded by a quantity $X$, and with discriminant a fixed square
integer $d$, is $cX\log X+O(X(\log X)^{3/4})$. This result was obtained by the use of ergodic
methods. Here we use the method of shifted convolution sums of Fourier coefficients of
certain automorphic forms to obtain a sharpened result of a related asymptotic, obtaining
a second main term and an error of $O(X^{1/2})$.
Álvaro Lozano-Robledo, University of
Connecticut
On the minimal degree of definition of $p$-primary torsion subgroups of elliptic
curves
In this talk, we discuss the minimal degree $[K(T):K]$ of a $p$-subgroup
$T\subseteq E(\overline{K})_\text{tors}$ for an elliptic curve $E/K$ defined
over a number field $K$. Our results depend on the shape of the image of the
$p$-adic Galois representation
$\rho_{E,p^\infty}:\operatorname{Gal}(\overline{K}/K)\to \operatorname{GL}(2,\mathbf{Z}_p)$. However,
we are able to show that there are certain uniform bounds for the minimal degree
of definition of $T$. When the results are applied to $K=\mathbb{Q}$ and $p=2$,
we obtain a divisibility condition on the minimal degree of definition of any
subgroup of $E[2^n]$ that is best possible. This is joint work with Enrique
González-Jiménez (UAM).
Christian Maire, Besançon / CRM / University of Montréal
Cohomological dimension, number fields and ramification
In this talk I will discuss on pro-$p$-extensions of number fields for which the
cohomological dimension is finite. I will show how to produce some (very different)
arithmetic situations where the dimension is at most 2.
Sarah Manski (Kalamazoo College)
A Ramsey Theoretic Approach to Finite Fields and Quaternions.
Ramsey theory concerns itself with how large a set needs to be for a certain
structure to arise. We concentrate on sets avoiding 3-term geometric
progressions. Previous work studied this problem in the integers and number
fields; we resolve analogous problems for polynomials over finite fields and in
the Hurwitz Quaternions. New features emerge in the function field case -- the
proofs are distinctly combinatorial, a feature not seen in other cases. We take
advantage of the combinatorics arising from finite characteristic through
counting $q$-smooth elements and the number of irreducible polynomials. In the
Hurwitz Quaternions, the loss of commutativity greatly complicates the arguments
and affects the limiting behavior. We construct maximally sized sets of Hurwitz
quaternions that avoid geometric progressions up to units and bound their
densities, while adjusting for unique properties of the ring. The proofs involve
a mix of the algebra of the quaternions with an analysis of the resulting
infinite products. This work is joint with Megumi Asada, Eva Fourakis, Eli
Goldstein, Gwyn Moreland, Nathan McNew and Steven J. Miller.
Kenneth McMurdy, Ramapo College of New Jersey
Elliptic curves with non-abelian entanglements.
Let $K$ be a number field. An elliptic curve $E/K$ is said to have a non-abelian
entanglement if there are relatively prime positive integers, $m_1$ and $m_2$,
such that $K(E[m_1])\cap K(E[m_2])$ is a non-abelian Galois extension of K. In
this talk, we will discuss our ongoing efforts to classify, using explicit
methods, all infinite families of elliptic curves $E/K$, for a fixed $K$, with
non-abelian entanglements. This problem is closely related to that of
determining when the image of $\rho_E$ in $\operatorname{GL}_2(\hat{Z})$ is maximal, and to the
study of correction factors for various conjectural constants for elliptic
curves over $\mathbf{Q}$. This is joint work with Nathan Jones.
Steven Miller (Williams) and Kevin Yang (Harvard)
Biases in Moments of Satake Parameters and Models for $L$-function Zeros.
We report on two related projects concerning zeros of $L$-functions. The
Katz-Sarnak philosophy states that the statistics of zeros of families of
$L$-functions agree with those of eigenvalues of classical compact Lie groups in
the limit of large conductor (resp. dimension). In 2006, Miller discovered a
discrepancy in these statistics for elliptic curve $L$-functions of finite
conductor; in 2012 Duenez, Huynh, Keating, Miller and Snaith explained this
behavior through their Excised Orthogonal Ensemble, which depends on an
effective matrix size which incorporates the discreteness of central values of
the $L$-functions. We investigate a similar effective matrix size construction
for quadratic Dirichlet $L$-functions and quadratic twists of $L$-functions
given by symmetric squares of holomorphic cusp forms, both symplectic families.
We find a negative effective matrix size, indicating that for bounded conductor,
such families of $L$-functions statistically resemble the group SO(odd) for
suitable dimension, and also provide a formal justification for improved
statistics provided by the excised ensemble.
The second project involves the moments of the Satake parameters of
$L$-functions; these values play an important role in detecting arithmetic in
lower order terms in the 1-level density of families of $L$-functions, which
help explain the behavior of zeros at or near the central point. Recently, it
was observed that the second moments of the Satake parameters exhibit a bias in
many families of of elliptic curve $L$-functions, and it was conjectured that
similar biases exist in other families. We resolve the conjecture for various
families of Dirichlet $L$-functions and families of symmetric lifts of
holomorphic newforms on GL(2) by exploiting powerful trace formulas and
orthogonality relations. These projects are joint with Megumi Asada, Owen
Barrett, Eva Fourakis, Gwyn Moreland and Blaine Talbut.
Dan Nichols, University of Massachusetts
Analog of the Collatz Conjecture in Polynomial Rings of Characteristic 2
The Collatz conjecture (also known as the $3n+1$ problem) concerns the behavior
of integer sequences $\left\{ T^k(n) \right\}_{k=0}^\infty$, where $T$ is a
particular transformation map on the positive integers. In this talk we will
discuss an analog of the Collatz conjecture in polynomial rings of
characteristic 2 which exhibits some interesting properties. We will outline
some theoretical results, including a theorem concerning distribution of
stopping times analogous to one proved by Terras and Everett for the original
$3n+1$ problem. We will also present experimental data on stopping times and
cycle lengths.
Carl Pomerance, Dartmouth College
The ranges of some familiar arithmetic functions
We consider $4$ functions from elementary
number theory:
$\sigma$ (the sum-of-divisors function), $\varphi$ (Euler's function),
$\lambda$ (Carmichael's universal exponent function), and $s$ (the
sum-of-proper-divisors function. In particular we discuss the
distribution of the values of these functions, and coincidences
of values. Most of the problems considered have a fairly long
history, some over 80 years. We report on recent progress.
(Various parts of this work are joint with Kevin Ford, Tristan
Freiberg, Florian Luca, and Paul Pollack.)
Gautier Ponsinet, Université Laval
Functional equation for multi-signed Selmer groups
A. Lei and K. Büyükboduk
have recently defined a signed Selmer group for abelian variety at supersingular
prime. We will motivate and show a functional equation for this Selmer group,
generalizing a result of B.D. Kim for elliptic curves.
David Rohrlich, Boston University
Almost abelian Artin representations
The purpose of this talk is to point out a
connection between two seemingly unrelated achievements:
Anderson's paper (Duke Math. J. 114, 2002)
extending the Kronecker-Weber theorem
to the case of "almost abelian" extensions of $\mathbf{Q}$
and Shintani's paper (J. Math. Soc. Japan 30, 1978)
proving certain cases of
Stark's conjecture for real quadratic fields.
Adriana Salerno, Bates College
Multiple zeta values: A combinatorial approach to structure.
Multiple zeta functions are a multivariate version of the Riemann zeta
function. There are many open problems concerning these values, for
example, it's not even known if these numbers are rational or even
algebraic (although it is strongly suspected that they are transcendental).
However, these values satisfy many interesting algebraic relations between
them. A new approach to understanding multiple zetas is to study purely
their algebraic structure. I will talk about a few spaces (which turn out
to have the nice structure of a Lie algebra) that are essentially
equivalent to a formal version of these zetas, and where all the
interesting questions turn into combinatorial questions.
Jonathan Sands, University of Vermont
Derivatives of L-functions and annihilation of ideal class groups.
We consider certain abelian Galois extensions K/F of relative degree 6
with F real quadratic.
Letting G denote the Galois group and S the set of ramified primes, we first
computationally verify a version of Stark's conjecture that the derivative of
the S-imprimitive equivariant L-function, when multiplied by an equivariant
regulator, yields an element of the rational group ring of G. Subsequent
multiplication by an annihilator of roots of unity then produces an element of
the integral group ring, and further computation confirms that this element
annihilates the ideal class group of K. This provides evidence for an analog of
the Brumer-Stark conjecture that uses the leading coefficients of L-functions at
the origin and not just the values. This is joint work with Brett Tangedal.
Andrew Schultz, Wellesley College
Deformations of the Weyl character formula via ice models.
By assigning polynomial weights to certain symmetry classes of alternating sign
matrices, Okada gave deformations of the Weyl denominator formula in types $B$,
$C$ and $D$. In this talk we generalize these results by replacing Okada's
families of alternating sign matrices with more general families of ice models
from statistical mechanics. This allows us to use some local conservation rules
--- particularly the Yang-Baxter equation --- to evaluate the corresponding
partition functions. We will see that a certain specialization of these
polynomials returns the Weyl character formula.
Ari Shnidman, Boston College
Counting cusps via isogeny volcanoes
I will present solutions to various counting problems in number theory,
topology, and algebraic geometry. The key tool is a refinement of an
unpublished result of Lichtenbaum describing Ext groups of CM elliptic curves.
The general formula involves taking walks along certain graphs called isogeny
volcanoes. This is joint work with Julian Rosen.
Naomi Tanabe, Dartmouth College
Determining Hilbert modular forms by central values of
Rankin-Selberg convolutions.
Identifying an automorphic forms by studying the special
values of L-functions of its twists has been discussed in various setting.
In joint work with Alia Hamieh, we generalize these results to Hilbert
modular forms by analyzing a twisted first moment of Rankin-Selberg
convolutions.
Karen Taylor, Bronx Community College
Hyperbolic Fourier Expansions of Modular Forms on the Full Modular Group
Classically, modular forms are described by their (parabolic) Fourier expansions. In this
talk, we discuss hyperbolic Fourier expansions of modular forms. We give an explicit exact formula
for the $n$-th hyperbolic Fourier coefficient of a modular form on the full modular group.
Bianca Thompson, Smith College
A very elementary proof of a conjecture of B. and M. Shapiro for cubic rational
functions
Using essentially only algebra, we give a proof that a cubic rational function over
$\mathbb{C}$ with only real critical points is equivalent to a real rational function.
We also determine all fields $\mathbb{Q}_p$ over which a reasonable generalization holds.
Christelle Vincent, University of Vermont
Computing equations of hyperelliptic curves whose Jacobian has CM.
It is known that given a totally imaginary sextic field with totally real cubic
subfield (a so-called CM sextic field) there exists a non-empty finite set of
abelian varieties of dimension 3 that have CM by this field. In this talk we
present an algorithm that takes as input such a field, and outputs a period
matrix for such an abelian variety. We then check computationally if the abelian
variety is the Jacobian of a hyperelliptic curve, and compute an equation for
the curve if this is the case.
This is joint work with J. Balakrishnan, S. Ionica and K. Lauter.
Jan Vonk, McGill University
Stable models of Hecke operators
Many linear operators in number theory arise as linearisations of
correspondences between varieties. We will discuss how to postpone this
linearisation as long as possible, and study correspondences between curves as
geometric objects. After presenting a potential-semistability theorem for
correspondences analogous to the Deligne--Mumford theorem for curves, we indicate
how to extract spectral information from the combinatorics of the special fibre
of a semi-stable correspondence, and revisit some classical methods known for
Hecke operators.
Shou-Wu Zhang
Faltings heights and Zariski density of CM abelian varieties
The moduli of CM abelian varieties are the simplest objects in the category of Shimura
varieties, and have been intensively studied related to Hilbert's 12th problem and the BSD
conjecture. In this talk, I will discuss some recent progress on two different kinds of
problems:
1) Colmez' conjecture on Faltings' heights of CM abelian varieties in terms of Artin
L-functions, and
2) André-Oort's conjecture on Zariski density of CM points on Shimura varieties.
Michael Zieve, University of Michigan
Arithmetic of prime-degree functions
I will present several arithmetic results about prime-degree functions. For
instance, I will describe the possible sizes of the image $f(\mathbb{F}_q)$ where $f(x)$ is a prime-degree
rational function over $\mathbb{F}_q$. I will also describe all irreducible polynomials $f(x,y)\in \mathbb{Q}[x,y]$
with prime y-degree for which there are infinitely many rational numbers c such that f(x,c) is
reducible. Finally, I will describe all prime-degree rational functions $f(x) \in \mathbb{Q}(x)$ for which
the induced function on $P^1(\mathbb{Q})$ is noninjective over infinitely many values, and explain how
this result suggests a vast generalization of Mazur's theorem on uniform boundedness of
rational torsion on elliptic curves. The proofs rely on studying the possibilities for the
fundamental invariants (i.e., the monodromy group and ramification type) of morphisms of
complex curves.
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List of Participants
Harry Altman, University of Michigan
Angelica Babei, Dartmouth College (grad)
Jonathan Bayless, Husson University
David Bradley, University of Maine
Daniel Buck, University of Maine (grad)
Kenneth Bundy, University of Maine (grad)
David Burt, Williams College (undergrad)
Hugo Chapdelaine, Université Laval
Sara Chari, Dartmouth College (grad)
Liubomir Chiriac, University of Massachusetts
Michael Chou, University of Connecticut (grad)
Harris Daniels, Amherst College
Henri Darmon, McGill University
Chantal David, Concordia University
Danielle David, University of Maine (grad)
Daniel Disegni, McGill University
David Dummit, University of Vermont
Taylor Dupuy, Hebrew University / University of Vermont
Zhenguang Gao, Framingham State University
Eva Goedhart, Smith College
Leo Goldmakher, Williams College
Fernando Gouvêa, Colby College
Theodore Halnon, Pennsylvania State University (undergrad)
Jeffrey Hatley, Union College
Jeffrey Hoffstein, Brown University
Thomas Hulse, Colby College
Dubi Kelmer, Boston College
Ju-Lee Kim, Massachusetts Institute of Technology
Paul Kinlaw, Husson University
Hershy Kisilevsky, Concordia University
Andrew Knightly, University of Maine
Dimitris Koukoulopoulos, University of Montréal
Juergen Kritschgau, Bates College (undergrad)
David Krumm, Colby College
Chan Ieong Kuan, University of Maine
Prateek Kunwar, University of Maine (grad)
John Larson, University of Maine (undergrad)
Hao (Billy) Lee, McGill University
Antonio Lei, Université Laval
Patrick Letendre, Université Laval (grad)
Claude Levesque, Université Laval
Li-Mei Lim, Bard College at Simon's Rock
David Lowry-Duda, Brown University (grad)
Álvaro Lozano-Robledo, University of Connecticut
Sergey Lvin, University of Maine
Christian Maire, Besançon / CRM / University of Montréal
Ayesha Maliwal, University of Maine (grad)
Sarah Manski, Kalamazoo College (undergrad)
Kenneth McMurdy, Ramapo College of New Jersey
Steven Miller, Williams College
Daniel Nichols, University of Massachusetts (grad)
Robert Niemeyer, University of Maine
Vincent Ouellet, Université Laval (grad)
Nigel Pitt, University of Maine
Carl Pomerance, Dartmouth College
Gautier Ponsinet, Université Laval (grad)
Yannan Qiu, University of Maine
Caroline Reno, University of Maine (grad)
David Rohrlich, Boston University
Thomas Sacchetti, Bates College (undergrad)
Adriana Salerno, Bates College
Jonathan Sands, University of Vermont
Andrew Schultz, Wellesley College
Ariel Shnidman, Boston College
Chip Snyder, University of Maine
Blaine Talbut, University of Chicago (undergrad)
Naomi Tanabe, Dartmouth College
Karen Taylor, Bronx Community Collge
Ervin Thiagalingam, McGill University (grad)
Bianca Thompson, Smith College
Daniel Vallières, University of Maine
Christelle Vincent, University of Vermont
Jan Vonk, McGill University
Alexander Walker, Brown University (grad)
Benjamin Weiss, University of Maine
Alice Wise, University of Maine (grad)
Siman Wong, University of Massachusetts
Kevin Yang, Harvard University (undergrad)
Joshua Zelinsky, University of Maine
Shou-Wu Zhang, Princeton University
Michael Zieve, University of Michigan
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