Number Theory Seminar : Past Events
Past Seminars
The following seminars have already happened, you may instead view upcoming seminars in this series.
Sept. 17, 2008
Alina Cojocaru :
2 p.m. in SEO 636
Abstract
Let $E$ be an elliptic curve over $\mathbb{Q}$ and let $E_p$ be its reduction modulo a prime $p$. The talk will focus on the problem of finding an asymptotic formula for the number of primes $p < x$ for which the order of the group of points of $E_p$ is squarefree, in the case that the global curve $E/\mathbb{Q}$ has Complex Multiplication.
Sept. 24, 2008
Alina Carmen Cojocaru :
2 p.m. in SEO 636
Oct. 1, 2008
Alina Carmen Cojocaru :
2 p.m. in SEO 636
Oct. 8, 2008
Ramin Takloo-Bighash :
2 p.m. in SEO 636
Abstract
This will be an (accessible) exposition of a result joint with Dominic Lanphier which characterizes when the Rankin-Cohen bracket of two eigenforms of full
level is an eigenform. We will show that this happens only for a finite number of cases, and identify every possibility. Our theorem generalizes a
theorem of Bill Duke dealing with products of eigenforms.
Oct. 15, 2008
Ramin Takloo-Bighash :
2 p.m. in SEO 636
Oct. 29, 2008
Izzet Coskun :
2 p.m. in SEO 636
Abstract
Conjecturally there is a very close connection between the existence of rational points on a variety X and the existence of rational curves and abelian subvarieties on X. In this talk, I will discuss some geometric conditions, such as rationality, unirationality and rational connectedness, that guarantee that X has many rational curves. I will explain some results and questions about rational curves on hypersurfaces and their implications for existence of points over function fields of curves.
Nov. 12, 2008
Djordje Milicevic :
2 p.m. in SEO 636
Nov. 19, 2008
Alex Gamburd :
2 p.m. in SEO 636
Dec. 3, 2008
Jon Hanke :
2 p.m. in SEO 636
Feb. 4, 2009
Anatoly Libgober :
4 p.m. in SEO 427
Abstract
I will discuss an extenstion of the ring of Jacobi forms which is an analog the ring of Kaneko-Zagier quasimodular
forms. A motivation coming from the study of elliptic genus also will be discussed.
March 4, 2009
Alexandru Popa :
4 p.m. in SEO 427
Abstract
In a 1984 paper, Kohnen and Zagier developed the theory of modular
forms with rational periods for the full modular group. I recall some
of this theory, and present a new result decomposing an arbitrary cusp
form into forms with rational even (or odd) periods. As a consequence,
I discuss a decomposition of Hecke eigenforms in terms of forms with
rational Fourier coefficients, given by Rankin-Cohen brackets of
Eisenstein series.
March 11, 2009
Yiannis Sakellaridis :
4 p.m. in SEO 427
Abstract
Motivated by the relative trace formula of Jacquet and experience on period integrals of automorphic forms, we take the first steps towards formulating a "relative" Langlands program, i.e. a set of conjectures on H-distinguished representations of a reductive group G (both locally and globally), where H is a spherical subgroup of G. We prove several results in this direction. Locally, the spectrum of H\G is described with the help of the dual group associated to any spherical variety by Gaitsgory and Nadler. Globally, period integrals are conjectured to be Euler products of explicit local functionals, which we compute at unramified places and show that they are equal to quotients of L-values. If time permits, I will also discuss an approach which shows that different integral techniques for representing L-functions (e.g. Tate integrals, Rankin-Selberg integrals, period integrals) are, in fact, the same. This is in part joint work with Akshay Venkatesh.
March 18, 2009
No speaker :
4 p.m. in SEO 427
April 22, 2009
Mahdi Asgari :
4 p.m. in SEO 427
Aug. 26, 2009
Nathan Kaplan :
3:30 p.m. in SEO 427
Sept. 16, 2009
Ellen Eischen :
4 p.m. in SEO 712
Abstract
At certain special points, the values of the Riemann zeta function and many other L-functions are algebraic, up to a well-determined transcendental factor. G. Shimura, H. Maass, and M. Harris extensively studied a class of differential operators on automorphic forms; these differential operators play an important role in proofs of algebraicity properties of many $L$-functions.
Building on work of N. Katz, we introduce a p-adic analogue of these differential operators, which should be similarly significant in the study of many p-adic L-functions, in particular p-adic L-functions attached to families of p-adic automorphic forms on unitary groups.
Oct. 21, 2009
Byungchul Cha :
3:30 p.m. in SEO 712
Abstract
The Linear Independence (LI) assumption states that the set of nonnegative ordinates of critical zeros of zeta/L-functions is linearly independent over rational. In this talk, we present the function field version of LI and its applications to prime number races using techniques of Rubinstein and Sarnak. In the second part of this talk, we will study another application of LI in connection with the growth rate of the summatory function of Moebius function in the function field setting.
Oct. 28, 2009
Matt Satriano :
3:30 p.m. in SEO 427
Abstract
Given a singular scheme X, one way to study it is through a resolution of singularities, which is oftentimes hard to control. In certain cases, however, one can construct a smooth stack which well-approximates X and can serve as a replacement for the resolution of singularities. In this talk, I describe two cases where such a stack exists and give applications to Invariant Theory, Hodge Theory, and toric Artin stacks.
Nov. 4, 2009
Nick Ramsey :
3:30 p.m. in SEO 427
Abstract
In the 1970's, Lenstra generalized the notion of a Euclidean ring to that
of a ring with a Euclidean ideal. In the context of Dedekind domains, the
consequence of the existence of such an ideal is the cyclicity of the
class group in much the same way that the consequence of the existence
of a Euclidean algorithm is the triviality of the class group. In this talk,
I'll discuss Lentra's notion in light of some recent developments of Hester Graves.
In particular, I'll discuss a joint result with Graves classifying the quadratic imaginary
fields (which play a rather exception role in the theory) that have a Euclidean ideal.
Nov. 11, 2009
Eliot Brenner :
3:30 p.m. in SEO 427
Abstract
We partially generalize the results of Kudla and Rallis on the poles of degenerate, Siegel-parabolic Eisenstein series to residual-data Eisenstein series. In particular, for $a,b$ integers greater than 1, we show that poles of the Eisenstein series induced from the Speh representation on the Levi component of the Siegel parabolic of $\mathrm{Sp}_{2ab}$ are located in a particular "segment" of half integers $X_{b}$ between a "right endpoint" and its negative, inclusive of endpoints. We study the automorphic forms $\Phi_{i}^{(b)}$ obtained as residues at the points $s_i^{(b)}$ (defined precisely in the paper) by calculating their cuspidal exponents in certain cases. In the case of the "endpoint" $s_0^{(b)}$ and `first interior point' $s_1^{(b)}$ in the segment of singularity points, we are able to determine a set containing \textit{all possible} cuspidal exponents of $\Phi_0^{(b)}$ and $\Phi_1^{(b)}$ precisely for all $a$ and $b$. In these cases, we use the result of the calculation to deduce that the residual automorphic forms lie in $L^2(G(k)\backslash G(\mathbf{A}))$. In a more precise sense, our result establishes a relationship between, on the one hand, the actually occurring cuspidal exponents of $\Phi_i^{(b)}$, residues at interior points which lie to the right of the origin, and, on the other hand, the "analytic properties" of the original residual-data Eisenstein series at the origin. If time permits we will discuss further analytic properties such as wave-front sets of the residual automorphic forms, and applications of our calculations.
Nov. 18, 2009
Ramin Takloo-Bighash :
3:30 p.m. in SEO 427
Feb. 24, 2010
Lior Silberman :
4:15 p.m. in SEO 427
Sept. 8, 2010
Joey Lee :
3 p.m. in SEO 612
Abstract
In 1950, Northcott proved the height inequality for
morphisms on projective spaces. Let $f:P^n\to P^n$ be a morphism. Then,
there is a constant $C$ depending on the given morphism such that the
height function $h$ satisfies
$h(f(P)) + C > \deg f \, h(P) > h(f(P)) - C$
for all points $P \in P^n$.
Unfortunately, the upper bound of the inequality does not hold
for rational maps. However, we can find weaker inequality by 1)
restricting points on an affine open set and 2) enlarging the upper
bound a little bit. In this talk, I will introduce the D-ratio for
a rational map on projective space and prove the Weak Northcott
property for rational maps.
Sept. 22, 2010
Mehrdad M. Shahshahani :
2 p.m. in SEO 427
Abstract
Let $M$ be a compact orientable topological surface and $M^a$ a Riemann
surface whose underlying surface is $M$. Then as an algebraic
curve $M^a$ can be de fined over a number fi eld if and only if it admits of
a non-constant meromorphic function
$f : M^a \to {\mathbb C}{\mathbb P}(1)$ with at most
three critical values according to theorems of Belyi and Weil. The critical
values may be normalized to be $0$, $1$ and $\infty$ and such a normalized
meromorphic function is referred to as a Belyi function and the set
$f^{-1}[0; 1]$ is a graph (called \emph{dessin}) on $M$. Fixing the the equations
defi ning the algebraic curve $M^a$ and the Belyi function $f$, the absolute
Galois group $G = Gal(\bar{{\mathbb Q}}/{\mathbb Q})$ acts on the coefficients of the equations
defi ning $M^a$ and the Belyi function $f$. Inspired by this remarkable
theorem, Grothendieck suggested that studying graphs, satisfying certain
conditions, on orientable topological surfaces and the action of the
absolute Galois group accordingly provides a geometric and combinatorial
framework for gaining a deeper understanding of $G$ which is known
essentially only through its action on algebraic numbers. The seminal
works of Drinfeld and Ihara on this subject embeds the absolute Galois
group into the Grothendieck-Teichmuller group and was the starting
point for research by a number of other mathematicians.
After a general introduction, in this lecture I will discuss the work
of my student, Ali Kamalinejad, on this program and specifi cally
(1) The construction of dessins on curves of arbitrary genus.
(2) The construction of families of dessins, the associated cartographic
groups, and the corresponding Galois groups.
(3) Relation with Jenkins-Strebel differentials.
Sept. 29, 2010
Xinyi Yuan :
3 p.m. in SEO 612
Abstract
The uniqueness part of the Calabi-Yau theorem asserts
that the Monge-Ampere measure on a (complex) positive Hermitian line
bundle determines the Hermitian metric up to constant. Here we
introduce a p-adic analogue of the theorem. Combined with the
equidistribution theory, we obtain the rigidity of preperiodic points
for algebraic dynamical systems.
Oct. 27, 2010
Jing Long Hoelscher :
2 p.m. in SEO 427
Abstract
This talk will concern the divisibility of class numbers of cyclotomic
function fields and their relation with Bernoulli numbers for rational
function fields over finite fields. In number fields, the Herbrand-Ribet
theorem gives a precise relation between the divisibility of class numbers
of cyclotomic number fields and Bernoulli numbers. In function fields, the
Herbrand's direction has been proven to be true, but the other direction
has obvious counter-examples. Gekeler reformulated a conjecture similar to
Ribet's theorem. This talk will report some recent progress towards the
Gekeler's conjecture.
Nov. 3, 2010
Masoud Kamgarpour :
2 p.m. in SEO 427
Abstract
In geometric representation theory, one often wishes to
describe representations realized on spaces of invariant functions as
trace functions of equivariant perverse sheaves. In the
case of principal series representations of GL(N) over a local field
F, i.e., representations obtained by parabolic induction from
characters of the torus, there is a beautiful description of families
of these representations realized on spaces of functions on GL(N)
invariant under the translation action of the Iwahori subgroup, or a
suitable smaller compact open subgroup, studied by Howe, Roche,
Bushnell-Kutzko, and others. Based on conjectures of Drinfeld, I will
define and describe perverse sheaves whose traces
recover these families. This is joint work with Travis Schedler.
Nov. 10, 2010
Alina Carmen Cojocaru :
2 p.m. in SEO 636
Abstract
Let E be an elliptic curve over Q and let Q(E[n]) be its n-th division field. In 1972, Serre showed that if E is without complex
multiplication, then the Galois group of Q(E[n])/Q is as large as possible, that is, GL_2(Z/n Z), for all integers n coprime to a
constant integer c(E, Q) depending (at most) on E/Q. Serre also showed that the best one can hope for is to have
|GL_2(Z/n Z) : Gal(Q(E[n])/Q)| at most 2 for all nonzero integers n.
I will discuss the frequency of this optimal situation in a one-parameter family of elliptic curves over Q.
This is joint work with David Grant and Nathan Jones.
Dec. 1, 2010
Jing Long Hoelscher :
2 p.m. in SEO 427
April 13, 2011
Eyal Goren :
3 p.m. in SEO 1227
Abstract
Our story begins more than a century ago with Kronecker's Jugendtraum and Hilbert's 12th problem, where complex multiplication appears as a way to understand Galois extensions of number fields - a problem still at the heart of number theory.
Our lecture will have a strong historical flavour; we shall attempt to survey the development of the theory of complex multiplication and the philosophy behind it.
On this background, we will present some exciting recent results, some of which build in an essential way on Borcherds' theory.
Finally, we shall sketch some of the key challenges of the theory of complex multiplication today and future directions.
Sept. 12, 2011
Anatoly Libgober :
3 p.m. in SEO 612
Sept. 19, 2011
Ramin Takloo-Bighash :
3 p.m. in SEO 612
Abstract
In this talk I will explain a method, based on p-adic integration, to count the number of subrings of ${\mathbb Z}^n$ of bounded additive index.
The method yields exact asymptotic formulae for small $n$, and gives non-trivial bounds for all $n$. At the end I will explain how this method could be
applied to the problem of counting orders in quintic fields. This is joint work with Nathan Kaplan.
Sept. 28, 2011
Alina Bucur :
10 a.m. in SEO 427
Abstract
I will give a quick overview of some new developments in counting points on curves over finite fields. Then we will concentrate on giving a probabilistic model for the number of rational points on a complete intersection of hypersufaces in projective n-space. A somewhat surprising corollary is that the number of rational points on a random smooth intersection of two curves in projective 3-space is strictly less than the number of points on the projective line. This is joint work with K. Kedlaya.
Oct. 3, 2011
A.C. Cojocaru :
3 p.m. in SEO 612
Abstract
Many remarkable questions about prime numbers have natural analogues in the
context of elliptic curves. Among them, Artin's primitive root conjecture, the
twin prime conjecture, and the Schinzel hypothesis have inspired a broad family of
conjectures regarding the behaviour of the reductions modulo primes of an elliptic
curve over Q. I will give an overview of such questions and progress made towards
their resolution.
Oct. 10, 2011
ChongGyu (Joey) Lee :
3 p.m. in SEO 636
Abstract
Since the Weil height functions work over number fields,
arithmetic dynamics only works for endomorphisms defined over number
fields. Because of the Moriwaki's height, we can build the Moriwaki's
height machine, which gives arithmetic functions on a finitely
generated field over Q. It allows to adapt the results in arithmetic
dynamics to complex dynamics.
Oct. 17, 2011
David Savitt :
3 p.m. in SEO 612
Oct. 24, 2011
Matthew Morrow :
3 p.m. in SEO 612
Abstract
The ring of adeles of a number field is self-dual, offering an arithmetic analogue of Serre duality. I will show that arithmetic surfaces satisfy similar dualities, using the higher adeles introduced by A. Parshin and A. Beilinson.
Nov. 14, 2011
Liang Xiao :
3 p.m. in SEO 612
Abstract
The parity conjecture is a weak version of BSD Conjecture or more
generally, Beilinson-Bloch-Kato Conjecture. It is conjectured that
the order of the L-function at the central point has the same parity
as the dimension of the Bloch-Kato Selmer group. I will explain an
approach to this conjecture for modular forms by varying the modular
form in a p-adic family. This is a joint work with Kiran Kedlaya and
Jay Pottharst.
April 27, 2012
Nathan Jones :
noon in SEO 427
Abstract
Given an elliptic curve E defined over a number field K, consider the action of the absolute Galois group of K on the n-torsion of E.
This defines a Galois representation into GL_2(Z/n Z). It is of fundamental interest to understand the image of this representation, as
illustrated by the now classical complex multiplication theory and by Serre's study of ell-adic representations.
In this talk I will survey some results on this topic, highlighting the situation for a ``typical'' elliptic curve. I will also discuss how this
typical behavior allows for applications to studies of families of elliptic curves.
Oct. 2, 2012
Majid Hadian :
3 p.m. in SEO 636
Oct. 9, 2012
Ramin Takloo-Bighash :
3 p.m. in SEO 636
Abstract
In this talk I will explain a recent joint work with Nathan Kaplan in which we have formulated a precise conjecture
about the distribution of orders in quintic extensions of ${\mathbb Q}$. I will also describe a possible approach involving
p-adic integration to proving the conjecture.
Oct. 23, 2012
Rachel Davis :
3 p.m. in SEO 512
Abstract
For $\ell$-adic Galois representations associated to elliptic curves, there are theorems
concerning when the images are surjective. For example, Serre proved that for a fixed non-CM elliptic
curve $E/{\mathbb Q}$, for all but finitely many primes $\ell$, the $\ell$-adic Galois representation is surjective.
Grothendieck and others have developed a theory of outer Galois representations. These are representations
from the absolute Galois group to an outer automorphism group of a free pro-$\ell$-group. In this case,
there is less known about the size of the images.
The goal of this research is to understand more tangibly Galois representations to automorphism
groups of non-abelian groups. Let $E$ be a semistable elliptic curve over ${\mathbb Q}$ with good supersingular
reduction at $2$. Associated to $E$, there is a Galois representation to a subgroup of the automorphism
group of a metabelian group. I conjecture that there is a Galois representation surjecting to this
subgroup (with the right ramification) and give evidence for this conjecture. Then, I compute some
conjugacy invariants for the images of the Frobenius elements. This will give rise to new arithmetic
information analogous to the traces of Frobenius for the $\ell$-adic representation.
Oct. 30, 2012
Masoud Kamgarpour :
3 p.m. in SEO 636
Abstract
I will discuss two topics from geometric representation theory which are related to the geometric Langlands program. The first topic is geometric class field theory due to Deligne. The second is geometric Satake Isomorphism due to Lusztig, Ginzburg, Drinfeld, Mirkovic and Vilonen.
Nov. 5, 2012
Xin Wan :
3 p.m. in SEO 427
Abstract
In this talk I will first formulate the Iwasawa main conjecture for unitary groups. Then I'll discuss the proof of one divisibility for two different kinds of Rankin-Selberg p-adic L-functions. This is a generalization of an earlier work of Skinner-Urban.
Nov. 15, 2012
Jonathan Pottharst :
3 p.m. in SEO 636
Abstract
Given an elliptic curve $E/{\mathbb Q}$ with good reduction at p, Iwasawa theory studies its arithmetic over all the fields of $p^n$-th roots of unity.
For example, there are nontrivial relations among the participants in the Birch–Swinnerton-Dyer conjectures for $E$ over each layer in this tower of
fields. If the reduction of $E \mod p$ is ordinary, then have had a satisfactory description of the scenario for quite some time. But if the reduction of
$E \mod p$ is supersingular, the correct description has required new advances in $p$-adic Hodge theory. We will discuss the background and what
is now known, and then, time permitting, describe the new tools and how they fit into an emerging larger picture of $p$-adic number
theory.
Nov. 20, 2012
Henri Gillet :
3 p.m. in SEO 636
Abstract
The function field analog of the Bombieri-Lang conjecture asserts that if $X$ is a smooth projective variety over a
function field $K/k$, and $X$ is of general type, then if $X(K)$ is dense in $X$, then $X$ is isotrivial
(defined over the ground field $k$, roughly speaking). Grauert proved this for curves in characteristic zero, and
Samuel extended Grauert's result to characteristic $p>0$ (these are the function field analogs of the Mordell conjecture).
Noguchi, and independently Martin-Deschamps extended Grauert's argument to the case where $X$ is of arbitrary dimension
with ample cotangent sheaf. I will discuss to what extent one can adapt the arguments of
Noguchi, Martin-Deschamps, and Samuel to varieties in positive characteristic.
Nov. 27, 2012
Ana Caraiani :
3 p.m. in SEO 636
Abstract
Given a cuspidal automorphic representation of $GL(n)$ over a CM field, which is regular algebraic and conjugate self-dual,
one can associate to it a Galois representations. This Galois representation is known in almost all cases to be
compatible with local Langlands.
I will prove the last missing case of the compatibility by identifying the monodromy operators when $l=p$ and n is even.
Ana Caraiani :
3 p.m. in SEO 636
Dec. 4, 2012
Yifeng Liu :
4 p.m. in SEO 636
Abstract
In this talk, I will introduce a class of divisors on Shimura
curves associated to certain unitary groups, which consist of the unitary
version of the Heegner points. The idea comes from the theta lifting, a key
construction in the theory of automorphic representations. We show that the
height of those divisors is related to the central L-derivative of
automorphic forms of unitary groups of 2 variables. In fact, these
construction and relation can be generalized to unitary groups of higher
ranks, which correspond to higher dimensional Shimura varieties, as well. A
precise formulation of the height formula will be given.
Dec. 10, 2012
Yuri Tschinkel :
3 p.m. in SEO 636
Abstract
Anabelian geometry techniques allow the construction of
explicit universal spaces which capture birational properties of
algebraic varieties. I will describe this theory and its applications.
Joint with F. Bogomolov.
Jan. 22, 2013
Simon Marshall :
4:15 p.m. in SEO 636
Abstract
I will discuss some ideas related to the theory of p-adically
completed cohomology developed by Frank Calegari and Matthew Emerton.
If F is a number field that is not totally real, I will use these
ideas to prove a strong upper bound for the dimension of the space of
cohomological automorphic forms on GL_2 over F that have fixed level
and growing weight.
Jan. 29, 2013
Matthew Morrow :
3 p.m. in SEO 636
Abstract
It has been known since the early days of algebraic K-theory that it fails to satisfy excision
for example, the difference between the K-theory of a singular curve and its normalisation is not determined by the
conductor ideal and it is now known that the precise obstruction to this may be described using cyclic or topological
cyclic homology. However, this obstruction vanishes in all cases of interest in algebraic and arithmetic geometry if one
passes to formal infinitesimal thickenings in a suitable sense, and this is sufficient for many applications.
I will begin with an introduction to algebraic K-theory, then explain this infinitesimal excision result, and finally
indicate the applications within algebraic geometry (which are joint with Amalendu Krishna).
Feb. 5, 2013
Christophe SOULÉ :
3 p.m. in SEO 636
Abstract
Given a auclidean lattice, we describe a method
to compute its successive minima.
We apply this to the lattice of sections
of a line bundle on an arithmetic surface.
Feb. 11, 2013
Moshe Adrian :
3 p.m. in SEO 427
Abstract
A well known result of Borel says that category of modules over the Iwahori-Hecke algebra of a semisimple p-adic group G describes the Bernstein component associated to the unramified principal series of G. We consider Bernstein components associated to principal series induced from "simple supercuspidal representations", recently discovered by Benedict Gross and Mark Reeder. We define and study Hecke algebras that describe these Bernstein components for symplectic groups. This investigation has direct applications to the local Langlands correspondence and lifting, which we will also discuss.
Feb. 12, 2013
See February 11 :
3 p.m. in SEO 636
Feb. 26, 2013
Paul Herman :
3 p.m. in SEO 636
Abstract
It is well known that for an exponential sum with a prime modulus the best bound for the sum comes from Weil's famous estimation. In this talk, we discuss when this bound can be improved on average over integral modulus in a number field. Investigations into exponential sums on average, or sums of exponential sums, have many applications including the Riemann hypothesis and the Ramanujan conjecture for automorphic forms.
In particular, we will get an asymptotic for sums of quartic exponential sums over the Gaussian integers. Tools we will use to get this asymptotic include automorphic forms and the trace formula.
March 5, 2013
Mihran Papikian :
3 p.m. in SEO 636
Abstract
Let $X$ be a Mumford curve. We say that an elliptic curve is an optimal quotient of $X$ is there is a finite
morphism $X\to E$ such that the homomorphism $\pi: Jac(X)\to E$ induced by the Albanese functoriality has connected and
reduced kernel. We consider the functorially induced map $\pi_\ast: \Phi_X\to \Phi_E$ on component groups of the Neron
models of $Jac(X)$ and $E$. We show that in general this map need not be surjective, which answers negatively a question of
Ribet and Takahashi. Using rigid-analytic techniques, we give some conditions under which $\pi_\ast$ is surjective, and discuss
arithmetic applications to modular curves. This is a joint work with Joe Rabinoff.
March 12, 2013
Derek Garton :
1 p.m. in SEO 636
Abstract
The Cohen-Lenstra-Martinet heuristics predict the frequency with
which a fixed finite abelian group appears as an ideal class group of an
extension of number fields, for certain sets of extensions of a base
field. Recently, Malle found numerical evidence suggesting that their
proposed frequency is incorrect when there are unexpected roots of unity
in the base field of these extensions. Moreover, Malle proposed a new
frequency, which is a much better match for his data. I will explain a
random matrix heuristic (coming from function fields) that leads to a
function field version of Malle's conjecture (as well as generalizations
of it).
March 19, 2013
Liang Xiao :
1 p.m. in SEO 636
Abstract
The special fiber of a modular curve admits a stratification by the
ordinary locus and the supersingular locus, where the latter is given
by the zero locus of the Hasse invariant. Goren and Oort generalized
this picture to define a stratification of the special fiber of
Hilbert modular varieties, given by the zero locus of partial Hasse
invariants. I will discuss the geometric structure of this
stratification and various applications of the structure theorem. If
time permits, I will explain certain (mostly conjectural)
generalizations of the construction to PEL type Shimura varieties.
This is an ongoing joint project with David Helm and Yichao Tian.
April 9, 2013
A.C. Cojocaru :
1 p.m. in SEO 636
Abstract
Let $\psi$ be a generic Drinfeld module of rank $r \geq 2$. We study the first elementary divisor
$d_{1, \wp}(\psi)$ of the reduction of $\psi$ modulo a prime $\wp$, as $\wp$ varies.
In particular, we obtain the density of the primes $\wp$ for which $d_{1, \wp} (\psi)$ is fixed.
For $r = 2$, we also study the second elementary divisor (the exponent) of the reduction of $\psi$ modulo $\wp$
and prove that, on average, it has a large norm. Our work is motivated by the study of J.-P. Serre of an elliptic curve analogue
of Artin's Primitive Root Conjecture, and, in particular, by refinements to Serre's study
developed by A.C. Cojocaru and M. R. Murty. This is joint work with Drew Shulman.
April 16, 2013
Erik Wallace :
1 p.m. in SEO 636
Abstract
We introduce Galois representations on Abelian varieties, and discuss a theorem of Duke for elliptic curves over Q,
as well as various extensions of his theorem to other number fields, and to higher dimensions. The general flavor
of these theorems is that most of the time we can expect the Galois representations to be surjective for all or almost
all primes
l.
April 23, 2013
Nathan Jones :
1:15 p.m. in SEO 636
Abstract
Let $f$ be a function from the set of natural numbers to itself. We call $f$
a global power map if $f(n) = n^k$ for some non-negative integer exponent $k$.
For a set $S$ of prime numbers, we call f {a local power map at $S$ if for each prime $p$ in $S$,
$f$ induces a well-defined group homomorphism on the multiplicative group
$(Z/pZ)^*$. In this talk, I will motivate the conjecture that if $f$
is a local power map at an infinite set $S$ of primes, then $f$ must be a global power map.
I will also discuss some progress towards this conjecture.
April 30, 2013
Davide Reduzzi :
1 p.m. in SEO 636
Abstract
I will discuss congruences modulo a prime p between modular forms arising
from geometric (PEL) and algebraic settings. In particular I will explain
how, by allowing changes in the weights, the (mod p) Hecke eigenforms
associated to some reductive groups G can be seen as eigenforms associated
to an inner form of G which is compact modulo center at the infinite
places. This type of considerations were originally made by J-P. Serre for
elliptic modular forms.
Sept. 24, 2013
Majid Hadian :
1 p.m. in SEO 636
Oct. 1, 2013
Ramin Takloo-Bighash :
1 p.m. in SEO 636
Abstract
In this talk, based on an ongoing joint work with Kumar Murty, I will describe a curious Siegel modular form that shows up
in connection with the modularity of a certain Abelian surface.
Oct. 15, 2013
Ramin Takloo-Bighash :
1 p.m. in SEO 636
Abstract
In this elementary talk I will cover some of the background necessary for Apoorva Khare's lecture on Wednesday.
Oct. 16, 2013
Apoorva Khare :
2:30 p.m. in SEO 1227
Abstract
Verma modules over a complex semisimple Lie algebra, as well as their simple quotients are important and well-studied
objects in representation theory. We present three formulas to compute the set of weights of all such simple highest weight
modules (and others) over a complex semisimple Lie algebra ${\mathfrak g}$. These formulas are direct and do not involve
cancellations. Our results extend the notion of the Weyl polytope to general highest weight ${\mathfrak g}$-modules $V^\mu$.
We also show that for all such simple modules, the convex hull of the weights is a $W_J$-invariant polyhedron for some parabolic
subgroup $W_J$. We compute its vertices, faces, and symmetries - more generally, we do so for all parabolic Verma modules, and
for all modules $V^\mu$ with $\mu$ not on a simple root hyperplane. Our techniques also enable us to completely classify
inclusions between "weak faces" of arbitrary $V^\mu$, in the process extending results of Vinberg, Chari, Cellini, and others
from finite-dimensional modules to all highest weight modules.
Oct. 22, 2013
Luis Lomeli :
1 p.m. in SEO 612
Nov. 5, 2013
Robert Harron :
1 p.m. in SEO 636
Abstract
A fruitful way to study the arithmetic significance of special values of L-functions is via their interpolation by p-adic L-functions.
In this talk, I will discuss the phenomenon of L-invariants, which arise when the interpolation property provides no immediate information.
Specifically, the value of the p-adic L-function may vanish even when the value of the original L-function does not. Beginning with the work of Mazur–Tate–Teitelbaum
on a p-adic Birch–Swinnerton-Dyer conjecture, it has been conjectured that the value of the derivative of the p-adic L-function should relate to the original L-value,
up to the introduction of a new factor: the${\mathcal L}$-invariant. I will give an overview of the subject and what is known before discussing joint work with Andrei Jorza where we
obtain formulas for the ${\mathcal L}$-invariants of symmetric powers of modular forms.
Nov. 12, 2013
Patrick Allen :
1 p.m. in SEO 636
Abstract
Modularity lifting theorems are a tool for proving that p-adic
Galois representations come from modular (or automorphic) forms using the
assumption that its associated mod p representation comes from a modular
form. Due to their technical nature, the method encounters difficulties if
p divides the dimension of the representation, or when the mod p
representation has small image. We show how the 2-adic patching method of
Khare and Wintenberger combined with the strategy of Skinner and Wiles,
using Hida familes, can be used to prove modularity of some two
dimensional, 2-adic Galois representations over totally real fields that
are nearly ordinary and residually dihedral. As an application we deduce
modularity of some elliptic curves over totally real fields that have good
ordinary or multiplicative reduction at places above 2.
Nov. 13, 2013
Dani Szpruch :
11 a.m. in SEO 1227
Abstract
In this talk we shall survey the construction of genuine principal series representations of covering groups and their Whittaker functionals emphasizing the differences between the linear and the metaplectic cases. We shall then move to the special case of the metaplectic double cover of GSp(2n,F). We shall present a symmetric Casselman-Shalika formula for this group exploiting the uniqueness of Whittaker model for the metaplectic double cover of Sp(2n,F) and a surprising connection between representation theory of these two groups. As an application we shall describe a family of reducible unitary unramified principal series representations.
Nov. 20, 2013
Benjamin Bakker :
2:30 p.m. in SEO 636
Abstract
A crucial step in the proof of Fermat's last theorem was Frey's insight that a nontrivial solution would yield an elliptic curve with modular p-torsion but which was itself not modular. The connection between an elliptic curve and its p-torsion is very deep: a conjecture of Frey and Mazur, stating that the p-torsion group scheme actually determines the elliptic curve up to isogeny (at least when p>13), implies an asymptotic generalization of Fermat's last theorem. We study a geometric analog of this conjecture, and show that the map from isogeny classes of "fake elliptic curves"---abelian surfaces with quaternionic multiplication---to their p-torsion is at most two-to-one, and one-to-one in special cases. Our proof involves understanding curves on a certain Shimura surface, and fundamentally uses the interaction between its hyperbolic and algebraic properties. This is joint work with Jacob Tsimerman.
Dec. 3, 2013
Li-Mei Lim :
1 p.m. in SEO 636
Abstract
It is well-known that the sum of special values of the
$GL(2)$ Eisenstein series is equal to (up to some factors)
$\zeta(s)L(s,\chi_d)$, where $\zeta(s)$ is the Riemann zeta function and $L(s, \chi_d)$
is the quadratic Dirichlet $L$-series. In this talk, I will describe a new proof for this
old result, using a multiple Dirichlet series associated to binary quadratic forms.
In addition, I will describe generalizations and extensions.
Sept. 2, 2014
Nathan Jones :
11 a.m. in SEO 427
Abstract
There is a modular curve X'(6) of level 6 defined over Q whose rational points correspond to j-invariants of elliptic curves E over Q for which Q(E[2]) is a subfield of Q(E[3]). In this talk I will characterize the j-invariants of elliptic curves with this property by exhibiting an explicit model of X'(6). The motivation is two-fold: on the one hand, X'(6) belongs to the list of modular curves which parametrize non-Serre curves (and is not well-known), and on the other hand, the set of rational points of X'(6) gives an infinite family of examples of elliptic curves with non-abelian ``entanglement fields,'' which is relevant to the systematic study of correction factors of various conjectural constants for elliptic curves over Q. This is based on joint work with J. Brau (Cambridge University, UK).
Sept. 9, 2014
A.C. Cojocaru :
11 a.m. in SEO 427
Abstract
Given a finite Galois extension L/K of global fields and a conjugacy class C of Gal(L/K), a fundamental problem is that of describing the (unramified) primes p of K for which the conjugacy class of the Frobenius at p is C. The Chebotarev Density Theorem provides the density of these primes, while, in general, the characterization of the primes themselves is a finer and deeper question. We focus on unraveling this question for the division fields of a generic Drinfeld module. For Drinfeld modules of rank 2, we obtain an explicit global description of the Frobenius. We apply this description to derive a criterion for the splitting modulo primes of a class of non-solvable polynomials and to study the frequency with which the reductions of Drinfeld modules have small endomorphism rings. We also generalize some of these results to higher rank Drinfeld modules and prove CM-lifting theorems for Drinfeld modules. This is joint work with Mihran Papikian (Pennsylvania State University, USA).
Sept. 16, 2014
Ramin Takloo-Bighash :
11 a.m. in SEO 427
Sept. 23, 2014
See next day. :
11 a.m. in SEO 427
Sept. 24, 2014
Ali Rajaei :
11 a.m. in SEO 427
Sept. 30, 2014
Nathan Jones :
11 a.m. in SEO 427
Abstract
Let f be a function from the set of integers into itself. We call f a global power map if there exists a non-negative integer k so that f(x) = x^k for every integer x. We call f a local power map at the prime number p if f induces a well-defined group homomorphism on the multiplicative group of integers modulo p. It has been conjectured that, if f is a local power map at infinitely many primes p, then f is a global power map. In this talk, I will discuss a theorem implying that, if f is a local power map at all primes p in a set with positive upper density relative to the set of all primes, then f must be a global power map.
Oct. 7, 2014
Ramin Takloo-Bighash :
11 a.m. in SEO 427
Abstract
In this talk I will report on a recent work on the distribution of orders in number fields. In particular, I will sketch the proof of an asymptotic formula for the number of orders of bounded discriminant in a given quintic number field emphasizing the role played by p-adic (and motivic) integration. This is joint work with Nathan Kaplan (Yale) and Jake Marcinek (Caltech).
Oct. 14, 2014
A.C. Cojocaru :
11 a.m. in SEO 427
Oct. 21, 2014
Abel Castillo :
11 a.m. in SEO 427
Oct. 28, 2014
Ramin Takloo-Bighash :
2:30 p.m. in SEO 636
Nov. 4, 2014
Ramin Takloo-Bighash :
11 a.m. in SEO 427
Nov. 11, 2014
Ramin Takloo-Bighash :
11 a.m. in SEO 427
Abstract
This talk will focus on the Shimura construction in the theory of modular forms.
Nov. 14, 2014
Dipendra Prasad :
4:15 p.m. in SEO 636
Abstract
The decomposition of a representation of a group when restricted to a subgroup has been studied in many instances. In this lecture, we will look at a variation on these questions involving
concepts in homological algebra which aims to finds simpler and more flexible theorems.
Nov. 18, 2014
Sho Tanimoto :
11 a.m. in SEO 427
Nov. 25, 2014
Nathan Jones :
11 a.m. in SEO 427
Dec. 2, 2014
A.C. Cojocaru :
11 a.m. in SEO 427
Feb. 10, 2015
Julia Gordon :
11 a.m. in SEO 427
Abstract
It is a well-known theorem of Harish-Chandra that the orbital integrals, normalized by the square root of the discriminant,
are bounded (for a fixed test function).
However, it is not easy to see how this bound behaves if we let the
$p$-adic field vary (for example, if the group $G$ is defined over a
number field $F$, and we consider the family of groups
$G_v=G(F_v)$, as $v$ runs over the set of finite places of $F$), and how it varies for a family of test functions.
Using a method based on model theory and motivic integration, we
prove that for a fixed test function, the bound on orbital integrals can be taken to be a fixed power (depending on $G$) of the cardinality of the residue field, and also obtain a uniform bound for the family of generators of the spherical Hecke algebra playing the role of the test functions. This
statement has an application to the recent work of S.-W. Shin and N.
Templier on counting zeroes of L-functions.
This project is joint work with R. Cluckers and I. Halupczok.
Feb. 12, 2015
Efrat Bank :
10:30 a.m. in SEO 427
Abstract
In this talk I will present a function field analogue of a conjecture in number theory. This conjecture is a combination of several famous conjectures, including the Hardy-Littlewood prime tuple conjecture, conjectures on the number of primes in arithmetic progressions and in short intervals, and the Goldbach conjecture. I prove an asymptotic formula for the number of simultaneous prime values of n linear functions, in the limit of a large finite field.
A key role is played by the computation of some Galois groups.
Feb. 17, 2015
Nathan Jones :
11 a.m. in SEO 427
Abstract
If the class number of an imaginary quadratic field behaved like a random positive integer, then the asymptotic proportion of class numbers which
are divisible by 3 would be 1/3, or about 33.3%. However, this proportion is observed to be around 43%. In this talk, I will survey heuristic
reasoning due to Cohen-Lenstra from the 1980s which explains this phenomenon and makes various other precise statistical predictions about
class groups.
March 3, 2015
Nathan Jones :
11 a.m. in SEO 427
Abstract
For a positive integer h, let F(h) denote the number of imaginary quadratic fields with class number equal to h. In this talk, I will survey a result of Soundararajan which,
using a theorem on the distribution of values of L(1,\chi_d), proves an asymptotic formula for the average of F(h).
March 10, 2015
Frank Thorne :
11 a.m. in SEO 427
Abstract
How many number fields K are there with |Disc(K)| < X, as a function of X? In the first part of the talk, I will give a historical overview of this question and describe work beginning with Minkowski and Hermite. It will emerge that enumerating quadratic fields, while "trivial", anticipates the methods used to count higher degree fields. We will also see that both analytic and algebraic methods have a lot to offer concerning this problem.
In the second part of the talk, I will discuss some of my recent and ongoing work addressing these questions, from both analytic and algebraic points of view. I will not give a complete account, but will instead focus on some of the most interesting technical questions -- how they arise and how we address them. This is (in various permutations) joint work with Bhargava, Cohen, Cojocaru, Lemke Oliver, Rubinstein-Salzedo, Taniguchi, and Xiong.
March 17, 2015
Majid Hadian :
11 a.m. in SEO 427
Abstract
In this lecture I will explain two applications of dessins (or children's drawings). The first application demonstrates how dessins are an effective tool for making explicit calculations that may have been difficult or impossible otherwise. A second application is more theoretical in nature. The notion of flat refinement of a dessin is introduced. This functorial notion enables one to assign to a given dessin infinite families of canonically defined dessins. From these geometric towers one constructs infinite towers of number fields with controlled ramification in the sense that the set of ramified primes is stable (This lecture represents joint work with Ali Kamalinejad and Mehrdad Shahshahani).
April 21, 2015
Nathan Jones :
11 a.m. in SEO 427
Abstract
Which abelian groups occur as the class group of some imaginary quadratic field? Inspecting tables of M. Watkins on imaginary quadratic fields of class
number up to 100, one finds that some abelian groups do not occur as the class group of any imaginary quadratic field (for instance (Z/3Z)^3 does not).
In this talk, I will combine heuristics of Cohen-Lenstra together with a refinement of a conjecture of Soundararajan to make precise predictions about the
asymptotic distribution of imaginary quadratic class groups, partially addressing the above question. I will also present some numerical evidence of the
resulting conjectures. This is based on joint work with S. Holmin, P. Kurlberg, C. Macleman, and K. Petersen.
April 28, 2015
David Zureick-Brown :
11 a.m. in SEO 427
Abstract
Let $X$ be a curve of genus $g$ over a number field $F$ of degree $d = [F:\mathbb{Q}]$. The conjectural existence of a uniform bound $N(g,d)$ on the number
$\#X(F)$ of $F$-rational points of $X$ is an outstanding open problem in arithmetic geometry, to follow from the Bomberi--Lang conjecture. We prove a special
case of this conjecture -- we give an explicit uniform bound when $X$ has Mordell--Weil rank $r\leq g-3$. This generalizes recent work of Stoll on uniform
bounds for hyperelliptic curves. Using the same techniques, we give an explicit, unconditional uniform bound on the number of $F$-rational torsion points of
$J$ lying on the image of $X$ under an Abel--Jacobi map. We also give an explicit uniform bound on the number of geometric torsion points of $J$ lying on
$X$ when the reduction type of $X$ is highly degenerate.
Our methods combine Chabauty--Coleman's $p$-adic integration, non-Archimedean harmonic analysis on Berkovich curves, and the theory of linear systems
and divisors on metric graphs.
This is joint work with Joe Rabinoff and Eric Katz.
Aug. 24, 2015
Mihran Papikian :
11 a.m. in SEO 427
Abstract
The Eisenstein ideal for modular curves over Q was introduced by Mazur in his seminal paper
``Modular curves and the Eisenstein ideal'' and since then the Eisenstein ideal has become an indispensable
tool in various problems related to modular curves.
In this talk I will discuss a joint work with Fu-Tsun Wei where we study the properties of the Eisenstein ideal
acting on Drinfeld modular curves of composite levels, with the goal of producing explicit Jacquet-Langlands isogenies.
Sept. 29, 2015
Nathan Jones :
11 a.m. in SEO 427
Oct. 6, 2015
Nathan Jones :
11 a.m. in SEO 427
Oct. 13, 2015
Ramin Takloo-Bighash :
11 a.m. in SEO 427
Oct. 20, 2015
Ramin Takloo-Bighash :
11 a.m. in SEO 427
Abstract
In this talk I will describe some recent results about the distribution of rational points of bounded height on
blowups of P^3. The varieties we consider can be thought of as compactificaions of the group PGL(2), and
we will use harmonic analysis on the non-commutative group PGL(2) to obtain our results. In an old work,
joint with Shalika and Tschinkel, we had studied rational points on smooth bi-equivariant compactifications
of arbitrary semi-simple groups. The varieties under consideration here are only one sided compactifications.
In concrete terms, this means that our height functions are not left and right invariant under the action of
compact subgroups of PGL(2). This introduces delicate technical difficulties that require new ideas. This is
joint work with Sho Tanimoto, and with Sho Tanimoto and Yuri Tschinkel.
Oct. 27, 2015
Ramin Takloo-Bighash :
11 a.m. in SEO 427
Abstract
This talk will be a continuation of the discussion from the previous week.
Nov. 3, 2015
Ramin Takloo-Bighash :
11 a.m. in SEO 427
Abstract
This is a first of a series of lectures on the Bloch-Kato conjecture. We will start the series with some general foundational
material. We will also choose speakers and topics for the future lectures.
Nov. 10, 2015
Ramin Takloo-Bighash :
11 a.m. in SEO 427
Abstract
This is a second of a series of lectures on the Bloch-Kato conjecture, and is a continuation of last week's
lecture on general foundational material.
Nov. 17, 2015
Ramin Takloo-Bighash :
11 a.m. in SEO 427
Abstract
In this talk I will continue discussing the foundational material for the Bloch-Kato conjecture.
Dec. 1, 2015
Ramin Takloo-Bighash :
11 a.m. in SEO 427
Abstract
In this talk I will continue discussing the foundational material for the Bloch-Kato conjecture.
Feb. 16, 2016
Nathan Jones :
11 a.m. in SEO 427
Abstract
This talk is one of a short series that will survey the Lang-Trotter conjecture for fixed Frobenius fields and
sketch the proof of an average version.
Feb. 24, 2016
Nathan Jones :
1 p.m. in SEO 1227
Abstract
This talk is the second of a short series in which we will discuss the conjecture of Lang-Trotter on Frobenius
Fields.
April 20, 2016
Joseph Gunther :
1 p.m. in SEO 1227
Abstract
We'll present two new applications of Poonen's closed-point sieve over finite fields. The first is that the
obvious local obstruction to embedding a curve in a smooth surface is the only global obstruction, over
any perfect field. The second is a proof of a conjecture of Vakil and Wood, on the probability that a random
hypersurface slice of a smooth variety will have a given number of singularities.
May 4, 2016
Jennifer Park :
1 p.m. in SEO 1227
Abstract
I will discuss a heuristic that predicts that the ranks of all but finitely many elliptic curves defined over Q are
bounded above by 21. This is joint work with Bjorn Poonen, John Voight, and Melanie Matchett Wood.
May 6, 2016
Vlad Serban :
11 a.m. in SEO 427
Abstract
Let $G$ be an abelian variety or a product of multiplicative groups
$\mathbb{G}_m^n$ and let $C$ be an embedded curve. The Manin-Mumford
conjecture (a theorem by work of Lang, Raynaud et al.) states that only
finitely many torsion points of $G$ can lie on $C$ unless $C$ is in
fact
a subgroup of $G$. I will show how these purely algebraic statements
extend to suitable analytic functions on open $p$-adic unit poly-disks.
These disks occur naturally as weight spaces parametrizing families of
$p$-adic automorphic forms for $GL(2)$ over a number field $F$. When
$F=\mathbb{Q}$, the "Hida families" in question play a crucial role in
the study of modular forms. When $F$ is imaginary quadratic, I will
explain how our results imply that Bianchi modular forms are sparse in
these $p$-adic families.
Oct. 13, 2016
Lilian Pierce :
10 a.m. in SEO 427
Abstract
Each number field has a positive integer associated to it called the class number, defined to be the cardinality of the class group of the field. Class numbers are important objects that arise naturally in many contexts in number theory: for example, Gauss famously investigated class numbers of quadratic fields, in the context of classifying the representation of integers by binary quadratic forms. Today, many deep open questions remain about the structure of class groups and the growth and divisibility properties of class numbers as fields vary over an appropriate infinite family. This talk will focus on the size of the p-torsion subgroup of the class group: it is conjectured that for any number field and any rational prime p, the p-torsion part of the class group of the field should be very small, in a suitable sense, relative to the discriminant of the field. This talk will present recent progress on bounding p-torsion in class groups of number fields of degree 2, 3, 4, 5.
Oct. 18, 2016
Joseph Gunther :
10 a.m. in SEO 427
Abstract
Recent work of Bhargava, Shankar, and Wang extended results on counting number fields of bounded
discriminant and low degree over the rationals, to allow any global field as the base field. Their work
uses geometry of numbers for both number fields and function fields. We'll explain how, in the function
field case, one can instead give algebro-geometric proofs, which suggest new avenues for research and
shed light on the geometry of the number field situation.
Feb. 7, 2017
Rahul Krishna :
11 a.m. in SEO 612
Abstract
I present a new trace formula approach to Waldspurger's formula for toric periods of automorphic forms on $PGL_2$. The method is motivated by interpreting Waldspurger's result as a period relation on $SO_2 \times SO_3$, which leads to a strange comparison of relative trace formulas. I will explain the local results needed to carry out this comparison, and discuss some optimistic dreams for extending these results to high rank orthogonal groups.
Feb. 14, 2017
Ramin Takloo-Bighash :
11 a.m. in SEO 612
Abstract
We consider the problem of counting the number of rational points of
bounded height in the zero-loci of Brauer group elements on semi-simple algebraic
groups over number fields. We obtain asymptotic formulae for the counting problem
for wonderful compactifications using the spectral theory of automorphic forms.
Applications include asymptotic formulae for the number of matrices over Q whose
determinant is a sum of two squares. These results provide a positive answer to some
cases of a question of Serre concerning such counting problems. This is joint work with Daniel Loughran
and Sho Tanimoto.
Feb. 21, 2017
Ramin Takloo-Bighash :
11 a.m. in SEO 612
Abstract
This is the continuation of last week's seminar. In this talk I will explain some ideas of the proof of the main theorem. Joint work with Daniel Loughran and Sho Tanimoto.
March 14, 2017
Jack Shotton :
11 a.m. in SEO 612
Abstract
Let $G = {\rm Gal}\,(\overline{\mathbb Q}_p/\mathbb Q_p)$. The Breuil-Mézard conjecture relates the
complexity of
deformation rings for mod $p$ Galois representations of $G$ with prescribed $p$-adic Hodge type
to the reduction mod $p$ of representations of $GL_n(\mathbb Z_p)$ associated to that type. It has been
important in the $p$-adic Langlands program and in first proof of the Fontaine-Mazur conjecture
for $GL_2$. We develop an analogous conjecture for mod $l$ representations of $G$ when $l \neq p$,
and explain how it can be proved with global methods.
April 4, 2017
Jack Shotton :
11 a.m. in SEO 612
Abstract
Let $G={\rm Gal}(\overline{{\mathbb Q}}_p/{\mathbb Q}_p)$. The Breuil-Mézard conjecture relates the
complexity of deformation rings for mod $p$
Galois representations of $G$ with prescribed $p$-adic Hodge type to the reduction mod p of representations of
$GL_n({\mathbb Z}_p)$ associated to that type. It has been important in the $p$-adic Langlands program and in
first proof of
the Fontaine-Mazur conjecture for $GL_2$. We develop an analogous conjecture for mod l representations of $G$
when $l \ne p$, and explain how it can be proved with global methods.
April 11, 2017
Silas Johnson :
11 a.m. in SEO 612
Abstract
The Malle-Bhargava heuristics give asymptotic predictions for the density of number fields of bounded
discriminant with a given Galois group G, in terms of the number of G-extensions of p-adic fields $\mathbb Q_p$.
These heuristics can also be applied when the discriminant is replaced by any of a wide variety of other
“counting functions”. Motivated by field-counting heuristics, I'll introduce the idea of such alternate counting
functions and how to build them, and discuss results on global mass formulas for alternate counting functions.
Jan. 19, 2018
A.C. Cojocaru :
11 a.m. in SEO 612
Jan. 26, 2018
McKinley Meyer :
11 a.m. in SEO 612
Feb. 2, 2018
McKinley Meyer :
11 a.m. in SEO 612
Feb. 9, 2018
McKinley Meyer :
11 a.m. in SEO 612
Feb. 16, 2018
McKinley Meyer :
11 a.m. in SEO 612
Feb. 23, 2018
Jacob Maybe :
11 a.m. in SEO 612
March 2, 2018
Matthew Fitzpatrick :
11 a.m. in SEO 612
March 9, 2018
Nathan Jones :
11 a.m. in SEO 612
March 16, 2018
Nathan Jones :
11 a.m. in SEO 612
March 23, 2018
Vlad Serban :
11 a.m. in SEO 612
April 6, 2018
Jacob Mayle :
11 a.m. in SEO 612
April 13, 2018
Kevin Vissuet :
11 a.m. in SEO 612
April 20, 2018
Arda Huseyin Demirhan :
11 a.m. in SEO 612
April 27, 2018
Seoyoung Kim :
noon in SEO 612
Abstract
Nagao's conjecture relates the rank of an elliptic surface to a limit formula arising from a weighted average of fibral Frobenius traces, and it is further generalized for smooth irreducible projective surfaces by M. Hindry and A. Pacheco. We show that the Sato-Tate conjecture based on the random matrix model implies Nagao's conjecture for certain twist families of elliptic curves and hyperelliptic curves.
Adrian Diaconu :
11 a.m. in SEO 612
Abstract
Let $\overline{\mathscr{M}}_{g, n}$ be the moduli space of $n$-pointed stable genus
$g$ curves, and let $\mathscr{M}_{g, n}$ be the moduli space of $n$-pointed smooth curves of genus $g.$ In this talk, I will discuss an asymptotic expansion for the characteristic of the free modular operad $\mathbb{M}\mathcal{V}$ generated by a stable $\mathbb{S}$-module $\mathcal{V},$ allowing to effectively compute $\mathbb{S}_{n}$-equivariant Euler characteristics of $\overline{\mathscr{M}}_{g, n}$ in terms of $\mathbb{S}_{n'}$-equivariant Euler characteristics of $\mathscr{M}_{g'\!, n'}$ with $0\le g' \le g,$ $\textrm{max}\{0, 3 - 2g' \} \le n' \le 2(g - g') + n.$ This answers a question posed by Getzler and Kapranov by making their integral representation
of the characteristic of the modular operad $\mathbb{M}\mathcal{V}$ effective. I will also discuss some applications.
May 4, 2018
Bharathwaj Palvannan :
11 a.m. in SEO 612
Abstract
A recent paper of Bleher, Chinburg, Greenberg, Kakde, Pappas, Sharifi and Taylor has initiated the topic of
higher codimension Iwasawa theory. As a generalization of the classical Iwasawa main conjecture, they prove
a relationship between analytic objects (two Katz's 2-variable p-adic L-functions) and algebraic objects
(two ``everywhere unramified'' Iwasawa modules) involving codimension two cycles. The talk will describe
an analogous result by considering the restriction, to an imaginary quadratic field K where a prime p splits, of
an elliptic curve E defined over Q with good supersingular reduction at p. This is joint work with Antonio Lei.
June 19, 2018
Jacob Mayle :
10 a.m. in SEO 427
June 21, 2018
Matthew Fitzpatrick :
10 a.m. in SEO 427
June 26, 2018
McKinley Meyer :
10 a.m. in SEO 427
June 28, 2018
Jacob Mayle :
10 a.m. in SEO 427
July 3, 2018
Matthew Fitzpatrick :
10 a.m. in SEO 427
July 5, 2018
McKinley Meyer :
10 a.m. in SEO 427
July 10, 2018
Jacob Mayle :
10 a.m. in SEO 427
July 12, 2018
Matthew Fitzpatrick :
10 a.m. in SEO 427
July 17, 2018
McKinley Meyer :
10 a.m. in SEO 427
July 19, 2018
Jacob Mayle :
10 a.m. in SEO 427
July 24, 2018
Matthew Fitzpatrick :
10 a.m. in SEO 427
July 26, 2018
McKinley Meyer :
10 a.m. in SEO 427
July 31, 2018
Jacob Mayle :
10 a.m. in 427 SEO
Aug. 2, 2018
McKinley Meyer :
10 a.m. in 427 SEO
Aug. 9, 2018
Jacob Mayle :
10 a.m. in 427 SEO
Dec. 20, 2018
Ken McMurdy :
10 a.m. in 427 SEO
Sept. 6, 2019
Nathan Jones :
10 a.m. in 1227 SEO
Abstract
In this talk, I will survey various results dealing with the nature of the the image of the Galois representation
on the torsion of an elliptic curve over a number field.
Sept. 13, 2019
Nathan Jones :
10 a.m. in 1227 SEO
Abstract
I will continue to survey various results dealing with the nature of the the image of the Galois representation
on the torsion of an elliptic curve over a number field.
Sept. 20, 2019
A.C. Cojocaru :
9:30 a.m. in 1227 SEO
Abstract
Inspired by the similarities between the group of units of the finite field $\mathbb{F}_p$ with p elements
and the group of $\mathbb{F}_p$-rational points of an elliptic curve, I will give an overview of
old and new results pertaining to the cyclicity of the groups defined by the reductions
modulo primes of an elliptic curve over $\mathbb{Q}$.
Sept. 27, 2019
A.C. Cojocaru :
9:30 a.m. in 1227 SEO
Abstract
Let $E/\mathbb{Q}$ be an elliptic curve defined over the field of rational numbers
and let $p$ be a prime of good reduction for $E$. We discuss the growth in $p$ of the absolute discriminant
of the ring of $\mathbb{F}_p$-endomorphisms of the reduction of $E$ modulo $p$.
Oct. 4, 2019
Matthew Fitzpatrick :
9:30 a.m. in 1227 SEO
Abstract
Let $E/\mathbb{Q}$ be an elliptic curve defined over the field of rational numbers
and let $p$ be a prime of good reduction for $E$. We discuss recent results on the growth in $p$ of the absolute discriminant
of the ring of $\mathbb{F}_p$-endomorphisms of the reduction of $E$ modulo $p$, valid for a density 1 of primes $p$.
This is joint work with A.C. Cojocaru.
Oct. 11, 2019
Sarthak Chimni :
9:30 a.m. in 1227 SEO
Abstract
In this talk I will report on a joint work with Gautam Chinta and Ramin Takloo-Bighash
in which we study subrings of $\mathbb{Z}^{n+k}$, of $(n+k)$-tuples of integers, of co-rank $k$.
We relate the number of subrings $R$ such that the torsion subgroup of $\mathbb{Z}^{n+k}/R$ is of size $r$
to the number of full rank subrings of $\mathbb{Z}^n$ of index $r$.
Oct. 18, 2019
A.C. Cojocaru :
9:30 a.m. in 1227 SEO
Abstract
We discuss CM lifting theorems for Drinfeld modules and their role in understanding the endomorphism rings of
the reductions of a generic Drinfeld module of rank 2. This is joint work with Mihran Papikian.
Oct. 25, 2019
Stephanie N. Reyes :
9:30 a.m. in 1227 SEO
Nov. 1, 2019
Simon L. Marshall :
9:30 a.m. in 1227 SEO
Abstract
I will recall the subconvexity problem for L-functions on $\text{GL}_n$,
and describe some of its applications.
I will then state a theorem of mine that is work in progress,
and gives a subconvex bound for a family of L-functions on $\text{GL}(n) \times \ \text{GL}(n+1)$
for arbitrary $n$.
Nov. 8, 2019
Heidi Goodson :
9:30 a.m. in 1227 SEO
Abstract
Let $C_m: y^2=x^m+c$ be a smooth projective curve defined over $\mathbb Q$.
We would like to study the limiting distributions of the coefficients of the normalized L-polynomial for $C_m$.
To determine the distributions, we study the Sato-Tate groups of the Jacobians of the curves. In this talk,
I will give both general results and explicit examples of Sato-Tate groups for certain curves $C_m$.
I will then use these groups to determine the limiting distributions of the coefficients of the normalized
L-polynomial. This is joint work with M. Emory.
Nov. 15, 2019
Sumita Garai :
9:30 a.m. in 1227 SEO
Abstract
For a generic, rank $r$ Drinfeld module, we discuss its reductions and their endomorphism rings.
We discuss the significance of the Frobenius indices and prove that they can be arbitrarily large.
We also give an algorithm to compute these indices explicitly and a basis of the endomorphism ring.
This is joint work with Mihran Papikian.
Nov. 22, 2019
Frank Thorne :
9:30 a.m. in 1227 SEO
Abstract
In a famous paper, Ellenberg and Venkatesh studied the number
of number fields $K$ with $[K : \mathbb{Q}] = n$ and $|\text{Disc}(K)| < X$.
A folk conjecture establishes that this quantity is asymptotic to a
constant (depending on $n$) times $X$; although this problem remains open
for $n > 5$, Ellenberg and Venkatesh obtained both upper and lower
bounds.
In the talk, I will give an overview of their proofs and of several
new results on number field counting which their work inspired. This
is joint work with Robert Lemke Oliver and (in part) Aaron Landesman.
Dec. 6, 2019
Tian Wang :
9:30 a.m. in 1227 SEO
Jan. 24, 2020
Dylon Chow :
1 p.m. in 1227 SEO
Jan. 14, 2022
Jacob Mayle :
1 p.m. in Zoom
Abstract
Let E/Q be an elliptic curve. For each prime number ell, one considers
the mod ell Galois representation of E, which encodes the action of the
absolute Galois group of Q on the ell-torsion subgroup of E. In 1972,
Serre showed that if E is without complex multiplication, then the mod
ell Galois representation is surjective whenever ell is sufficiently
large. In this talk, we'll discuss the theorem, as well as recent
computational and theoretical progress related to it.
Jan. 28, 2022
Nathan Jones :
1 p.m. in Zoom
Abstract
Given an elliptic curve E over a number field K, we say that E has an
entanglement if the intersection of two division fields of E of coprime
level is larger than K. Mazur's Program B, which asks for a
classification of the elliptic curves whose adelic Galois
representation lands inside a given fixed open subgroup of the
group of (finite) adelic points of GL2, falls naturally into two
parts: first, to classify all of the p-adic images for each prime p;
and second, to classify the entanglements. In this talk, I will
discuss motivating examples and survey various recent results in
this area, some of which are based on joint work of mine with K.
McMurdy and H. Daniels, and also with K. Vissuet and with S.M. Lee.
Feb. 4, 2022
Damaris Schindler :
1 p.m. in Zoom
Abstract
In this talk I will discuss joint work with Shuntaro Yamagishi where we
establish an asymptotic formula for the number of rational points, with
bounded denominators, within a given distance to a compact submanifold M
of R^n. We show that under certain curvature conditions we obtain
stronger results for manifolds in higher codimension than for
hypersurfaces, and discuss relations to Serre's dimension growth
conjecture.
Feb. 11, 2022
Tung T. Nguyen :
1 p.m. in Zoom
Abstract
Fekete polynomials play an important role in the study of special values of L-functions of quadratic fields.
While their analytic properties are well-studied in the literature, little is known about their arithmetics. In this
talk, we will discuss some surprising arithmetical properties of these polynomials. In particular, we will see
that special values of Fekete polynomials contain some rich information about the class numbers of quadratic
fields. Furthermore, their Galois groups seem to follow a rather simple pattern. Time permitting, I will discuss
some recent progress on generalized Fekete polynomials. This is based on joint work with Jan Minac and
Nguyen Duy Tan.
Feb. 18, 2022
Eun Hye Lee :
1 p.m. in Zoom
Abstract
Subconvexity problem is one of the central interests in analytic number theory, and it has been studied actively for over a century. In this talk, I will introduce the subconvexity problem with some known results, and then, I will survey the recent results of myself and R. Hough on the subconvexity of Shintani zeta functions.
Feb. 25, 2022
Kelly Isham :
1 p.m. in Zoom
Abstract
We can define a zeta function of a group (or ring) to be the Dirichlet series associated to the sequence that counts the number of subgroups (or subrings) of a given index. The subgroup zeta function over Z^n is well-understood, as is the asymptotic growth of subgroups in Z^n. Much less is known about the subring zeta function over Z^n and the asymptotic growth of subrings in Z^n. In this talk, we discuss the progress toward answering this question and we give new lower bounds on the asymptotic growth of subrings in Z^n. We then define a similar zeta function corresponding to subrings of corank at most k in Z^n. While the proportion of subgroups in Z^n of corank k is positive for each k, we show this is not the case for subrings in Z^n of corank k when n is sufficiently larger than k. If there is time, we will make connections to orders in number fields. Part of this work is joint with Nathan Kaplan.
March 4, 2022
Florin Boca :
1 p.m. in Zoom
Abstract
The large sieve inequality provides an estimate for the largest eigenvalue
of a matrix A*A, where A is a Vandermonde type rectangular matrix defined
by the roots of unity of order at most Q. The talk will discuss some
aspects concerning the behavior of the eigenvalues of these matrices when
N ~ cQ^2, with Q large and c>0 constant. In particular, we are interested
in asymptotic formulas for their moments, and in establishing the
existence of a limiting distribution as a function of c. This is joint
work with Maksym Radziwill.
March 11, 2022
Álvaro Lozano-Robledo :
1 p.m. in Zoom
Abstract
In a recent preprint, Rouse, Sutherland, and Zureick-Brown have given a
(conjectural) explicit classification of all the $\ell$-adic Galois
representations (up to conjugation) attached to elliptic curves over
$\mathbb{Q}$ without complex multiplication. In this talk, we apply a
recent classification (by the speaker) of Galois representations attached
to curves with CM to give a complete and explicit classification of all
the $\ell$-adic Galois representations in the CM case over $\mathbb{Q}$.
In particular, we will describe how many different representations appear
for each rational j-invariant with CM.
March 18, 2022
Chien-Hua Chen :
1 p.m. in Zoom
Abstract
In 2009, Pink and Rütsche proved the Drinfeld module analogue of the
``open image theorem''. Based on the open image theorem, it is natural
to ask the function field analogue of two questions raised by Serre in
the context of elliptic curves. The first question is whether there is
a Drinfeld module whose adelic Galois representation is surjective. The
other one is the analogue of the famous ``Serre's uniformity problem''.
In this talk, I will discuss my result on a certain class of Drinfeld
modules having adelic surjective Galois representation and a partial
result of the uniformity problem.
April 1, 2022
Pete Clark :
1 p.m. in Zoom
Abstract
The Chevalley-Warning Theorem says that for a system of polynomials of
"low degree"(the sum of the degrees is less than the number of variables)
over a finite field of characteristic p, the number of solutions is a
multiple of p. The Ax-Katz Theorem refines this to the best p-adic
divisibility on the size of the solution set in terms of the degrees of
the polynomials and the number of variables. Recently Aichinger-Moosbauer
developed a calculus of finite differences for maps between commutative
groups and used this to generalize Chevalley-Warning to any finite
rng of prime power order. Remarkably, they give a purely group-theoretic
result from which their ring-theoretic result follows. I will present
an extension of the Ax-Katz Theorem to finite rings of characteristic p,
which again follows from a purely group-theoretic result stated in terms
of the Aichinger-Moosbauer calculus. This is joint with U. Schauz. If
time permits, I will mention work towards extending this result to all
finite p-groups, joint with Schauz and N. Triantafillou.
Zoom link:
https://uic.zoom.us/j/88173268700?pwd=aEhmTGpSOVhidWE4L1VWUnNhNVlvUT09
April 22, 2022
Chantal David :
1 p.m. in Zoom
Abstract
Let E be an elliptic curve over Q, and let $\chi$ be a Dirichlet character
of order $\ell$ for some prime $\ell \geq 3$. Heuristics based on the
distribution of modular symbols and random matrix theory have led to
conjectures predicting that the vanishing of the twisted L-functions
$L(E, \chi, s)$ at $s = 1$ is a very rare event (David-Fearnley-Kisilevsky
and Mazur-Rubin). In particular, it is conjectured that there are only
finitely many characters of order $\ell > 5$ such that $L(E, \chi, 1) = 0$
for a fixed curve E.
We investigate in this talk the case of elliptic curves over function
fields. For Dirichlet L-functions over function fields, Li and Donepudi-Li
have shown how to use the geometry to produce infinitely many characters
of order $\ell \geq 2$ such that the Dirichlet L-function $L(\chi, s)$
vanishes at s = 1/2, contradicting (the function field analogue of)
Chowla’s conjecture. We show that their work can be generalized to
isotrivial curves E/Fq(t), and we show that if there is one Dirichlet
character $\chi$ of order $\ell$ such that $L(E, \chi, 1) = 0$, then
there are infinitely many, leading to some specific examples
contradicting (the function field analogue of) the number field
conjectures on the vanishing of twisted L-functions. Such a dichotomy does
not seem to exists for general (non-isotrivial) curves over Fq(t), and we
produce empirical evidence which suggests that the conjectures over
number fields also hold over function fields for non-isotrivial E/Fq(t).
May 13, 2022
Alex Gamburd :
1 p.m. in Zoom
Abstract
The Markoff equation x^2+y^2+z^2=3xyz, which arose in his spectacular
thesis (1879), is ubiquitous in a tremendous variety of contexts. After
reviewing some of these, we will discuss joint work with Bourgain and
Sarnak establishing forms of strong approximation on varieties of Markoff
type, as well as ensuing implications, diophantine and dynamical.
Feb. 24, 2023
Auden Hinz :
noon in 427 SEO
Abstract
I will explain an argument that gives an upper bound on the number of primes for which a pair of non-isogenous non-CM elliptic curves have the same Frobenius Field.
The argument presented is an application of the square sieve, matrix counting arguments, and the Chebotarev Density Theorem.
March 3, 2023
Nathan Jones :
noon in 427 SEO
Abstract
Given an elliptic curve E over a number field K, we say that E has an
entanglement if the intersection of two division fields of E of coprime
level is larger than K. Mazur's Program B, which asks for a
classification of the elliptic curves whose adelic Galois representation
lands inside a given fixed open subgroup of the group of (finite) adelic
points of GL2, falls naturally into two parts: first, to classify all
of the p-adic images for each prime p; and second, to classify the
entanglements. In this talk, I will discuss motivating examples and
survey various recent results in this area, some of which are based on
joint work of mine with K. McMurdy, and also with K. Vissuet and
with S. M. Lee.
March 10, 2023
Chirag Singhal :
noon in 612 SEO
Abstract
Let $A = \mathbb{F}_q[T]$ be the ring of polynomials in the indeterminate $T$ and
with coefficients in the finite field $\mathbb{F}_q$ with $q$ elements,
and let $K$ be an $A$-field.
Let $\phi$ be a Drinfeld $A$-module over $K$, of rank $r \geq 2$.
Denote by $j_{k_1, \ldots, k_{l}}^{s_1, \ldots, s_{l}}(\phi)$
the family of $j$-invariants associated to $\phi$ by I.Y. Potemine.
Motivated by an argument of G. Shimura used in the context of abelian varieties,
we show that
$\mathbb{F}_q(T, j_{k_1, \ldots, k_{l}}^{s_1, \ldots, s_{l}}(\phi))$
is a field of definition for $\phi$.
March 17, 2023
Soumya Sankar :
1 p.m. in Zoom
Abstract
The classical problem of counting elliptic curves with a rational N-isogeny can be phrased in terms of
counting rational points on certain moduli stacks of elliptic curves. Counting points on stacks poses various
challenges, and I will discuss these along with a few ways to overcome them. I will also talk about the theory
of heights on stacks developed in work of Ellenberg, Satriano and Zureick-Brown, among others, and use it to
count elliptic curves with an N-isogeny for certain N. The talk assumes no prior knowledge of stacks and is
based on joint work with Brandon Boggess. Time permitting, I will also talk about ongoing work with Jennifer
Park on counting elliptic curves of higher degree.
Chirag Singhal :
noon in 612 SEO
Abstract
Let $A = \mathbb{F}_q[T]$ be the ring of polynomials in the indeterminate $T$ and
with coefficients in the finite field $\mathbb{F}_q$ with $q$ elements,
and let $K$ be an $A$-field.
Let $\phi$ be a Drinfeld $A$-module over $K$, of rank $r \geq 2$.
Denote by $j_{k_1, \ldots, k_{l}}^{s_1, \ldots, s_{l}}(\phi)$
the family of $j$-invariants associated to $\phi$ by I.Y. Potemine.
Motivated by an argument of G. Shimura used in the context of abelian varieties,
we show that
$\mathbb{F}_q(T, j_{k_1, \ldots, k_{l}}^{s_1, \ldots, s_{l}}(\phi))$
is a field of definition for $\phi$.
March 31, 2023
Tian Wang :
noon in 612 SEO
Abstract
Given an abelian variety $A$ defined over $\mathbb{Q}$,
we look at its reduction $A_p$ at an arbitrary prime $p$
and study the distribution of $A_p$ with specific properties.
For example, the Lang-Trotter Conjecture about
an abelian variety $A$ of dimension $1$
and its generalizations
to an abelian variety $A$ of higher dimension concern
the distribution of $A_p$ having a prescribed Frobenius trace, while
the Murty-Patankar Conjecture about an absolutely simple abelian variety
$A$
concerns the distribution of $A_p$ having split reduction.
In this talk, I will discuss new upper bounds for
the counting functions related to the aforementioned conjectures.
April 14, 2023
Frederick Vincent Saia :
noon in 612 SEO
Abstract
A CM component of the $\ell$-isogeny graph of elliptic curves has
a particular structure, that of an $\ell$-volcano,
at least away from certain CM orders.
The structure of “isogeny volcanoes’’ has seen much use in the study
of CM elliptic curves over finite fields,
originating with the 1996 PhD thesis work of Kohel.
Recent work of Clark—Saia leverages infinite depth versions of
these graphs to study moduli of isogenies of CM elliptic curves
over $\overline{\mathbb{Q}}$.
We will discuss an analogue of this work for abelian surfaces with
quaternionic multiplication. A main result includes an algorithm
to compute the $\mathfrak{o}$-CM locus on the Shimura curve
$X_0^D(N)$ over $\mathbb{Q}$, for $\mathfrak{o}$ any imaginary
quadratic order and $\text{gcd}(D,N) = 1$.
As an application, we give an explicit list of pairs $(D,N)$ for
which the Shimura curves $X_0^D(N)$ and $X_1^D(N)$ may fail to have
a sporadic CM point.
April 21, 2023
Andreea Iorga :
noon in 612 SEO
Abstract
In this talk, I will show that,
under a specific assumption,
any semi-direct product of a $p$-group $G$ with a group
$\Phi$ of
order prime-to-$p$ can appear as the Galois group of
a tower of extensions $H/F/E$ with the property that $H$
is the maximal pro-$p$ extension of $F$ that is
unramified everywhere and $\text{Gal}(H/F) = G$.
At the end, I will show that a nice consequence of this is
that any local ring admitting a surjection to $\mathbb{Z}_5$ or
$\mathbb{Z}_7$
with finite kernel can be written as a universal everywhere
unramified deformation ring.
May 2, 2023
Pedro-Jose Cazorla Garcia :
8:45 a.m. in 612 SEO
Abstract
The Lebesgue-Ramanujan-Nagell equation
$x^2 + D = y^n$
has been studied extensively by number theorists
during the last century,
both for its intrinsic interest and as a generalisation
to Catalan's conjecture.
With the advent of the modular methodology developed by
Wiles, Ribet and others and employed in the proof
of Fermat's last theorem,
along with the evolution of computational number theory techniques,
it is now feasible to consider
the generalised Lebesgue-Ramanujan-Nagell equation
$C_1 x^2 + C_2 = y^n$.
In this talk, we will discuss a variety of techniques
which allow us to solve the aforementioned equation in the range
$1 \leq C_1, C_2 \leq 20$,
involving the modularity of Galois representations and Thue equations.
Aug. 25, 2023
Auden Hinz :
1 p.m. in 427 SEO
Sept. 8, 2023
Freddy Saia :
1 p.m. in 427 SEO
Abstract
In this second talk, we'll discuss moduli interpretations and canonical models, with CM points playing a key role.
Sept. 15, 2023
John Sung Min Lee :
1 p.m. in 427 SEO
Abstract
Given an elliptic curve $E/\mathbb{Q}$ and a prime $p$ of good reduction for $E$, let $\tilde{E}_p$ denote the reduction of $E$ modulo $p$. If $\tilde{E}_p(\mathbb{F}_p)$ forms a cyclic group, we call $p$ a prime of cyclic reduction for $E$. In this talk, we study the issue of which arithmetic progressions $k \pmod n$ have
the property that, for all but finitely many primes $p \equiv k \pmod n$, the group $\tilde{E}_p(\mathbb{F}_p)$ is not cyclic, answering a question of Akbal and G\"{u}lo\v{g}lu. Also, we show that primes of cyclic reduction are statistically biased modulo $n$, refining Banks and Shparlinski's results on average density of primes of cyclic reduction. The first part of this talk is a joint work with Nathan Jones.
Sept. 29, 2023
John Sung Min Lee :
1 p.m. in 427 SEO
Abstract
Given a prime power $q$, let $A = \mathbb{F}_q[T]$ and $k = \mathbb{F}_q(T)$. Take a finite extension $K/k$ and $\psi$ a generic Drinfeld $A$-module over $K$ of rank $r \geq 2$. Given a prime $\wp$ of good reduction for $\psi$, the reduction $\psi_\wp(\mathbb{F}_\wp)$ forms a finite $A$-module of rank at most $r$. Let us denote the first invariant factor of $\psi_\wp(\mathbb{F}_\wp)$ by $d_{1,\wp}(\psi)$. Kuo and Liu determined the density of primes of $K$ for which $d_{1,\wp}(\psi) = 1$, given $\psi$ has a trivial endomorphism ring. Cojocaru and Shulman largely generalized their results and determined the density of primes of $K$ for which $d_{1,\wp}(\psi) = d$ without any assumption. In this talk, we add a congruence class condition on their results, i.e., we study the distribution of primes $\mathfrak{p}$ of $k$ that lie in an arithmetic progression and $d_{1,\mathfrak{p}}(\psi) = d$.
Feb. 28, 2024
John Voight :
1 p.m. in 636 SEO
Abstract
We begin by briefly surveying the problem of counting elliptic
curves defined over the rationals with prescribed level structure by
height. We then discuss recent joint work with Grant Molnar where we
count elliptic curves with a 7-isogeny.
March 8, 2024
Tian Wang :
1 p.m. in 427 SEO
Abstract
Let $ E$ over $\bf{Q}$ be an elliptic curve without complex multiplication.
Lang and Trotter made a conjecture regarding the number of primes $p$ up to $x$
for which the reduction of $E$ at $p$ is supersingular.
Though the conjecture is still open, we now have unconditional upper and lower bounds,
thanks to the work of several mathematicians in the past few decades.
However, much less has been studied for the distribution of supersingular primes for
abelian surfaces (even conjecturally).
In this talk, I will present my recent work on unconditional upper bounds for the number of primes
$p$ up to $x$ for which the reduction of a fixed abelian surface at $p$ is supersingular.
April 26, 2024
Ciaran Schembri :
1 p.m. in 1227 SEO
Abstract
In a celebrated work Mazur classified which torsion subgroups can occur for elliptic curves defined over the rationals. A natural analogue is to consider surfaces with geometric endomorphisms by a quaternion order (PQM surfaces), since the associated moduli space is 1-dimensional. In this talk I will discuss progress towards classifying which torsion subgroups are possible for these surfaces. We also give a description of the moduli problem for PQM surfaces. This is joint work with Eran Assaf, Jef Laga, Freddy Saia, Ari Shnidman, Jacob Swenberg and John Voight.
Oct. 4, 2024
-- :
9 a.m. in 636 SEO
Oct. 11, 2024
Freddy Saia :
9 a.m. in 636 SEO
Abstract
Since Mazur's work on rational isogenies and rational torsion of elliptic curves over $\mathbb{Q}$, there has been concerted effort towards understanding low degree points on modular curves such as $X_0(N)$ and $X_1(N)$. Considerably less is known for Shimura curves, which parameterize abelian surfaces with quaternionic multiplication and analogous torsion structures. By a result of Shimura, these curves have no real points, hence no odd degree points, so we focus first on degree 2 points. We will discuss the determination of the Shimura curves $X_0^D(N)$ with infinitely many quadratic points, resulting from a study of the bielliptic curves in this family. This is based on joint work with Oana Padurariu.
Oct. 15, 2024
Chris Keyes :
2 p.m. in 636 SEO
Abstract
A cubic hypersurface in $\mathbb{P}^n$ defined over $\mathbb{Q}$ is given by the vanishing locus of an integral cubic form in $n+1$ variables. For $n \geq 4$, it is conjectured that these varieties satisfy the Hasse principle. Recent work of Browning, Le Boudec, and Sawin shows that this conjecture holds on average, in the sense that the density of soluble cubic forms is equal to that of the everywhere locally soluble ones. But what do these densities actually look like? We give exact formulae in terms of the probability that a cubic hypersurface has $p$-adic points for each prime $p$. These local densities are rational functions, uniform in $p$, recovering a result of Bhargava, Cremona, and Fisher in the $n=2$ case. This is joint work with Lea Beneish.
Oct. 18, 2024
Tristan Phillips :
9 a.m. in 636 SEO
Abstract
Let $E$ be an elliptic curve over a number field $K$. By the Mordell-Weil Theorem, the set of rational points $E(K)$ forms a finitely generated abelian group, which can be expressed as $E(K) \cong E(K)_\text{tors} \times \mathbb{Z}^r$, where $E(K)_{\text{tors}}$ is the finite torsion subgroup and $r$ is the rank of $E$. In this talk, I will present results on the frequency with which elliptic curves exhibit a prescribed torsion subgroup, and how to establish bounds on the average analytic rank of elliptic curves over number fields. A key approach underlying these results involves employing techniques from Diophantine geometry to count points of bounded height on genus zero modular curves.
Oct. 25, 2024
Hui Xu :
9 a.m. in 636 SEO
Abstract
In this talk, we discuss explicit spanning sets for two vector spaces. One is the subspace generated by integral-weight Hecke eigenforms with nonzero central L- values. The other is a subspace generated by half-integral weight Hecke eigenforms with nonvanishing first Fourier coefficients. We also show that these two spaces are isomorphic via the Shimura lift.
Oct. 29, 2024
Dani Szpruch :
2 p.m. in 636 SEO
Abstract
The Hasse-Davenport product relation is an identity involving products of
Gauss sums. In this talk we shall introduce a generalization of this
classical result for ε-factors defined on a p-adic field. We will then
introduce an analog of Shahidi local coefficients for covering groups
and discuss the relation between this local factor and the generalized
Hasse-Davenport product relation.
Nov. 15, 2024
Ramin Takloo-Bighash :
9 a.m. in 636 SEO
Abstract
In this talk I will discuss a recent work in which Eun Hye Lee and I prove the analytic continuation of the Shintani zeta function of the prehomogeneous vector space of binary cubic forms twisted by automorphic forms over an arbitrary number field, generalizing works by Hough and Hough-Lee. Time allowing, I will also discuss some possible arithmetic applications.
Dec. 6, 2024
Santiago Arango-Piñeros :
9 a.m. in 636 SEO
Abstract
I will discuss the multiplicative group generated by the roots of a monic polynomial with integer coefficients. In particular, I will focus on polynomials that arise as the characteristic polynomial of the Frobenius endomorphism of an abelian variety defined over a finite field. I will discuss some recent results and applications obtained in joint work with Bhamidipati and Sankar (https://arxiv.org/abs/2306.02237), and with Frengley and Vemulapalli (https://sarangop1728.github.io/GaloisGroupsFrobPolys_version0.pdf).
Jan. 17, 2025
Organizational Meeting :
1 p.m. in 636 SEO
Jan. 31, 2025
Ramin Takloo-Bighash :
1 p.m. in 636 SEO
Abstract
In this talk I will explain a new work joint with Dylon Chow, Daniel
Loughran, and Sho Tanimoto in which we formulate a conjecture about
the distribution of Campana points on Fano varietieis. I will explain
how our conjecture is compatible with results in literature and present
a new class of new examples where the conjecture holds. The latter class
consists of wonderful compactifications of semisimple groups of adjoint
type over an arbitrary number field.
Feb. 14, 2025
Freddy Saia :
1 p.m. in 636 SEO
Abstract
It is natural to study the torsion of an elliptic curve via its Galois representations, which encode the action of Galois on torsion points. Serre’s Open Image Theorem tells us that for an elliptic curve E over a number field K, the mod-$\ell$ Galois representation is surjective for all sufficiently large primes $\ell$. Since Serre’s work, considerable effort has gone towards obtaining extensions to other classes of abelian varieties and also towards making “sufficiently large” effective.
I will discuss joint work with Alina Cojocaru, in which we obtain an explicit surjectivity result for products of elliptic curves over function fields. This comes from careful use of techniques of Masser—Wüstholz from the number field setting, in combination with a result of Cojocaru—Hall for elliptic curves over function fields and recent isogeny bounds for elliptic curves over function fields due to Griffon—Pazuki. As an application, we prove that most specializations of certain families of products of elliptic curves over the rationals have no small exceptional primes.
Feb. 21, 2025
Jinyue Luo :
1 p.m. in 636 SEO
Abstract
In the study of connections between Galois representations and modular forms, one often seeks an R=T theorem, which asserts that the deformation ring R is isomorphic to a (localized) Hecke algebra T. However, sometimes only the framed deformation ring exists. With the framing variables, it is obviously larger than the Hecke algebra. Pseudorepresentations, which is a generalization of the notion of traces of representations, was invented to get around this issue. We will introduce the notion of pseudorepresentations and discuss the criteria for pseudorepresentations to arise from genuine representations. Next, we will introduce the algorithm used to explicitly compute usual deformation rings and pseudodeformation rings for finitely presented groups, which leads to the discovery of a counterexample.
April 11, 2025
Tyler Genao :
1 p.m. in 636 SEO
Abstract
In 1996, Merel showed that for any elliptic curve $E$ defined over a number field $F$ of degree $d\in\mathbb{Z}^+$,
the size of the torsion group of $E$ over $F$ is bounded by a constant $B:=B(d)$ which depends only on $d$, and that conjecturally is in fact a polynomial in $d$.
In this talk, I will discuss recent joint work with Abbey Bourdon which shows that $B$ is polynomial in $d$ for torsion from the family $\mathcal{I}_{\mathbb{Q}}$ of elliptic curves which are geometrically isogenous to at least one rational elliptic curve. For torsion from the subfamily $\mathcal{F}_{\mathbb{Q}}$ of elliptic curves with rational $j$-invariant, our results strengthen prior work of Clark and Pollack.
April 18, 2025
Micah Milinovich :
1 p.m. in 636 SEO
Abstract
There are many situations where one imposes certain conditions on a
function and its Fourier transform and then attempts to optimize a
certain quantity. I will describe how two such Fourier optimization
frameworks can be used to study classical problems in number theory:
bounding the maximum gap between consecutive primes assuming the
Riemann hypothesis and bounding for the size of the least quadratic
non-residue modulo a prime assuming the generalized Riemann hypothesis
(GRH) for Dirichlet L-functions. The resulting extremal problems in
analysis can be stated in accessible terms, but finding the exact
answer appears to be rather subtle. However, we can experimentally
find upper and lower bounds for our desired quantity that are
numerically close. If time allows, I will discuss how a similar
Fourier optimization framework can be used to bound the size of the
least prime in an arithmetic progression on GRH. This is based upon
joint works with E. Carneiro (ICTP), E. Quesada-Herrera (Lethbridge),
A. Ramos (SISSA), and K. Soundararajan (Stanford).
Aug. 29, 2025
-- :
noon in 636 SEO
Sept. 12, 2025
Ramin Takloo-Bighash :
noon in 636 SEO
Abstract
In this talk I will explain a new result joint with Eun Hye Lee in which we prove the analytic continuation of the Shintani zeta function of the prehomogeneous space of binary cubic forms twisted with an automorphic form over an arbitrary number field to the domain $\Re s > 3/4$, independent of all data. The main application of this result is the effective equidistribution of shapes of cubic extensions of arbitrary number fields.
Sept. 19, 2025
Freddy Saia :
noon in 636 SEO
Abstract
This talk, based on joint work with Oana Padurariu, will concern quotients of the Shimura curves $X_0^D(N)$ over $\mathbb{Q}$ by Atkin—Lehner involutions, which parameterize abelian surfaces with potential quaternionic multiplication. We prove that there are exactly $3711$ quotients $X_0^D(N)/W$, with $D>1$ and $W$ a non-trivial Atkin—Lehner subgroup, having genus $g \leq 2$. We also investigate the arithmetic of these curves; we determine isomorphism classes over $\mathbb{Q}$ of the Jacobians of genus $1$ quotients, we prove a theorem on infinitude of rational points on Atkin—Lehner quotients, and we produce equations for over $500$ non-elliptic genus $1$ and bielliptic genus $2$ quotients.
One of our main tools is an algorithm we implement to compute, when $N$ is squarefree, the dual graph of the minimal regular model of $X_0^D(N)/W$ over $\mathbb{Z}_p$ for each prime divisor $p$ of $D$ using the theory of Cerednik--Drinfeld reductions. We will aim to give an approachable overview of this theory in this context, with concrete examples along the way.
Sept. 26, 2025
Tian An Wong :
noon in 636 SEO
Abstract
In 2000, Langlands proposed a method to weight the Arthur-Selberg trace formula with automorphic L-functions, whose analytic behaviour is expected to detect when an automorphic form is a functorial transfer from a smaller group. An immediate obstruction that arises is the presence of nontempered representations, i.e., representations that do not satisfy the Ramanujan conjecture. It was later proposed in 2010 that an appropriate Poisson summation might be used to remove the contribution of such representations. So far, this has been carried out successfully in limited cases involving GL(2). In this talk I will introduce a general method for GL(2) and its connection to prehomogeneous vector spaces. Time permitting, I will also discuss potential consequences for the analytic behaviour of GL(2) L-functions.
Oct. 24, 2025
Alex Smith :
noon in 636 SEO
Abstract
Among the nondegenerate C^4 hypersurfaces, we characterize the rational quadrics as the hypersurfaces that are the least well approximated by rational points. For all other hypersurfaces, we give a heuristically sharp lower bound for the number of rational points near them, improving the sensitivity of prior results of Beresnevich and Huang. Our methods are dynamical, involving the application of Ratner's theorems for unipotent orbits, and we will show how our work relates to the dynamical resolution of the Oppenheim conjecture by Margulis.
Nov. 21, 2025
Shiva Chidambaram :
noon in 636 SEO
Abstract
Let $C$ be a genus 3 curve whose Jacobian is geometrically simple and has imaginary multiplication. I will discuss an algorithm, developed jointly with Pip Goodman, to compute the set of primes $\ell$ for which the image of the associated mod-$\ell$ Galois representation is not maximal. There are two natural families of genus 3 Jacobians with imaginary multiplication by $\mathbb{Z}[i]$ and $\mathbb{Z}[\zeta_3]$, coming from curves with a $\mu_4$ or $\mu_6$ action. Can we construct any new families? How do we rigorously, and not just numerically, certify these extra endomorphisms when they don't come from automorphisms on the curve? I will also discuss these questions. These are ongoing joint works with Pip Goodman and Francesc Fite.
March 13, 2026
Zachary Porat :
noon in 1227 SEO
Abstract
Ash, Grayson, and Green computed the action of Hecke operators on the cuspidal cohomology of congruence subgroups $\Gamma_0(3, p) \subseteq \mathrm{SL}(3, \mathbb{Z})$ for small $p$. A natural question to ask is for what other congruence subgroups of $\mathrm{SL}(3, \mathbb{Z})$ can one perform analogous computations. In this talk, we detail techniques for working with congruence subgroups that are Iwahori at $p$, providing a framework for understanding the action of Hecke operators on the corresponding cohomology. If time permits, we will discuss some improvements for the $\Gamma_0(3, p)$ setting as well. $$$$
Note the room change!
March 20, 2026
Eun Hye Lee :
noon in 636 SEO
Abstract
Shintani zeta functions arise as generating functions for number of cubic fields. In this talk, we will be exploring the analytic properties of Shintani zeta functions and their automorphic form twists. As an application, we will also look at the shape of cubic fields, using the orbital integrals. This is based o joint works with R. Hough.
April 27, 2026
Harald Helfgott :
noon in 612 SEO
Abstract
Let $A(s) = \sum_n a_n n^{-s}$ be a Dirichlet series with meromorphic continuation. Say we are given information on the poles of $A(s)$ with $|\Im s| \leq T$ for some large constant $T$. What is the best way to use such finite spectral data to give explicit estimates for sums $\sum_{n\leq x} a_n$?
The problem of giving explicit bounds on the Mertens function $M(x) = \sum_{n\leq x} \mu(n)$ illustrates how open this basic question was. Bounding $M(x)$ might seem equivalent to estimating $\psi(x) = \sum_{n\leq x} \Lambda(n)$ or the number of primes $\leq x$. However, we have long had fairly good explicit bounds on prime counts, while bounding $M(x)$ remained a notoriously stubborn problem.
We prove a sharp, general result on sums $\sum_{n\leq x} a_n n^{-\sigma}$ for $a_n$ bounded, giving a optimal way to use information on the poles of $A(s)$ with $|\Im s|\leq T$ and no data on the poles above.
Our bounds on $M(x)$ are stronger than previous ones by many orders of magnitude. We also give a sharp result on such sums for a_n non-negative and not necessarily bounded, and apply it to obtain optimal bounds on psi(x)-x given finite verifications of RH.
Our proofs mixe a Fourier-analytic approach in the style of Wiener--Ikehara with contour-shifting, using optimal approximants of Beurling--Selberg type as in (Graham--Vaaler, 1981) and (Carneiro--Littmann, 2013). While our approach does not depend on existing explicit work in number theory, our method has an important step in common with work on another problem by (Ramana–Ramare, 2020).