Collaborative Number Theory Seminar at the CUNY
Graduate Center
Coorganizers: Gautam Chinta, Clayton Petsche, Maria Sabitova, Lucien Szpiro.
Fall 2010 Schedule:
September 24: No meeting this week.
October 1: Jonas B. Rasmussen (City College of New
York).
Title: Higher congruences between modular forms.
Abstract: Given a prime number p and two modular forms
on a congruence
subgroup that have coefficients in the ring of
integers of some number field
K, we will discuss the problem of deciding whether
these forms have
coefficients that are congruent mod P^m (where P is a
fixed prime ideal of K
over p and m is a positive integer). The main focus
will be on eigenforms on Gamma_1(N), and determining
necessary
and sufficient computable criteria for two such forms
to be congruent mod P^m
(possibly outside the prime divisors of Np). One of
the primary applications
of determining such criteria is the ability to
determine if the associated mod
P^m Galois representations are isomorphic. Using these
computable criteria, we have systematically computed a
wealth of
examples of such higher congruences, and based on
these examples we discuss
conjectures on maximal congruences between newforms on
Gamma_0(N). This is based on joint work with I. Chen
and I. Kiming.
October 8: No meeting this week.
October 15: Jeffrey Hoffstein (Brown University).
Title: Shifted multiple Dirichlet Series and moments
of Lseries.
Abstract: I'll explain what shifted multiple Dirichlet
Series are and show how to obtain their meromorphic
continuation. I'll give a sample application to mean
values of products of two GL(2) Lseries averaged over
character twists.
October 22: Oliver Lorscheid (City College of New
York).
Title: Toroidal automorphic forms.
Abstract: A formula of Erich Hecke in an article from
1917 laid a connection between a sum of values of an
Eisenstein series E(,s) with the value zeta(s) of the
zeta function. We call an automorphic form toroidal if
the corresponding sum (or integral in its adelic
formulation) vanishes for all right translates. The
importance of this definition lies in a reformulation
of the Riemann hypothesis in terms of the space of
toroidal automorphic forms as observed by Don Zagier.
Namely, the Eisenstein series E(,s) lies in a
tempered representation if and only if s has real part
1/2, and by Hecke's formula, E(,s) is toroidal if s
is a zero of the zeta functions. In order to reverse
the latter statement, nonvanishing results has to be
shown for the factors occuring in Hecke's formula. In
a joint work with Gunther Cornelissen, double
Dirichlet series are used for this purpose. In this
talk, we will introduce into the theory of toroidal
automorphic
forms and give an overview of results in this
direction. Then we will explain how to use double
Dirichlet series to show nonvanishing results.
October 29: Phil Williams (CUNY Graduate Center).
Title: Resultant and conductor: minimality and
semistability.
Abstract: The minimal discriminant is a divisor that
has support at the points of bad reduction of
an elliptic curve over a function field or number
field. Szpiro's Theorem bounds the degree
of the minimal discriminant divisor of an elliptic
curve over a function field in terms of the
genus of the function field and the number of points
of bad reduction. The corresponding
statement for number fields is equivalent to one
formulation of the ABCconjecture.
For selfmaps of the projective line over a function
field, there is an analogous concept
of bad reduction. One can construct a divisor which
measures this: the minimal resultant.
We construct a condition for measuring the minimality
(at a point on the given curve)
of a given presentation of a dynamical system, and use
this to show a counterexample
to the natural dynamical analogue to Szpiro's Theorem
in this context. We also show
that a semistable presentation of a dynamical system
over a function field realizes the
conductor (the set of points of bad reduction). This
is joint work with Lucien Szpiro and Michael Tepper.
November 5: Takashi Taniguchi (Kobe University and
Princeton University).
Title: The secondary term in the counting function of
cubic field discriminants, and their distributions in
arithmetic progressions.
Abstract: In this joint work with Frank Thorne, we
present a proof of Roberts' conjecture that gives the
secondary term in the counting function for cubic
fields. Our work is independent of another proof of
the conjecture by Bhargava, Shankar and Tsimerman, and
uses the Shintani zeta functions. With the twists of
the Shintani zeta function by Dirichlet characters, we
can also study the distributions of cubic field
discriminants in arithmetic progressions. We show a
couple of nonequidistribution results in the
secondary term.
November 12: Cormac O'Sullivan (Bronx Community
College).
Title: Taylor coefficients of modular forms.
Abstract: The Fourier coefficients of modular forms
are wellknown to contain useful arithmetic
information. These Fourier coefficients may be thought
of as Taylor coefficients at infinity. In joint work
with Morten Risager, we begin to study the Taylor
expansions of modular forms at points in the upper
half plane, building on earlier work of Rodriguez
Villegas and Zagier. At CM points we show these Taylor
coefficients are nonzero and that they also have
interesting arithmetic properties, many still
unexplained.
November 19: Krzysztof Klosin (Queens College).
Title: Modularity of residually reducible Galois
representations.
Abstract: Proving that ``nice'' Galois representations
arise from automorphic forms has been a major theme in
number theory for well over a decade. We will discuss
an approach based on studying the ideal of
reducibility of a universal deformation that allows us
to prove new cases of modularity. Examples include
some 2dimensional Galois representations over
imaginary quadratic fields (unconditional) and
4dimensional Galois representations over the
rationals (conditional on some assumptions). This is
joint work with T. Berger.
November 26: Thanksgiving, the University is closed.
December 3: Amanda Folsom (Yale University)
Title: KacWakimoto characters, asymptotics, and mock
modular forms.
Abstract: Recently, Kac and Wakimoto established
specialized character formulas
for irreducible highest weight sl(m,1)^ modules, and
later works of the author
and BringmannOno show that these characters may be
realized as parts of
certain nonholomorphic modular functions. We will
describe this, and show in
joint work with Bringmann, how the ``modularity" of
these characters can be
exploited to obtain detailed asymptotics.
December 10: Rafe Jones (College of the Holy Cross).
Title: Galois theory of rational maps with
nontrivial automorphisms.
Abstract: I'll describe recent work investigating the
dynamical Galois representation attached to a degree2
rational function that commutes with a nontrivial
Mobius transformation. This is in some sense the
equivalent of the Galois representation attached to a
CM elliptic curve. The image of this dynamical Galois
representation must lie in a certain subgroup C of the
automorphism group of the complete infinite binary
rooted tree. In joint work with Michelle Manes, we
show that the image has finite index in C under fairly
weak hypotheses.
Spring 2010 Schedule:
February 5: Clayton Petsche (Hunter College).
Title: A nonArchimedean Weyl equidistribution
theorem.
Abstract: Weyl proved that if an Ndimensional real
vector v has linearly
independent coordinates over Q, then its integer
multiples v, 2v, 3v,
.... are uniformly distributed modulo 1. Stated
multiplicatively (via
the exponential map), this can be viewed as a
Haarequidistribution
result for the cyclic group generated by a point on
the Ndimensional
complex unit torus. I will discuss an analogue of this
result over a
nonArchimedean field K, in which the equidistribution
takes place on
the Ndimensional Berkovich unit torus over K. The
proof uses a
general criterion for nonArchimedean
equidistribution, along with a
theorem of MordellLang type for the group variety
G_m^N over the
residue field of K, which is due to Laurent.
February 12: Lincoln's Birthday, the University is
closed.
February 19: Maria Sabitova (Queens College).
Title: Local root numbers of abelian varieties.
Abstract: Root numbers are important fundamental
invariants that arise in
connection with several influential conjectures of
number theory and
representation theory, such as BirchSwinnertonDyer
conjecture,
conjectural functional equations for Lfunctions, and
the Langlands
program. In this talk I will discuss recent progress
on root numbers
attached to abelian varieties.
February 26: Benjamin Hutz (Amherst College).
*Canceled due to snow*
March 5: Andrew Obus (Columbia University).
Title: Ramification in Fields of Moduli.
Abstract: If G is a finite group, then the field of
moduli of a branched Gcover of the Riemann sphere is
the intersection of all fields of definition of the
Gcover. A result of Beckmann says that for any
3point GGalois cover of the Riemann sphere, if a
prime p does not divide the order of G, then p is
unramified in the field of moduli of the Gcover.
Wewers generalized this: if p exactly divides the
order of G, then p is tamely ramified in the field of
moduli. We will discuss extensions of this result,
contained in the speaker's thesis, involving more
general groups G with cyclic pSylow groups.
March 12: Anupam Bhatnagar (CUNY Graduate Center)
Title: Points of Canonical Height Zero.
Abstract: We give a parametrization of points of
canonical height zero of an algebraic dynamical system
defined over the function field of a curve. The main
theorem describes the relation between points of
canonical height zero and preperiodic isotrivial
subvarieties. This is joint work with Lucien Szpiro.
March 19: Kiryl Tsishchanka (NYU)
Title: On approximation of real numbers by algebraic
numbers of bounded degree.
March 26: Matt Papanikolas (Texas A&M)
Title: Special values of Goss Lfunctions for
function fields.
Abstract: Values of Dirichlet Lfunctions at positive
integers are expressible in terms of powers of pi and
values of polylogarithms at algebraic numbers. In this
talk we will focus on finding analogies of these
results over function fields of positive
characteristic. In particular, we will consider
special values of Goss Lfunctions for Dirichlet
characters, which take values in the completion of the
rational function field in one variable over a finite
field. Building on work of Anderson for the case of
L(1,chi), we deduce various power series identities on
tensor powers of the Carlitz module that are
"logalgebraic" and in turn use these formulas to
determine exact values of L(n,chi) for arbitrary n
> 0. Moreover, we relate these Lseries values to
powers of the Carlitz period and values of Carlitz
polylogarithms at algebraic points.
April 2: Spring Break, the University is closed.
April 9: Aaron Levin (Institute for Advanced Study)
Title: Towards Schmidt's Theorem for Algebraic Points
of Bounded Degree.
Abstract: The Schmidt subspace theorem is a deep
generalization of Roth's theorem in Diophantine
approximation to the setting of hyperplanes in
projective space. Another wellknown generalization of
Roth's theorem is the theorem of Wirsing, which
extends Roth's theorem from rational points to
algebraic points of bounded degree. In a similar way,
I will discuss some results giving a version of
Schmidt's theorem for algebraic points of bounded
degree.
April 16: Yiannis Petridis (University College
London)
Title: Dissolving cusp forms into resonances: Higher
order Fermi's Golden Rules.
Abstract: For a hyperbolic surface embedded
eigenvalues are unstable and tend to become
resonances. The sufficient dissolving condition was
identified by PhillipsSarnak and is elegantly
expressed in Fermi's Golden Rule. We prove formulas
for higher approximations and obtain necessary and
sufficient conditions for dissolving a cusp form into
a resonance. We relate the result to the special
values of Lseries involving a RankinSelberg
convolution of the cusp form with higher order
automorphic forms. This is joint work with Morten S.
Risager.
April 23: Michael Tepper (Penn State Abington)
Title: Isotrivial is equivalent to potential good
reduction.
Abstract: Let K = k(C) be the function field of a
complete nonsingular curve C over an arbitrary field
k. We prove, in two different ways, an endomorphism of
projective nspace over K is isotrivial if and only if
it has potential good reduction at all places v of K.
The first proof uses algebraic geometry and geometric
invariant theory; the second uses nonarchimedean
analysis and dynamics. While both proofs use dynamics,
the second more directly, generalize results of
Benedetto for polynomial maps and Baker for arbitrary
rational maps when n=1. This is joint work with
Clayton Petsche and Lucien Szpiro.
April 30: No speaker this week.
May 7: No speaker this week.
May 14: DoubleHeader! *Please note the special
times*
3:004:00: Benjamin Hutz (Amherst College)
Title: Good Reduction of Rational Periodic Points.
Abstract: We define the notion of a (discrete)
dynamical system and the study of rational periodic
points. We then state the MortonSilverman conjecture
on the uniform boundedness of rational preperiodic
points and define a notion of good reduction. The main
theorem addresses the local decomposition of the
primitive period of a rational periodic point at a
prime of good reduction. Throughout we present
explicit examples and give instances where the
theorems are applied to address related problems.
4:305:30: Umberto Zannier (Scuola Normale
Superiore)
Title: Simultaneous torsion values of sections of an
elliptic pencil.
Abstract: The talk shall illustrate the proof, joint
with David
Masser, of the following conjecture of his: Consider
the Legendre
family of elliptic curves $E_\lambda:
y^2=x(x1)(x\lambda)$, and pick
two points $P_\lambda,Q_\lambda$ on it, with abscissas
resp. $2,3$.
There are only finitely many complex values of
$\lambda$ such that
both points become torsion. I shall also discuss
variations on this problem, which turns out to be
a case of very general conjectures of Pink.
