Collaborative Number Theory Seminar at the CUNY
Graduate Center
Coorganizers: Gautam Chinta, Brooke Feigon, Maria Sabitova,
Lucien Szpiro.
The seminar currently meets Fridays 4:00 to 5:30 PM in
Room 4422. The CUNY Graduate Center is located on Fifth
Avenue, on the east side of the street, between 34th and
35th Streets in midtown Manhattan. For further
information, please contact Maria Sabitova.
Fall 2011 Schedule:
September 2: No meeting this week.
September 9: No meeting this week.
September 16: LiangChung Hsia (National Taiwan Normal
University)
Title: Preperiodic points for family of rational maps
Abstract
September 23: Marc Masdeu Sabaté
(Columbia University)
Title:
A padic approach to explicit automorphic forms on
Shimura curves
Abstract: The explicit
computation of spaces of automorphic forms is a very
active and useful area of research in computational
number theory. Several authors have described
algorithms to compute these Hecke modules: to name a
few, John Voight, Matt Greenberg and Lassina
Dembélé have described algorithms that
work in different levels of generality. In joint work
with Cameron Franc, we propose a padic approach to
this problem, via understanding quotients of the
BruhatTits tree.
In this talk we will give an
introduction to this subject and explain our approach
to these explicit computations, as well as possible
applications of our approach.
September 30: No meeting this week.
October 7: No meeting this week.
October 14: No meeting this week.
October 21: *Please note the special time*
23:30:
Zhengyu Mao ()
Title: FourierWhittaker coefficients of automorphic
forms on Mp(2n)
October 28: Gerard
Freixas
i Montplet (C.N.R.S.  Institut de
Mathématiques
de Jussieu)
Title: On the height of the fixed point set of
AtkinLehner involutions
Abstract: The
arithmetic RiemannRoch theorem of GilletSoule in
Arakelov geometry does not apply to natural examples
as modular curves and bundles of modular forms with
their Petersson metrics, due to the singularities of
the metrics near cusps and elliptic fixed points. I
will present a variant of the theorem which applies to
this situation, and I will explain how this can be
used to derive expressions for heights of fixed point
sets of AtkinLehner involutions in terms of special
values of twisted Selberg zeta functions. In genus 0,
these formulas together with the ChowlaSelberg
formula provide new Selberg zeta values. Part of this
talk will be based on joint work with Anna von Pippich
(Humboldt University, Berlin).
November 4: Adam
Towsley
(University of Rochester)
Title: Reduction of Orbits
Abstract : We will look at several results
about the behavior of orbits of rational
maps when reduced modulo a prime ideal. For instance,
if we look at a rational map taking a number
field to itself we will consider for what proportion
of primes p will a wandering point
become periodic modulo p. Additionally we will look
at the function field analogs of these
results.
November 11: *Please note the double header
and special times*
4:005:00: Thomas Tucker (University
of Rochester)
Title:
Towards a dynamical relative ManinMumford conjecture
Abstract:
Question:
Given two points a, b in C and a family F of rational
maps, when can there be infinitely many members f of F
such that a and b are both preperiodic for f?
MasserZannier answered this question or Lattes
maps and BakerDeMarco have answered it for the family
x^d + c (where c varies over C). Here, we adapt
BakerDeMarco's method to a more general situation,
using recent results of Yuan and Zhang. This work
is joint with L.C. Hsia and D. Ghioca.
5:156:15: Daniel Fiorilli (IAS)
Title: On how the first term
of an arithmetic progression can influence the
distribution of an arithmetic sequence
Abstract:
In this talk we will show that many arithmetic sequences
have asymetries in their distribution amongst the
progressions mod q. The general phenomenon is that if we
fix an integer a having some arithmetic properties
(these properties depend on the sequence), then the
progressions a mod q will tend to contain fewer elements
of the arithmetic sequence than other progressions a mod
q, on average over q. The observed phenomenon is for
quite small arithmetic progressions, and the maximal
size of the progressions is fixed by the nature of the
sequence. Examples of sequences falling in our range of
application are the sequence of primes, the sequence of
integers which can be written as the sum of two squares
(with or without multiplicity), the sequence of twin
primes (under HardyLittlewood) and the sequence of
integers free of small prime factors. We will focus on
these examples as they are quite fun and enlightening.
November 18: *Please note the double
header and special times*
4:005:00: Reinier
Bröker (Brown University)
Title: Computing Gauss sums and general
theta series
Abstract: It is
a classical problem to compute Gauss sums
efficiently. Besides Gauss' result for the
quadratic sum, no easy formula is known. In this
talk we view Gauss sums as coefficients of a
general theta series and we explain a method to
efficiently approximate the residue of the theta
series. Many examples will be given.
5:156:15: John T. Cullinan (Bard
College)
Title: Divisibility properties of torsion subgroups
of abelian surfaces
Abstract: Let A be an abelian variety defined over
a number field K and suppose m >1 is a positive
integer. Suppose further that the number of points
on the reduction mod p of A is divisible by m for
almost all primes p of K. Does there exist a
Kisogenous abelian variety A' whose torsion
subgroup (over K) has order divisible by m? This
question was asked by Lang and answered in the
affirmative by Katz for elliptic curves and for
abelian surfaces when m is a prime number. We will
discuss counterexamples in higher dimensions as well
as recent work on abelian surfaces when m is
composite.
November 25: Thanksgiving
Break,
the University is closed.
December 2: Jens
Funke
(University of Durham)
Title: Regularized Theta Liftings and periods of
modular functions
Abstract: In this talk,
we discuss regularized theta liftings to construct
weak Maass forms weight 1/2 as lifts of weak Maass
forms of weight 0. As a special case we give a new
proof of some of recent results of Duke, Toth and
Imamoglu on cycle integrals of the modular jinvariant
and extend these to any congruence subgroup. Moreover,
our methods allow us to settle the open question of a
geometric interpretation for periods of j along
infinite geodesics in the upper half plane. In
particular, we give the `central value' of the
(nonexisting) `Lfunction' for j.
December 9: Anna
Haensch
(Wesleyan
University)
Title: Almost Universal Ternary Mixed Sums of Squares
and Polygonal
Numbers
Abstract: An inhomogeneous quadratic form is a sum
of a quadratic form and a linear form; it is
called almost universal if it represents
all but finitely many positive integers.
Examples of inhomogeneous quadratic forms are
mixed sums of squares and generalized polygonal
numbers. In this talk we will present a
characterization of positive definite almost
universal ternary mixed sums of squares and
triangular numbers, and extend this idea to mixed
sums of squares and mgonal numbers, where m 
2 = 2p, for a prime p. This generalizes the
recent work by Chan and Oh on almost universal
ternary sums of triangular numbers. If time
permits, we will also discuss a characterization of
positive definite almost universal ternary
inhomogeneous quadratic forms which satisfy some
mild arithmetic conditions.
