Collaborative Number Theory Seminar at the CUNY
Graduate Center
Coorganizers: Gautam Chinta, Clayton Petsche, 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 Clayton Petsche.
Spring 2011 Schedule:
February 4: No meeting this week.
February 11: Lincoln's Birthday, the University is
closed.
February 18: Joel Bellaiche (Brandeis University)
Title: padic Lfunctions and the BlochKato
conjecture.
Abstract: Given a padic representation V of the
Galois group of Q, the BlochKato conjecture predicts
a mysterious relationship between two different
invariants of V: its BlochKato Selmer group, defined
algebraically using Galois cohomology, and its
Lfunction, a complex analytic function defined by a
converging Euler product and analytic continuation. In
this talk, I will explain this conjecture and discuss
in detail its
padic variant, where the Lfunction is replaced by a
padic L
function and the BlochKato Selmer group is suitably
modified.
I will then explain a strategy to prove the padic
variant of the BlochKato conjecture for
representations V attached to automorphic forms, and
the results (on both the variant and the original
conjecture) that has been obtained so far with that
strategy.
February 25: Yuri Zarhin (Penn State).
Title: Ranks of jacobians in towers of function
fields in characteristic zero.
Abstract: We discuss explicit examples of abelian
varieties of bounded rank in
infinite towers of fields of rational functions in one
variable over
the complex numbers. In many cases we compute the rank
at every layer
of the tower. This is a report on a joint work with
Douglas Ulmer
(Georgia Tech).
March 4: No meeting this week.
March 11: ***Double Header***
4:005:00: Tom Tucker (University of Rochester).
Title: Orbits of morphisms mod p.
Abstract: Let f be a rational function. Silverman
proved that if f^2
is not a polynomial, then for any x, there are at most
finitely many
iterates f^n(x) that are integers, i.e. there are at
most finitely
many iterates f^n(x) that do no meet the point at
infinity mod p for
any prime p. Using this as a starting point, we
discuss a number of
questions about orbits of maps (usually rational
functions, though we
can also say something for endomorphisms of abelian
varieties) mod p.
In many cases it is possible to say something about
the proportion of
primes at which some iterate of a point meets another
given point
(such as the point at infinity) mod p. In others, one
can explain the
set of primes one gets via the socalled BangZigmondy
principle.
5:156:15: Alon Levy (Columbia University).
Title: Semistable reduction over the space of
morphisms on P^n.
Abstract: TBA.
March 18: Mahesh Agarwal (University of Michigan,
Dearborn).
Title: Yoshida lifts and the BlochKato Conjecture
for convolution Lfunctions.
Abstract: Let f_1 (resp. f_2) denote two (elliptic)
newforms of prime level N, trivial character and
weight 2 (resp. k+2, where k is either 8 or 12). We
provide evidence for the BlochKato conjecture for the
motive M=rho_{f_1} x \rho_{f_2}(k/21) by proving
that under some assumptions, the pvaluation of the
order of the BlochKato Selmer group of M is bounded
below by the pvaluation of a special value of the
convolution Lfunction of f_1 and f_2. We achieve this
by constructing congruences between the Yoshida lift
Y(f_1xf_2)$ of f_1 and f_2 and Siegel modular forms
whose padic Galois representations are irreducible.
Our result is conditional upon the conjectural formula
for the Petersson norm of Y(f_1 \otimes f_2). This is
joint work with Krzysztof Klosin.
March 25: Ralph Greenberg (University of
Washington).
Title: Iwasawa Theory and Projective Modules.
Abstract: The theme of the talk is that properties of
projective
modules over a group ring can be used to study the
behavior of certain
invariants which occur naturally in Iwasawa theory for
an elliptic
curve E. Modular representation theory plays a crucial
role in this
study. It is necessary to make a certain assumption
about the
vanishing of a $\mu$invariant. We then study
$\lambda$invariants $\lambda E(\sigma)$, where
$\sigma$
varies over a family of absolutely irreducible Artin
representations.
We show that there are nontrivial relationships
between these
invariants under certain hypotheses.
April 1: Cecilia Salgado (University of Leiden).
Title: Zariski density of rational points on del
Pezzo surfaces of low degree.
Abstract: Let k be a nonalgebraically closed field
and X be a surface defined over k. An interesting
problem is to know whether the set of krational
points X(k) is Zariski dense in X. A lot of research
is done in this field but, surprisingly, this problem
is not completely solved for the simplest class of
surfaces, the rational, where one expects a positive
answer. In this lecture I will define del Pezzo
surfaces, a important subclass of rational surfaces. I
will talk about the cases already treated (mainly by
Manin), as well as the two cases left open, the del
Pezzo surfaces of degrees one and two, presenting
recent results (in progress) in the field.
April 8: ***Double Header***
4:005:00: Jens Funke (University of Durham).
Title: Spectacle cycles and modular forms.
Abstract: The classical Shintani lift is the adjoint
of the Shimura correspondence. It realizes periods of
even weight cusp forms as Fourier coefficients of a
halfintegral modular form. In this talk we revisit
the Shintani lift from a (co)homological perspective.
In particular, we extend the lift to Eisenstein series
and give a geometric interpretation of this extension.
This is joint work with John Millson.
5:156:15: Bruce Berndt (University of Illinois at
UrbanaChampaign).
Title: The Circle and Divisor Problems, Bessel
Function Series, and Weighted Divisor Sums.
Abstract: A page in Ramanujan's lost notebook contains
two
identities for trigonometric sums in terms of doubly
infinite
series of Bessel functions. One is related to the
famous ``circle
problem'' and the other to the equally famous
``divisor problem.''
These relations are discussed as well as various
attempts to prove
the identities. Our methods also yield new identities
for certain
trigonometric sums, for which analogues of the circle
and divisor
problems are proposed. The research to be described is
joint work
with Sun Kim and Alexandru Zaharescu.
April 15: Paul Fili (University of Rochester).
Title: A generalization of Dirichlet's Sunit
theorem.
Abstract: In this talk we will discuss our recent
generalization of
Dirichlet's Sunit theorem from the context of the
Sunits of a number
field to the group of all algebraic numbers with
nontrivial valuations
only at places lying above S. As this group is of
infinite rank, we
first explore how one should reformulate the statement
of the theorem
into a statement which makes sense for this larger
group, and then we
discuss the techniques used in the proof that this
generalization holds.
(Joint work with Z. Miner.)
April 22: Spring Break, the University is closed.
April 29: No meeting this week.
May 6: Neil Lyall (University of Georgia).
Title: Polynomial patterns in subsets of the
integers.
Abstract: It is a striking and elegant fact (proved
independently by Furstenberg and Sarkozy) that any
subset of the integers of positive upper density
necessarily contains two distinct elements whose
difference is given by a perfect square. We will
present a new proof of this result and if time permits
also discuss a number of variations, extensions and
generalizations.
May 13: No meeting this week.
