Collaborative Number Theory Seminar at the CUNY Graduate CenterCoorganizers: Gautam Chinta, Brooke Feigon, Krzysztof Klosin,
Maria Sabitova,
Lucien Szpiro. Spring 2014 Schedule:
February 7: Xin Wan
(Columbia University) Title:
IwasawaGreenberg main conjectures for
RankinSelberg Abstract: In this talk I will prove an IwasawaGreenberg main conjecture for RankinSelberg padic Lfunctions for a general modular form and a CM form such that the CM form has higher weight, using Eisenstein series on U(3,1), under the assumption that the CM form is ordinary (no ordinary conditions on the general modular form). This has many arithmetic applications including proving an anticylotomic main conjecture in the sign=1 case (formulated by PerrinRiou). In view of the BeilinsonFlach elements this gives one way of proving the Iwasawa main conjecture for supersingular elliptic curves formulated in different ways by Kato, Kobayashi and PerrinRiou. February 21: Nathan Kaplan (Yale University) Title:
Rational point counts for curves and surfaces over
finite fields via coding theory Abstract: We explain an approach of Elkies to counting points on varieties over finite fields. A vector space of polynomials gives a linear subspace of (F_q)^N, a linear code, by the evaluation map. Studying properties of this code and its dual gives information about the distribution of rational point counts for the family of varieties defined by these polynomials. We will describe how this approach works for families of genus one curves and del Pezzo surfaces in projective space over F_q and will mention how class numbers and Fourier coefficients of modular forms appear in these point counts. No previous familiarity with coding theory will be assumed.
March 7:
Daniele Turchetti (Institut de mathématiques de
Jussieu) Title: Lifting Galois covers to characteristic zero with nonArchimedean analytic geometry
Abstract March 21:
Maksym Radziwill (IAS)
Abstract: I will
explain joint work with Kannan Soundararajan, where we
find an "Lfunction analogue" of the BrunHooley
sieve. Essentially, our method allows us to work
analytically with long truncated Euler products inside
the critical strip. As a consequence we obtain several
new results on the distribution of the central values
of families of Lfunctions. In particular I'll focus
on consequences for the distribution of the
TateShafarevich group of (prime) twists of an
elliptic curve. April 4: Ian Whitehead (Columbia
University) Title: Axiomatic Multiple
Dirichlet Series Abstract: I will outline
an axiomatic description of multiple Dirichlet series
based upon work of Diaconu and Pasol. The axioms lead
to a canonical construction of multiple Dirichlet
series with (infinite) affine Weyl groups of
functional equations. This work is over function
fields, and has applications to arithmetic problems
including the distribution of point counts and
Lfunctions in families of curves over a fixed finite
field. May 9: Adriana
Salerno (Bates College) Title: Effective
computations in arithmetic mirror symmetry Abstract: In this talk, I
will talk about computational approaches to the
problem of arithmetic mirror symmetry. One of the
biggest questions facing string theorists is the one
of mirror symmetry. In arithmetic mirror symmetry, we
approach the conjecture from a number theoretic point
of view, namely by computing Zeta functions of mirror
pairs. I will define all of these terms and then
explain our work through a couple of examples of
families of K3 surfaces. This is joint work with Xenia
de la Ossa, Charles Doran, Tyler Kelly, Stephen
Sperber, John Voight, and Ursula Whitcher.
