Collaborative Number Theory Seminar at the CUNY Graduate Center

Co-organizers: Gautam Chinta, Brooke Feigon, Krzysztof Klosin, Maria Sabitova, Lucien Szpiro.

The seminar currently meets Fridays 2:00 to 3:00 PM in Room 3209. 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.

Spring 2014 Schedule:

February 7: Xin Wan (Columbia University)

Title: Iwasawa-Greenberg main conjectures for Rankin-Selberg
p-adic L-functions

Abstract: In this talk I will prove an Iwasawa-Greenberg main conjecture for Rankin-Selberg p-adic L-functions 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 Perrin-Riou). In view of the Beilinson-Flach elements this gives one way of proving the Iwasawa main conjecture for supersingular elliptic curves formulated in different ways by Kato, Kobayashi and Perrin-Riou.

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 non-Archimedean analytic geometry


March 21: Maksym Radziwill (IAS)

Title: L-functions, sieves and the Tate Shafarevich group

Abstract: I will explain joint work with Kannan Soundararajan, where we find an "L-function analogue" of the Brun-Hooley 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 L-functions. In particular I'll focus on consequences for the distribution of the Tate-Shafarevich 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 L-functions 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.