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 3:30 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 2016 Schedule:

February 5: Jim Brown (Clemson University and Queens College)

Title: Congruence primes for automorphic forms on unitary groups and applications to the arithmetic of Ikeda lifts.

Abstract: The problem of classifying congruences between automorphic forms has attracted a considerable amount of attention due not only to its inherent interest, but also because of the arithmetic applications of many of these congruences.  For instance, congruences between automorphic forms have played a crucial role in recent progress on the Birch and Swinnerton-Dyer conjecture.  In this talk we will discuss a sufficient condition for a prime to be a congruence prime for an automorphic form on \teU(n,n)(𝔸F𝔸_F) where FF/ is a totally real field. This sufficient condition is given in terms of the divisibility of a certain special LL-value of the automorphic form.  We then apply this result to the case of the Hermitian Ikeda lift.  This work is joint with Kris Klosin.

February 19: Joseph Gunther (CUNY)

Title: Random Hypersurfaces and Embedding Curves in Surfaces

Abstract: We'll present two new applications of Poonen's closed point sieve over finite fields.  The first is that the obvious local obstruction to embedding a curve in a smooth surface is the only global obstruction.  The second is a proof of a conjecture of Vakil and Wood on the asymptotic probability of hypersurface sections having a prescribed number of singularities.

February 26: Gerard Freixas (C.N.R.S. - Institut de Mathématiques de Jussieu)

Title: Arithmetic intersections on Hilbert modular surfaces and the Jacquet-Langlands correspondence

Abstract: In this talk I will review joint work with D. Eriksson (Chalmers Univ.) and S. Sankaran (McGill Univ.). The Riemann-Roch theorem in Arakelov geometry relates the determinant of the cohomology of a hermitian vector bundle on a proper arithmetic variety to some arithmetic intersection numbers. There are cases of interest to which the formula does not apply, like the Hilbert modular varieties we study here. It is not clear how to prove such a result, not even the right statement. Instead, we try to conjecture a sensible formula that is compatible with the Riemann-Roch formula on a compact Shimura curve, through the Jacquet-Langlands correspondence. In particular, we need to give a meaning to holomorphic analytic torsion, and relate it to the one of a compact Shimura curve. This complements very well with classical results of Shimura for norms of Petersson forms, needed as well in our discussion. I will try to motivate and present the main lines of this work.

March 11: Kathrin Maurischat (Heidelberg University)

Title: Holomorphic and phantom holomorphic projection

Abstract: For Siegel modular forms, Sturm-type formulas describe the holomorphic part of a nonholomorphic form generally when its weight is larger than twice the genus or in the  case of weight two for the classical genus one. We establish this holomorphic projection in case of genus two and scalar weight four. Applying the same method for weight three produces additional nonholomorphic terms which we call phantoms. We show that their occurrence is not a coincidence of our choices (Poincaré\'e series) but a property of Sturm's operator. (These results are partially joint with Rainer Weissauer.)

April 1: Beth Malmskog (Villanova University)

Title: Picard Curves with Good Reduction away from p=3

Abstract: Picard curves are genus 3 curves of the form y^3=f(x), where f(x) is a polynomial of degree 4. They are the simplest non-hyperelliptic curves. This talk will discuss recent work with Chris Rasmussen, in which we found all Picard curves defined over the rationals with good reduction at all primes except p=3. This work was inspired by Nigel Smart's enumeration of genus 2 curves with good reduction at all primes except p=2. As in many finiteness results, an essential step in finding these curves is solving the equation x+y=1 over S-units of a number field. This work is relevant to the study of modular curves and employs several powerful tools, including Baker's method and the LLL algorithm.

May 6: Lucien Szpiro (CUNY)

Title: Solution of the Shafarevich Problem for dynamical systems on 1ℙ^1 (work with Loyd West and Tom Tucker)


May 13: Yves Martin (Universidad de Chile)

Title: On the integral kernel for a multiple Dirichlet series associated to Siegel cusp forms