## Collaborative Number Theory Seminar at the CUNY Graduate Center

Co-organizers: 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: p-adic L-functions and the Bloch-Kato conjecture.

Abstract: Given a p-adic representation V of the Galois group of Q, the Bloch-Kato conjecture predicts a mysterious relationship between two different invariants of V: its Bloch-Kato Selmer group, defined algebraically using Galois cohomology, and its L-function, 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 p-adic variant, where the L-function is replaced by a p-adic L -function and the Bloch-Kato Selmer group is suitably modified. I will then explain a strategy to prove the p-adic variant of the Bloch-Kato 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.

4:00-5:00: 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 so-called Bang-Zigmondy principle.

5:15-6:15: 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 Bloch-Kato Conjecture for convolution L-functions.

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 Bloch-Kato conjecture for the motive M=rho_{f_1} x \rho_{f_2}(-k/2-1) by proving that under some assumptions, the p-valuation of the order of the Bloch-Kato Selmer group of M is bounded below by the p-valuation of a special value of the convolution L-function of f_1 and f_2. We achieve this by constructing congruences between the Yoshida lift Y(f_1xf_2)$of f_1 and f_2 and Siegel modular forms whose p-adic Galois representations are irreducible. Our result is conditional upon the conjectural formula for the Petersson norm of Y(f_1 \otimes f_2). This is joint work with Krzysztof Klosin. March 25: Ralph Greenberg (University of Washington). Title: Iwasawa Theory and Projective Modules. Abstract: The theme of the talk is that properties of projective modules over a group ring can be used to study the behavior of certain invariants which occur naturally in Iwasawa theory for an elliptic curve E. Modular representation theory plays a crucial role in this study. It is necessary to make a certain assumption about the vanishing of a$\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 non-trivial 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 non-algebraically closed field and X be a surface defined over k. An interesting problem is to know whether the set of k-rational 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.

4:00-5:00: 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 half-integral 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.

5:15-6:15: Bruce Berndt (University of Illinois at Urbana-Champaign).

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 S-unit theorem.

Abstract: In this talk we will discuss our recent generalization of Dirichlet's S-unit theorem from the context of the S-units 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.