Collaborative Number Theory Seminar at the CUNY Graduate Center

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

The seminar currently meets Fridays 12:30 to 1:45 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 2013 Schedule:

February 22: Yves Martin (Universidad de Chile)

Title: On the analogue of Cohen's kernel in the case of Jacobi forms


March 1: Ellen Eischen (University of North Carolina at Chapel Hill)

Title: p-adic families of vector-weight Eisenstein series

Abstract: One approach to constructing certain p-adic L-functions relies on construction of a p-adic family of Eisenstein series. I will explain how to construct such p-adic families (including the case of vector-weight automorphic forms) for certain unitary groups. This builds on my earlier work on scalar- weight families. As part of the talk, I will explain how to p-adically interpolate certain values of both holomorphic and non-holomorphic Eisenstein series. I will also mention some applications to number theory and beyond.

March 8: Matilde Lalin (Université de Montréal)

Title: Mahler measure and special values of L-functions

Abstract: The Mahler measure of a Laurent polynomial P is defined as the integral of log|P| over the unit torus with respect to the Haar measure. For multivariate polynomials, it often yields special values of L-functions. In this talk I will discuss some of these relationships and the meaning behind them.

March 15: Thomas Tucker (University of Rochester)

Title: Preperiodic portraits modul primes

Abstract: Let  F be a rational function of degree > 1 over a  number field K and let z be a point that is not preperiodic.  For any prime p of good reduction of F, the reduction of z is preperiodic since the residue field of p is finite.  Ingram and Silverman conjecture that for all but finitely many positive integers (m,n), there is a prime p such that z has exact preperiodic m and exact period n (we call this pair (m,n) the portrait of z modulo p).  We present a counterexample to this conjecture and show that a generalized form of abc implies that these are the only counterexamples for generic rational functions.  This represents joint work with several authors.

March 22: No meeting this week

March 29: Spring break

April 5: No meeting this week

April 12: *Please note the special time*

1:30--2:45 pm: Ju-Lee Kim (MIT)

Title: On the characters of unipotent representations of a semisimple p-adic group

Abstract: Let G be a semisimple almost simple algebraic group defined and split over a non-archimedean local field K and let V be a unipotent repre- sentation of G(K) (for example, an Iwahori-spherical representation). We calculate the character of V at compact very regular elements of G(K). This is a joint work with George Lusztig.

April 19: Jim Brown (Clemson University)

Title: The Bloch-Kato conjecture for modular forms of odd square-free level

Abstract: Let f be a newform of weight 2k-2 and level N with N odd and square-free.  In joint work with Mahesh Agarwal we show roughly half of the Bloch-Kato conjecture in this setting, namely, the size of the Shafarevich-Tate group of the Galois representation associated to f is bounded below by an appropriately normalized special value of the L-function associated to f.  We accomplish this by studying congruences among automorphic forms on GSp(4).  We present the theorem and discuss the necessary hypotheses.  We will also present a conjecture about twisting special values to ensure they are p-adic units as well as numerical evidence for this conjecture.

April 26: *Please note the special time*

1:30--2:45 pm: Amy DeCelles (University of St. Thomas)

Title: Automorphic Spectral Identities from Differential Equations

Abstract: In hindsight, subconvexity results of Diaconu, Garrett, and Goldfeld can be viewed as an application of a spectral identity obtained from an automorphic partial differential equation.  In this talk I will discuss examples of spectral identities  obtained in this way and the analytic framework necessary for legitimizing their derivations and applications.  This approach offers the possibility of operating under somewhat different hypotheses than one would usually take when using trace formula or relative trace formula methods: while trace formulas work best with very smooth data, we use a Poincare series whose data is not smooth nor compactly supported.

May 3: *Please note the special time and room*

4:00--5:15 room 5212: Joseph Hundley (Southern Illinois University)

Title: Eulerian Integrals in the Group F4

Abstract: This talk is based on joint work with David Ginzburg.  Motivated by known integral representations Langlands L-functions, we study a certain two-dimensional array of global integrals attached to any reductive algebraic group, indexed by maximal parabolic subgroups in one direction and by unipotent conjugacy classes in the other.  Fourier coefficients attached to unipotent classes, Gelfand-Kirillov dimension of automorphic representations, and an identity which, empirically, appears to constrain the unfolding process will be discussed, with examples selected from the exceptional group F4.  Two new Eulerian integrals are included among these examples.

May 10: Mihran Papikian (Pennsylvania State University)

Title: Quotients of Mumford curves and component groups

Abstract: We consider elliptic curves parametrized by Mumford curves and answer negatively a question of Ribet and Takahashi about the surjectivity of the induced maps on component groups. In case of modular curves this leads to some interesting questions about the Hecke algebra and Galois represen-tations. (This is a joint work with Joe Rabinoff.)

May 17: *Please note the special time*

1:30--2:45 pm: P. Edward Herman (University of Chicago)

Title: On Patterson's conjecture: sums of exponential sums