The aim of this seminar is to cover material not normally covered in graduate courses and related to Analytic Number Theory and Automorphic Forms. It should be accessible to graduate students that have successfully completed the basic graduate courses in Real Analysis, Complex Analysis and Algebra.

This semester the seminar meets on Fridays at 4:00 PM in Room 4214-03 (Mathematics Thesis Room). Among other speakers, this semester the following people have agreed to give a talk: Alina Bocur (IAS), Wenzhi Luo (Ohio State and IAS).

**September 8, Nikos Diamantis (U. of Nottingham)**Title: Cusp forms of order higher than 2.

Note Change in Room: Only for this week we will meet in 4429

**September 15, Jay Jorgenson (City College, CUNY)**Title: Relating L-functions of Maass forms and holomorphic forms (joint with Juerg Kramer)

**September 22, Wilson Zuniga (Barry Univ.)**Title: Local Zeta functions for Polynomial Mappings and Multivariate Exponential Sums

**September 29, Alina Ioana Bucur (IAS)**Multiple Dirichlet Series and Moments of L-functions

**October 6, Wenzhi Luo (IAS and Ohio State)**Equidistribution problems on Siegel modular varieties

**October 13, Jay Jorgenson (City College, CUNY)**Title: Basic properties of hyperbolic and elliptic Eisenstein series

Abstract: We will define hyperbolic and elliptic Eisenstein series on general finite volume hyperbolic Riemann surfaces, after which we will discuss results regarding fundamental questions regarding these series, including: Meromorphic continuation, location of poles, and Kronecker limit problem. As time permits, we will show that both hyperbolic and elliptic Eisenstein series limit to parabolic Eisenstein series through degeneration.

**October 20, Steven J. Miller (Brown Univ.)**Title: Identifying and breaking the symmetry group of zeros of families of L-functions

Abstract: Many studies have shown that the zeros near the central point of families of L-functions are well modeled by eigenvalues near one of a classical compact group. While the monodromy group in the function field case suggests what that group should be, determining the group in the number field case is often a matter of calculation (and sometimes the results seem surprising at first). We show, at least for families built from nice GL(2) L-functions, that the symmetry can be determined by the second moment of the Satake parameters, and behaves nicely under convolution. The universality is similar to that found by Rudnick and Sarnak for the high zeros. We conclude by studying one-parameter families of elliptic curves over Q, where we break this universality by finding lower order correction terms which depend on the arithmetic of the family.TBA

**October 27, Yiannis Petridis (Lehman College, CUNY)**Perturbation theory for Maass cusp forms and scattering poles

NOTE CHANGE IN ROOM: Only for this week we will meet in Room 4419

**November 3, Gautam Chinta (City College, CUNY)**Whittaker coefficients of metaplectic Eisenstein series.

**November 10, Yiannis Petridis (Lehman College, CUNY)**NOTE CHANGE IN ROOM: Only for this week we will meet in Room 4419

Heisenberg manifolds, lattice-point counting and exponential sums

Abstract: I will discuss the fundamentals of Heisenberg manifolds, their importance in Spectral Geometry, Weyl's Law, its relation to lattice-point counting and exponential sums.

**November 17, Ramin Takloo-Bighash (Princeton U.)**Rational points and automorphic forms

Abstract: In this talk I will discuss recent progress in the study of the distribution of rational points on symmetric varieties, and explain how the theory of automorphic forms enters the picture.

**December 1, Claus Sorensen (Princeton U.)**Eisenstein series for the Hilbert modular group

**December 6, Juerg Kramer (Humboldt U., Berlin)****SPECIAL TALK ON WEDNESDAY, 3:30-5:00 Room 4430**An integral characterizing the Andreotti-Mayer locus.

Abstract: Let A_n denote the moduli space of n-dimensional, principally polarized, abelian varieties and p:A->A_n the corresponding universal abelian variety with universal theta divisor Theta_A. The Andreotti-Mayer locus N_0 is then defined as the set of all points q in A such that the fiber Theta_q of Theta is singular. D. Mumford showed that N_0=div(F_n), where F_n is a Siegel modular form of weight k=n!(n+3)/2 (and some character) for the full Siegel modular group Sp_n(Z). Mumford raised the question, if there is an explicit polynomial in the theta constants, or other modular forms constructed from theta series (with quadratic forms and pluri- harmonic coefficients) whose zeros give N_0 with suitable multiplicities. An explicit representation of Mumford's modular form F_n is known for n <=4. In our talk we will report on joint work with R. Salvati Manni, in which we give an integral representation of Mumford's modular form F_n.

**December 8, Nicole Raulf (Princeton U.) in Room 4430**Class numbers of fields