This semester the seminar meets on Fridays from 4 - 5:30 PM in Room 8405. 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 Gautam Chinta (chinta@sci.ccny.cuny.edu). For information on the seminar in previous semesters, please see Collaborative Satellite Number Theory Seminar.
Title: The hyperbolic lattice-counting problem and modular symbols
Title: Minimal heights and regulators for elliptic curves and surfaces
Title: Moments of L-functions
Title: A survey of conjectures and results on small points for canonical heights
Title: Spectral convergence of elliptically degenerating hyperbolic Riemann surfaces
Title: Distribution of Angles in Hyperbolic Lattices
Title: Heat kernels on finitely generated groups
Title: A dynamical Mordell-Lang theorem for split quadratic polynomial maps
Abstract: We will prove the dynamical Mordell-Lang theorem for maps of the form f:A^n --> A^n where f acts as a quadratic polynomial in each coordinate. The theorem says that if a subvariety V of A^n intersects the orbit of a point P under the action of f in infinitely many points, then V must contain an f--periodic subvariety of positive dimension.
Title: Density of rational points on a cubic surface.
Abstract: A conjecture of Manin et al predicts the precise asymptotic behavior of the number of rational points on cubic surfaces. At present this conjecture is known to be true only for certain special singular cubic surfaces. In the smooth case the situation is even worse, and we only have rough upper bounds for the density. Recently, in a joint work with Henryk Iwaniec, we employed half dimensional sieves together with techniques from analytic number theory, to obtain almost sharp lower bound for the density of rational points for singular cubic surfaces with two conjugate singularities of type A_2. I will briefly discuss this work.
Title: The Hyperbolic Lattice Point Count in Infinite Volume with Applications to Sieves
Abstract: We develop novel techniques using only abstract operator theory to obtain asymptotic formulae for lattice counting problems on infinite-volume hyperbolic manifolds, with error terms which are uniform as the lattice moves through ``congruence'' subgroups. These methods have their origins in the work of Selberg, Lax-Phillips and Duke-Rudnick-Sarnak, and the uniformity relies on the spectral gap established in Bourgain-Gamburd and Bourgain-Gamburd-Sarnak. We give an application to the theory of affine linear sieves.
Title: The adjoint L function of GL_5
Abstract: We describe two Eulerian Rankin-Selberg integrals on the split exceptional group of type E8, and the strong evidence that one of them equals the adjoint L-function of GL_5. We also remark on the connections between the adjoint L function for GL_n and with two other families of L-functions: the so called $\wedge^2_0$ L-function for Spin_{2n+1} and the ratio $\zeta_K(s)/\zeta_F(s)$ where K is a degree n extension of F (which is a product of Artin L-functions). This latter connection was first remarked by Jacquet and Zagier, but our approach is more along the lines of Jiang and Rallis. This is joint with David Ginzburg.
Title: Averages of central L-values of Hilbert modular forms
Abstract: We use the relative trace formula to obtain exact formulas for central values of certain twisted quadratic base change L-functions averaged over Hilbert modular forms of a fixed weight and level. We apply these formulas to the subconvexity problem for these L-functions. This talk is based on joint work with David Whitehouse.
Title: Oscillatory Sequences: Revenge of the Trivial Zeros
Abstract: Under what conditions do the (possibly) complex coefficients of a general Dirichlet series exhibit oscillatory behavior? By coupling a seemingly little-known result of Laguerre with a rather well-known one of Landau, I shall provide a simple answer to this question. I shall also supply several interesting examples of oscillatory sequences that arise from L-functions, cusp forms, and Rankin-Selberg convolutions.
Title: Subconvexity of L-functions.
Abstract: In this talk we will give the first subconvexity bounds for GL(3) L-functions and GL(3)xGL(2) L-functions. The proof is a simple application of the Voronoi formula on GL(3).