Abstracts for Number theory lunch seminar

I consider expansion at a  CM-point for a Hecke-Maass cusp form. This leads
to a collection of (spherical) coefficients analogous to the classical (unipotent)
Fourier coefficients of automorphic functions on GL(2).  These coefficients were
introduced by H. Petersson, and are connected to special values of L-functions
via the theorem of Waldspurger on the torus period. We prove meromorphic
continuation for a  Dirichlet series build from these coefficients. For the Eisenstein

Borcherds lifts on $G=O(2,n+2)$ are meromorphic automorphic forms
whose divisors are Heegner divisors. In this talk, we discuss on
holomorphic automorphic forms
on  $G$ satisfying certain symmetries. We show that, with an additional
assumption,
these forms are Borcherds lifts. In the cases of $n=0$ and $n=1$, we can
remove this
additional condition and hence Borcherds lifts are characterized by the
symmetries.

Let $\rho$ be an $n$-dimensional continuous complex representation of $W'_F$, the Weil-Deligne group of a local field $F$. Denote by $\pi(\rho)$ the irreducible admissible representation of $GL_n(F)$ attached to $\rho$ by the local Langlands correspondence. Let $\Lambda^2$ be the exterior square representation of $GL_n(\Bbb C)$. Let $L(s,\Lambda^2\rho)$ and $\varepsilon(s,\Lambda^2\rho,\psi_F)$ be the Artin $L$-function and root number defined by $\Lambda^2\rho$.

Local Langlands program predicts a close relationship between
representations of the absolute Galois group of a
local field F and irreducible representations of reductive groups over F. A
geometric version of this program
attempts to relate vector bundles equipped with (flat) connections on the
complex puncture disk to
representations of the affine Kac-Moody algebras. After laying out the
process of going from classical
statements to geometric predictions, I will explain Frenkel-Gaitsgory's
picture in the unramified case. I will