I will discuss joint work with Akshay Venkatesh in which we use microlocalized test vectors (inspired by the orbit method) and Ratner theory to study mean values of L-functions on Gross--Prasad pairs. I will also indicate some further applications of these methods, such as to the quantum variance problem for Hecke--Maass eigenforms.
In this talk I will report on my joint work in progress with Samit Dasgupta on the tower of fields conjecture first formulated by Gross. This proves a conjecture of Dasgupta on explicit p-adic analytic formulae for Gross-Stark units. These units, when considered for all primes of a totally real number field F, generate the maximal abelian CM extension of F and therefore our work can be...
p-adic L-functions attached to automorphic representations and p-adic families of them, provide powerful tools to attack important problems such as Birch-Swinnerton-Dyer and Bloch-Kato conjetures. However, they are hard to construct and in fact beyond the case GL(2) the theory is poorly understood.
In this talk I will describe an approach based on the study of the overconvergent cohomology...
There is a general philosophy that the image of a Galois representation should be as large as possible, subject to its symmetries. This can be seen in Serre's open image theorem for non-CM elliptic curves, Ribet and Momose's work on Galois representations attached to modular forms, and recent work of the speak and Conti-Iovita-Tilouine on Galois representations attached to p-adic families of...
