Starting from the Poisson summation formula, I discuss spectral
summation formulae on GL(2) and GL(3) and present a variety of
applications to automorphic forms, analytic number theory, and
arithmetic.
The formal degree conjecture relates the formal degree of an
irreducible square-integrable representation of a reductive group
over a local field to the special value of the adjoint gamma-factor
of its L-parameter. We prove the formal degree...
The endoscopy theory provides a large class of examples of
Langlands functoriality, and it also plays an important role in the
classification of automorphic forms. The central part of this
theory are some conjectural identities of Harish-Chandra...
We prove a level raising mod $p = 2$ theorem for elliptic curves
over $\mathbb Q$, generalizing theorems of Ribet and
Diamond-Taylor. As an application, we show that the 2-Selmer rank
can be arbitrary in level raising families. We will begin by...
One of the major themes of the analytic theory of automorphic forms
is the connection between equidistribution and subconvexity. An
early example of this is the famous result of Duke showing the
equidistribution of Heegner points on the modular...
In the first part of the talk we will survey some recent results on
representations of finite (simple) groups. In the second part we
will discuss applications of these results to various problems in
number theory and algebraic geometry.
We give a geometric theory of vector-valued modular forms attached
to Weil representations of rank 1 lattices. More specifically, we
construct vector bundles over the moduli stack of elliptic curves,
whose sections over the complex numbers...
We consider a Rankin-Selberg integral representation of a cuspidal
(not necessarily generic) representation of the exceptional group
\(G_2\). Although the integral unfolds with a non-unique model, it
turns out to be Eulerian and represents the...
Given an elliptic curve \(E\) over a field \(k\), its \(p\)-torsion
\(E[p]\) gives a 2-dimensional representation of the Galois group
\(G_k\) over \(\mathbb F_p\). The Frey-Mazur conjecture asserts
that for \(k= \mathbb Q\) and \(p > 13\), \(E\) is...
Let \[ f(t)=\sum_{N n 2N}a_nn^{-it} \] be a Dirichlet polynomial.
We consider the weighted square mean value \[
I=\int_{-\infty}^{\infty}|f(t)|^2\exp\{-\Delta^{-2}(t-T)^2\}\,dt,
\] where \(T\) is a large paremeter and \[ \Delta = \frac{T}{\log
T}...
