Among the bounty of brilliancies bequeathed to humanity by
Srinivasa Ramanujan, the circle method and the notion of mock theta
functions strike wonder and spark intrigue in number theorists
fresh and seasoned alike. The former creation was honed...
We give a survey of recent results on conjectures of Heath-Brown
and Serre on the asymptotic density of rational points of bounded
height. The main tool in the proofs is a new global determinant
method inspired...
In the early 80's, Shimura made a precise conjecture relating
Petersson inner products of arithmetic automorphic forms on
quaternion algebras over totally real fields, up to algebraic
factors. This conjecture (which is a consequence of the Tate...
The $\lambda$-invariant is an invariant of an imaginary
quadratic field that measures the growth of class numbers in
cyclotomic towers over the field. It also measures the number of
zeroes of an associated $p$-adic L-function. In this talk, I
will...
This is a report on some joint work with Mark Reeder and
Jiu-Kang Yu. I will review the theory of parahoric subgroups and
consider the induced representation of a one-dimensional character
of the pro-unipotent radical. A surprising fact is that this...
In a recent paper, DeBacker and Reeder have constructed a piece
of the local Langlands correspondence for pure inner forms of
unramified p-adic groups and have shown that the corresponding
L-packets are stable. In this talk we are going to discuss...
Fontaine and Mazur have a remarkable conjecture that predicts
which (p-adic) Galois representations arise from geometry. In the
special case of two dimensional representations with distinct
Hodge-Tate weights, they further conjecture that these...
