A conjecture of Langlands-Rapoport predicts the structure of the
mod p points on a Shimura variety. The conjecture forms part of
Langlands' program to understand the zeta function of a Shimura
variety in terms of automorphic L-functions.
Let X a curve over F_q and G a semi-simple simply-connected
group. The initial observation is that the conjecture of Weil's
which says that the volume of the adelic quotient of G with respect
to the Tamagawa measure equals 1, is equivalent to the...
A fake projective plane is a smooth complex projective algebraic
surface whose Betti numbers are same as those of the complex
projective plane but which is not the complex projective plane. The
first fake projective plane was constructed by David...
A. Ghosh and P. Sarnak have recently initiated the study of
so-called real zeros of holomorphic Hecke cusp forms, that is zeros
on certain geodesic segments on which the cusp form (or a multiple
of it) takes real values. In the talk I'll first...
In this talk, I will present a formulation of the Gross-Zagier
formula over Shimura curves using automorphic representations with
algebraic coefficients. It is a joint work with Shou-wu Zhang and
Wei Zhang.
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...
