Title:
Applications of Modularity in Arithmetic Geometry and Number Theory
Abstract:
Modular forms are connected to several areas of arithmetic geometry and
number theory, for example through the Langlands program. They have the
useful property that they are amenable to computations. We give
an introduction to the theory of classical modular forms, their
relation to elliptic curves and abelian varieties and the Galois
representations attached to them, as well as how to perform
computations with them.
As an application we (1) show how Hilbert modular forms can be used to
solve new cases of the inverse Galois problem and (2) indicate how (a)
Hilbert and (b) Siegel modular forms can be used to establish (a) strong
BSD over totally real fields and (b) the BSD rank conjecture in
infinitely many generic cases in dimension 2 over the rationals,
respectively.