Lecture Room B, 4th Floor, The 3rd General Building, NTHU

(清華大學綜合三館 4樓B演講室)

Duality of Drinfeld modules and P-adic properties of Drinfeld modular forms

Shin Hattori (Kyushu University)

Abstract:

Let

be a rational prime,

a

-power and

a non-constant irreducible polynomial in

. The notion of Drinfeld modular form is an analogue over

of that of elliptic modular form. On the other hand, following the analogy with

-adic elliptic modular forms, Vincent defined

-adic Drinfeld modular forms as the

-adic limits of Fourier expansions of Drinfeld modular forms. Numerical computations suggest that Drinfeld modular forms should enjoy deep

-adic structures comparable to the elliptic analogue, while at present their

-adic properties are far less well understood than the

-adic elliptic case.

In this series of talks, I will explain how basic properties of

-adic Drinfeld modular forms are obtained in a geometric way, using the duality theories of Taguchi for Drinfeld modules and finite

-modules. Key ingredients are the theory of canonical subgroups of Drinfeld modules with ordinary reduction and Hodge-Tate-Taguchi maps, which give torsion comparison isomorphisms between the etale and de Rham sides.