Let be a rational prime, a -power and $P$ 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.