Abstract

Let be a rational prime, a -power integer and a non-constant irreducible polynomial in . The notion of Drinfeld modular form is an analogue over of that of elliptic modular form. The expectation is that there should be a deep -adic theory of Drinfeld modular forms which is comparable to the elliptic analogue.

However, -adic properties of Drinfeld modular forms are less well understood than the -adic elliptic case, and sometimes classical methods do not work or yield nothing useful (e.g. no eigenvariety so far).

In this talk, as an example of such unusual phenomena, I will explain that all Hecke operators act trivially on the space of ordinary Drinfeld cuspforms of level for any positive integer . It suggests that -adic Hida families of tame level one should be trivial. It also gives a counterexample to Gekeler's question, which asks if the weak multiplicity one holds for Drinfeld modular forms when the weight is fixed.