An explicit form of the canonical submodule of a Drinfeld module

Abstract

We define the canonical submodule of a Drinfeld module of rank greater than one over the affine line over a finite field. (This extends the definition of the level 1 canonical subgroup of Hattori for rank 2 with ordinary reduction.) We give a criterion for the existence of the canonical submodule in terms of a lift of the Hasse invariant. Then, we give an explicit form of the canonical submodule. The main tool is formal Drinfeld modules of Rosen. Recall that a canonical subgroup of an elliptic curve plays an important role in the theory of p-adic modular forms. An explicit form in this case is given by Coleman, and our result is its function field analogue.