An explicit form of the canonical submodule of a Drinfeld module

Satoshi Kondo; Yusuke Sugiyama

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arXiv · 1803.01981

An explicit form of the canonical submodule of a Drinfeld module

Satoshi Kondo·Yusuke Sugiyama

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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.

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Satoshi Kondo, Yusuke Sugiyama. 2018-03-06. An explicit form of the canonical submodule of a Drinfeld module. https://arxiv.org/abs/1803.01981

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