On Planar Algebraic Curves and Holonomic $\mathcal{D}$-modules in Positive Characteristic

TL;DR

This paper explores the relationship between cyclic modules over the Weyl algebra and planar algebraic curves in positive characteristic, potentially impacting the understanding of algebra automorphisms and polynomial symplectomorphisms.

Contribution

It establishes a correspondence between cyclic modules and algebraic curves in positive characteristic, suggesting a new approach to a longstanding conjecture.

Findings

Every planar algebraic curve has a preimage under a specific ind-scheme morphism.

The results may lead to an indirect proof of the isomorphism between automorphism groups and symplectomorphisms.

The study links algebraic modules with geometric curves in positive characteristic.

Abstract

In this paper we study a correspondence between cyclic modules over the first Weyl algebra and planar algebraic curves in positive characteristic. In particular, we show that any such curve has a preimage under a morphism of certain ind-schemes. This property might pave the way for an indirect proof of existence of a canonical isomorphism between the group of algebra automorphisms of the first Weyl algebra over the field complex numbers and the group of polynomial symplectomorphisms of $\mathbb{C}^2$.