On planar algebraic curves and holonomic \(\mathcal{D}\)-modules in positive characteristic (Q2816927)

There are several related conjectures around polynomial algebras \({\mathbb C}[x_1,\ldots,x_n]\), among them the Jacobian Conjecture for the invertibility of endomorphisms of \({\mathbb C}[x_1,\ldots,x_n]\) with invertible Jacobian matrix, the Dixmier Conjecture that the endomorphisms of the associative Weyl algebra \(A_{n,{\mathbb C}}\) of polynomial differential operators are automorphisms, the Belov-Kanel-Kontsevich Conjecture for automorphisms of polynomial algebras \(P_{n,{\mathbb C}}\) in \(2n\) variables with additional Poisson structure. The Belov-Kanel-Kontsevich Conjecture is true for \(n=1\). It follows from the description of the automorphism groups of \({\mathbb C}[x_1,x_2]\), the free associative algebra \({\mathbb C}\langle x_1,x_2\rangle\), and the Poisson algebra \({\mathbb C}\{x_1,x_2\}\) and relies essentially on the fact that all these automorphisms are tame.NEWLINENEWLINEIn the paper under review the authors try to find an indirect proof of existence of a canonical isomorphism between the group of algebra automorphisms of \(A_{n,{\mathbb C}}\) and the group of polynomial symplectomorphisms of \({\mathbb C}^2\). For this purpose they study a correspondence between cyclic modules over \(A_{n,{\mathbb C}}\) and planar algebraic curves in positive characteristic. In particular, the authors show that any such curve has a preimage under a morphism of certain ind-schemes. As a byproduct of the ideas of the paper the authors obtain a natural way of assigning, in the case of positive characteristic, to every planar algebraic curve a finite set of elements of the first Weyl algebra. This is an interesting way to assign noncommutative objects to commutative ones.