Endomorphisms of Weyl algebra and $p$-curvatures

Abstract

We first show that for each Weyl algebra over a positive characteristic field, we may obtain an affine space with a projectively flat connection on it. We give a set of differential equations which controls the behavior of the connection under endomorphism of the Weyl algebra. The key is the theory of $p$-curvatures.

Next we introduce a field $\mathbb{Q}_{\mathcal{U}}^{(\infty)}$ of characteristic zero as a limit of fields of positive characteristics. We need to fix an ultrafilter on the set of prime numbers to do this. The field is actually isomorphic to the field $\mathbb{C}$ of complex numbers.

Then we show that we may associate with a Weyl algebra over the field $\mathbb{Q}_{\mathcal{U}}^{(\infty)}$ an affine space with a symplectic form in a functorial way. That means, the association is done in such a way that an endomorphism of the Weyl algebra induces a symplectic map of the affine space.

As a result, we show that a solution of the Jacobian conjecture is sufficient for an affirmative answer to the Dixmier conjecture.