Introduction

Our aim in this paper is to gather some basic facts concerning algebra endomorphisms of Weyl algebras $A_n(k)$ for fields $k$ with positive characteristic $p$.

A study of such object may lead (Lemma 2) to some progress in the Dixmier conjecture (conjecture 1) which states that any algebra endomorphism of Weyl algebra over a field of characteristic zero is actually invertible.

It turns out that $A_n(k)$ is a free module of rank $p^{2n}$ over its center $Z(A_n(k))$ (Lemma 3) and that any $k$-algebra endomorphism $\phi$ of $A_n(k)$ sends central elements to central elements (Lemma4).

Thus the study of $\phi$ may be deeply related to the study of sheaves of matrix algebras over polynomial algebras $Z(A_n(k))$.

In particular, we may use traces to obtain a nice formula for candidate of inverse of $\phi$ (Proposition 1).

On the other hand, suppose we are given an algebra endomorphism $\phi$ of the Weyl algebra $A_n(\mathfrak{K})$ over an algebraic number field $\mathfrak{K}$. Then we have, for almost all (that is, for all except finite number of) prime ideal $\mathfrak{p}$ of $\mathfrak{K}$, an algebra endomorphism of $A_n(k(\mathfrak{p})$ over the residue field $k(\mathfrak{p})$. We prove that almost all such maps are injective and that geometric degree of these maps are bounded by a constant which is independent of $\mathfrak{p}$ (Proposition 2).