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