Connection between the Jacobian Conjecture and number theory conjectures

Question

In this paper E. Formanek says: ``The purpose of this paper is to point out a connection between certain differential equations which have arisen in attempts to establish the two-variable Jacobian Conjecture and the work of W. W. Stothers on the polynomial abc-conjecture".

Formanek applies results of T.-T. Moh and S. S. Abhyankar.

Is there any progress in the above direction? Moreover, are there other known connections between the (two-dimensional) Jacobian Conjecture and other number theory results/conjectures?

This question is relevant.

Thank you very much!

Answer 1

An interesting conjecture that has relation with the Jacobian Conjecture is the following:

Let $(\mathcal{O},\mathcal{M},k)$ be a complete discrete valuation ring with k finite.Let $f_{1},...,f_{n} \in \mathcal{O}[X_{1},...,X_{n}]$ with the jacobian condition: $\det J_{f} = 1$. Consider the scheme $$X = Spec(\mathcal{O}[X_{1},...,X_{n}]/(f_{1},...,f_{n}))$$ Then #$X(\mathcal{O})$ < #$k^{n}$.

If this is true for $\mathbb{Z}_{p}$ (for infinitley primes p), then Jacobian conjecture is true. Details in Essen-Lipton paper or in this paper. (Hensel Lemma ensures an bijection $X(\mathcal{O}) \cong X(k)$ and so #$X(\mathcal{O}) \leq $ #$k^{n}$.)