In [this paper](http://www.ams.org/journals/proc/2011-139-04/S0002-9939-2010-10523-3/S0002-9939-2010-10523-3.pdf) 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](https://mathoverflow.net/questions/294895/an-analogy-between-the-ring-of-polynomials-in-two-variables-and-another-commuta) is relevant.


Thank you very much!



Connection between the Jacobian Conjecture and number theory conjectures
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](https://www.sciencedirect.com/science/article/pii/S0022404914002564) or in this [paper](https://arxiv.org/abs/1802.04247). (Hensel Lemma ensures an bijection $X(\mathcal{O}) \cong X(k)$ and so #$X(\mathcal{O}) \leq $ #$k^{n}$.)


