Hilbert's tenth problem in Anticyclotomic towers of number fields

TL;DR

This paper explores conditions under which Hilbert's tenth problem becomes unsolvable in certain infinite extensions of imaginary quadratic fields, using properties of elliptic curves and diophantine definability.

Contribution

It establishes explicit criteria involving elliptic curves that make the anticyclotomic extension diophantine over the base field, leading to new cases of unsolvability of Hilbert's tenth problem.

Findings

Identifies conditions for diophantine definability in anticyclotomic towers.

Provides explicit example with p=3 and K=Q(√-5).

Shows Hilbert's tenth problem is unsolvable in certain infinite extensions.

Abstract

Let $K$ be an imaginary quadratic field and $p$ be an odd prime which splits in $K$. Let $E_1$ and $E_2$ be elliptic curves over $K$ such that the $Gal(\bar{K}/K)$-modules $E_1[p]$ and $E_2[p]$ are isomorphic. We show that under certain explicit additional conditions on $E_1$ and $E_2$, the anticyclotomic $\mathbb{Z}_p$-extension $K_{anti}$ of $K$ is integrally diophantine over $K$. When such conditions are satisfied, we deduce new cases of Hilbert's tenth problem. In greater detail, the conditions imply that Hilbert's tenth problem is unsolvable for all number fields that are contained in $K_{anti}$. We illustrate our results by constructing an explicit example for $p=3$ and $K=\mathbb{Q}(\sqrt{-5})$.