Hilbert's tenth problem in Anticyclotomic towers of number fields

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 $K=\mathbb{Q}(\sqrt{-5})$.

Version 2: Theorem 1.5 is now unconditional; paper accepted for publication in Transactions of the American Math Society

References in corpus (2)

• Remarks on Hilbert's tenth problem and the Iwasawa theory of elliptic curves
• Existential definability and diophantine stability

Cited by in corpus (1)

• Integers that are sums of two cubes in the cyclotomic ${\mathbb{Z}}_p$-extension