Remarks on Hilbert's tenth problem and the Iwasawa theory of elliptic curves

TL;DR

This paper explores conditions under which the rank of an elliptic curve remains constant in cyclotomic towers of number fields, using Iwasawa theory, and discusses implications for Hilbert's tenth problem.

Contribution

It provides explicit criteria for the constancy of elliptic curve ranks in cyclotomic extensions, linking Iwasawa theory to Hilbert's tenth problem.

Findings

Explicit conditions for rank constancy in cyclotomic towers

Application of Iwasawa theory techniques to elliptic curves

Potential implications for Hilbert's tenth problem

Abstract

Let $E$ be an elliptic curve with positive rank over a number field $K$ and let $p$ be an odd prime number. Let $K_{cyc}$ be the cyclotomic $\mathbb{Z}_p$-extension of $K$ and $K_n$ denote its $n$-th layer. The Mordell--Weil rank of $E$ is said to be constant in the cyclotomic tower of $K$ if for all $n$, the rank of $E(K_n)$ is equal to the rank of $E(K)$. We apply techniques in Iwasawa theory to obtain explicit conditions for the rank of an elliptic curve to be constant in the above sense. We then indicate the potential applications to Hilbert's tenth problem for number rings.