Hilbert's tenth problem for families of ${\mathbb{Z}}_p$-extensions of imaginary quadratic fields

Abstract

Via a novel application of Iwasawa theory, we study Hilbert's tenth problem for number fields occurring in ${\mathbb{Z}}_p$-towers of imaginary quadratic fields $K$. For a odd prime $p$, the lines $(a,b)\in {\mathbb{P}}^1({\mathbb{Z}}_p)$ are identified with ${\mathbb{Z}}_p$-extensions $K_{a,b}/K$. Under certain conditions on $K$ that involve explicit elliptic curves, we identify a line $(a_0,b_0)\in {\mathbb{P}}^1(\mathbb{Z}/p\mathbb{Z})$ such that for all $(a,b)\in {\mathbb{P}}^1({\mathbb{Z}}_p)$ with $(a,b)\not\equiv (a_0,b_0)(\mathrm{mod}p)$, Hilbert's tenth problem has a negative answer in all finite layers of $K_{a,b}$. Using results of Bhargava et al., we prove unconditionally that a positive proportion of imaginary quadratic fields meet our criterion when $p=3$. For $p=11,13,31,37$, the analogous conclusions obtained from the rank-zero twist families of Kriz--Li are conditional on the vanishing of the $p$-primary Tate--Shafarevich groups for a positive relative proportion of those twists.

Version 2: 33 pages; accepted for publication in the Journal of the Australian Math Society