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 $(a, b)\not\equiv (a_0, b_0)\pmod{p}$, Hilbert's tenth problem has a negative answer in all finite layers of $ K_{a,b} $. Using results of Kriz--Li and Bhargava et al., we demonstrate that for primes $ p = 3, 11, 13, 31, 37 $, a positive proportion of imaginary quadratic fields meet our criteria.