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

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 $K_{a,b}\mathrm{/}K$. Under certain conditions on $K$ that involve explicit elliptic curves, we identify a line $(a_0,b_0)\in {\mathbb{P}}^1(\mathbb{Z}\mathrm{/}p\mathbb{Z})$ such that for all $(a,b)\in {\mathbb{P}}^1({\mathbb{Z}}_p)$ with $(a,b)\equiv ̸(a_0,b_0)(\mathrm{m}\mathrm{o}\mathrm{d}p)$, 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.