Hilbert's tenth problem for families of Zₚ-extensions of imaginary quadratic fields

Via a novel application of Iwasawa theory, we study Hilbert's tenth problem for number fields occurring in Zₚ-towers of imaginary quadratic fields K. For a odd prime p, the lines (a,b) ∈ P¹(Zₚ) are identified with Zₚ-extensions Ka,b/K. Under certain conditions on K that involve explicit elliptic curves, we identify a line (a₀,b₀) ∈ P¹(Z/pZ) such that for all (a,b) ∈ P¹(Zₚ) with (a, b)≡ (a₀, b₀)p, Hilbert's tenth problem has a negative answer in all finite layers of Ka,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.