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

2024/06/03 by Katharina Müller, Anwesh Ray, Müller, Katharina +1
Mathematics · #11G05 #11R23 #11U05 #Advanced Differential Equations and Dynamical Systems #Analytic Number Theory Research #FOS: Mathematics #Meromorphic and Entire Functions #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2406.01443

openalex publication_date 2024/06/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Via a novel application of Iwasawa theory, we study Hilbert's tenth problem for number fields occurring in ℤ_p-towers of imaginary quadratic fields K. For a odd prime p, the lines (a,b) ∈ ℙ¹(ℤ_p) are identified with ℤ_p-extensions K_a,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(ℤ_p) with (a, b)\not≡ (a₀, b₀)\pmodp, 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.

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