Submission date: 8 February 2023

Abstract:

Let K be an imaginary quadratic field and p be an odd prime which splits in K. Let E_1 and E_2 be elliptic curves over K such that the Gal(K/K)-modules E_1[p] and E_2[p] are isomorphic. We show that under certain explicit additional conditions on E_1 and E_2, the anticyclotomic ℤ_p-extension K_{anti} of K is integrally diophantine over K. When such conditions are satisfied, we deduce new cases of Hilbert's tenth problem. In greater detail, the conditions imply that Hilbert's tenth problem is unsolvable for all number fields that are contained in K_{anti}. We illustrate our results by constructing an explicit example for p=3 and K=ℚ(√(-5)).

Mathematics Subject Classification: 11R23, 11U05

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Full text arXiv 2302.04157: pdf, ps.