Hilbert’s tenth problem in anticyclotomic towers of number fields

Abstract

Let K K be an imaginary quadratic field and p p be an odd prime which splits in K K . Let E 1 E_1 and E 2 E_2 be elliptic curves over K K such that the Gal ⁡ ( K ¯ / K ) \operatorname {Gal}(\bar {K}/K) -modules E 1 [ p ] E_1[p] and E 2 [ p ] E_2[p] are isomorphic. We show that under certain explicit additional conditions on E 1 E_1 and E 2 E_2 , the anticyclotomic Z p \mathbb {Z}_p -extension K anti K_{\operatorname {anti}} of K K is integrally diophantine over K 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 K_{\operatorname {anti}} . We illustrate our results by constructing an explicit example for p = 3 p=3 and K = Q ( − 5 ) K=\mathbb {Q}(\sqrt {-5}) .