Elliptic curves, L-functions, and Hilbert's tenth problem

Hilbert's tenth problem for rings of integers of number fields remains open in general, although a negative solution has been obtained by Mazur and Rubin conditional to a conjecture on Shafarevich-Tate groups. In this work we consider the problem from the point of view of analytic aspects of L-functions instead. We show that Hilbert's tenth problem for rings of integers of number fields is unsolvable, conditional to the following conjectures for L-functions of elliptic curves: the automorphy conjecture and the rank part of the Birch and Swinnerton-Dyer conjecture.