Birch and Swinnerton-Dyer conjecture for Jacobians of genus 2 curves over Q

Prove the conjecture of Birch and Swinnerton-Dyer for the Jacobian J of a genus 2 curve C over Q, relating the order of vanishing of L(J,s) at s=1 to the Mordell–Weil rank of J(Q) and expressing the leading Taylor coefficient in terms of standard arithmetic invariants.

Background

The paper’s computational framework targets L_p(T) values needed to form L(C,s) for genus 2 curves, which directly enters the formulation of the Birch and Swinnerton-Dyer conjecture for their Jacobians. The authors explicitly note BSD as one of the open conjectures concerning L(C,s).

Their algorithm provides a practical route to compute L(C,s) up to large bounds, supporting numerical exploration of BSD for abelian surfaces arising from genus 2 curves.