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.

References

The L-function $L(C,s)$ is the subject of many open conjectures in arithmetic geometry, including the paramodular conjecture, and generalizations of the Sato-Tate conjecture, the conjecture of Birch and Swinnerton-Dyer, and the Riemann hypothesis.

— Lifting $L$-polynomials of genus 2 curves  (2508.11028 - Shi, 14 Aug 2025) in Section 1 (Introduction)

We have that $$\prod_ {\substack{p x p \not \in S_X} \frac{N_p}{p} \sim C (\log x){rk (Jac(X)) -rk(NS(Jac(X)))+1}$$ as $x \to \infty$ for some constant $C$ depending on $X$.

— Products of point counts of higher genus curves over finite fields  (2608.18014 - Bucur et al., 18 Aug 2026) in Conjecture \ref{ourconj}, Section 1, subsection “The higher genus case”