On the Birch-Swinnerton-Dyer conjecture for modular abelian surfaces

Let $A$ be a modular abelian surface over $Q$ which either has trivial geometric endomorphism ring, or arises as the restriction of scalars of an elliptic curve over an imaginary quadratic field which is modular and is not a $Q$-curve. In the former case, assume that there exists an odd Dirichlet character $\chi$ such that $L(A,\chi,1)\neq 0$. We prove the following implication: if $L(A, 1) \ne 0$, and the $p$-adic eigenvariety for $GSp_4$ is smooth at the point corresponding to $A$ (and some auxiliary technical hypotheses hold), then $A(Q)$ is finite, as predicted by the Birch--Swinnerton-Dyer conjecture, and the $p$-part of the Tate--Shafarevich group is also finite. We also prove one inclusion of the cyclotomic Iwasawa Main Conjecture for $A$. Moreover, we also prove analogous results for cohomological automorphic representations of $GSp_4$, removing many of the restrictive hypotheses in our earlier work [2003.05960]; for cohomological representations we do not need to assume smoothness of the eigenvariety, since it is automatic in this case. The main ingredient in the proof is the Euler system attached to the spin representations of genus $2$ Siegel modular forms constructed in our earlier work with Skinner.