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The BSD conjecture

By [6] we now know that every elliptic curve over  Q is a quotient of the curve X₀(N) whose complex points are the isomorphism classes of pairs consisting of a (generalized) elliptic curve and a cyclic subgroup of order N. Let J₀(N) denote the Jacobian of X₀(N); this is an abelian variety of dimension equal to the genus of X₀(N) whose points correspond to the degree 0 divisor classes on X₀(N).

An optimal quotient of J₀(N) is a quotient by an abelian subvariety. Consider an optimal quotient A such that $L(A,1)\neq 0$. By [13],  A(Q) and  $\mbox{\cyr X}(A/\mathbf{Q})$ are both finite. The BSD conjecture asserts that

$$\frac{L(A,1)}{\Omega_A} = \frac{\char93 \mbox{\cyr X}(A/\math... ..._p} {\char93 A(\mathbf{Q})\cdot\char93 A^{\vee}(\mathbf{Q})}.$$

Here the Shafarevich-Tate group $\mbox{\cyr X}(A/\mathbf{Q})$ is a measure of the failure of the local-to-global principle; the Tamagawa numbers c_p are the orders of the component groups of A; the real number $\Omega_A$ is the volume of  A(R) with respect to a basis of differentials having everywhere nonzero good reduction; and $A^{\vee}$ is the dual of A. My goal is to verify the full conjecture for many specific abelian varieties on a case-by-case basis. This is the first step in a program to verify the above conjecture for an infinite family of quotients of J₀(N).