The Birch and Swinnerton-Dyer Conjecture

The conjecture of Birch and Swinnerton-Dyer makes sense for abelian varieties over fairly general global fields, but we only state a special case. This conjecture involves the $L$-function attached to $A=A_f$:

$$L(A,s) = \prod_{i=1}^d L(f^{(i)},s) = \prod_{i=1}^d \left(\sum_{n\geq 1} \frac{a_n^{(i)}}{n^{s}}\right),$$

where $a_n^{(i)}$ is the $i$th Galois conjugate of $a_n$. Hecke proved that $L(A,s)$ has an analytic continuation to the whole complex plane and satisfies a functional equation. Birch and Swinnerton-Dyer made the following conjecture, which relates the rank of $A$ to the order of vanishing of $L(A,s)$ at $s=1$.

Conjecture 2.1 (Birch and Swinnerton-Dyer)   The Mordell-Weil rank of $A$ is equal to the order of vanishing of $L(A,s)$ at $s=1$, i.e.,

$$\dim (A(\mathbf{Q})\otimes \mathbf{Q}) = \ord_{s=1} L(A,s).$$

Birch and Swinnerton-Dyer also furnished a conjectural formula for the order of the Shafarevich-Tate group

$${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontsh... ...,A) \longrightarrow \prod_{\text{all places $v$}} H^1(\mathbf{Q}_v,A)\right).$$

(They only made their conjecture for elliptic curves, but Tate [Tat95] reformulated it a functorial way which makes sense for abelian varieties. See also [Lan91, §III.5] for another formulation.) We now state their conjecture in the special case when $L(A,1)\neq 0$, where [KL89,KL92] implies that ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(A)$ is finite. The conjecture involves the Tamagawa numbers $c_p$ of $A$ (see Section 3.7), and the canonical volume $\Omega_A$ of $A(\mathbf{R})$ (see Section 4.2).

Conjecture 2.2 (Birch and Swinnerton-Dyer)   Suppose $L(A,1)\neq 0$. Then

$$\frac{L(A,1)}{\Omega_A} = \frac{\char93 {\mbox{{\fontencoding{OT2... ... c_p} {\char93 A(\mathbf{Q})_{\tor}\cdot\char93 A^{\vee}(\mathbf{Q})_{\tor}},$$

where $A^{\vee}$ is the abelian variety dual of $A$.

Remark 2.3   Since $L(A,1)\neq 0$, finiteness of ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(A)$ and the existence of the Cassels-Tate pairing implies that $\char93 {\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n... ...ng{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(A^{\vee})$, so Conjecture 2.2 can also be viewed as a formula for $\char93 {\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(A^{\vee})$.

The algorithms outlined in this paper leverage the fact that $A$ is attached to a newform in order to compute the conjectural order of ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(A)$ away from certain bad primes.