The BSD Conjecture

Let $E$ be an elliptic curve over  $\mathbb{Q}$ given by an equation

$$y^2 = x^3 + ax + b$$

with $a,b\in\mathbb{Z}$. For $p\nmid \Delta = -16(4a^3 + 27b^2)$, let $a_p = p+1 - \char93 E(\mathbb{Z}/p\mathbb{Z})$. Let

$$L(E,s) = \prod_{p\nmid\Delta} \frac{1}{1-a_p p^{-s} + p^{1-2s}}.$$

Theorem 1.1 (Breuil, Conrad, Diamond, Taylor, Wiles)  
$L(E,s)$ extends to an analytic function on all of $\mathbb{C}$.

Conjecture 1.2 (Birch and Swinnerton-Dyer)   The Taylor expansion of $L(E,s)$ at $s=1$ has the form

$$L(E,s) = c(s-1)^r +$$   higher order terms$$$$

with $c\neq 0$ and $E(\mathbb{Q})\approx \mathbb{Z}^r \times E(\mathbb{Q})_{\tor}$.

A special case of the conjecture is the assertion that $L(E,1)=0$ if and only if $E(\mathbb{Q})$ is infinite. The assertion ``$L(E,1)=0$ implies that $E(\mathbb{Q})$ is infinite'' is the part of the conjecture that secretely motives much of my own research.