What is the Birch and Swinnerton-Dyer Conjecture? This is probably a really silly question, but I was wondering if someone could explain the Birch and Swinnerton-Dyer conjecture to me in a simple way. I've read a lot about it, but cannot understand it. Maybe posting here will help.
What I do understand is the concept of ranks on an elliptic curve; for ever point that generates an infinite number of points on the elliptic curve, it contributes 1 to the rank. Further, I know that Mordell proved that every elliptic curve must have a finite rank.
Finally, with regards to BSD, I understand that the conjecture says that the "analytic" way of computing the rank is the same as the "algebraic" way of computing the rank. But what I cannot seem to understand is what exactly is the analytic way of computing the rank, or the algebraic way.
If anyone wants to help me out, that would be much appreciated! Thanks!
 A: Your phrasing of the problem, in terms of two ways to compute the rank (analytic and algebraic), is not quite correct.  Rather the conjecture states that two numbers are equal.
On the one hand there is the rank of the elliptic curve, which is as you said: the number of independent rational points of infinite order.  In the context of BSD, one calls this "the algebraic rank", $r_{alg}$, of the elliptic curve $E$.
Then, there is another quantity attached to $E$, also a non-negative integer, but of quite a different nature. To describe it, we begin by defining the $L$-function of $E$,
as a certain Euler product (a Dirichlet series given by taking a product
indexed by the primes), denoted $L(E,s)$, which is holomorphic in the half-plane $\Re s > 3/2$.   Thanks to the modularity results of Wiles et. al. we know that this function has analytic continuation to the whole complex plane, and so in particular, we can consider its order of vanishing at $s = 1$.  In the context of BSD, we call this "the analytic rank", $r_{an}$, of $E$.  (Note though that this is just a name; $r_{an}$ is not defined in terms of the rank of any abelian group, but rather is the order of vanishing at $s = 1$ of an entire function; the only reason for calling it a "rank" comes from the BSD conjecture.)
The BSD conjecture is then that $r_{an} = r_{alg}$; in other words, we can determine the rank of $E$ by determining the order of vanishing of $L(E,s)$ at $s = 1$.  To my mind the most striking consequence of this is that
$r_{an} > 0$ if and only if $r_{alg} > 0$; in other words, we (conjecturally) can determine whether or not $E$ has infinitely many rational points (i.e. has positive rank) by evaluating $L(E,1)$ and determining whether or not it equals zero.  
In fact one direction of this weaker form of the conjecture is actually known: it is known that if $L(E,1) \neq 0,$ then $E$ has only finitely many rational points; this is due to Gross, Zagier, and Kolyvagin, with another proof more recently by Kato.    

(I won't give the definition of $L(E,s)$ here; you can find it in many places,
including in Wiles's write-up I'm sure, or just by googling "L-function of elliptic curve".)
