Consequences of the Birch and Swinnerton-Dyer Conjecture? Before asking my short question I had made some research. Unfortunately I did not find a good reference with some examples. My question is the following 

What are the consequences of the Birch and Swinnerton-Dyer Conjecture ?

I have read the official statement of the conjecture in Clay Institute website and followed a very nice talk by Manjul Bhargava... 
Just to be clear enough, I'm not interested on the explanation of the conjecture, I would like to know what are the consequences. 
How can I make this question C.W. ?
 A: There is a theorem of Michael Stoll that there are no $c\in\mathbb Q$ such that the polynomial $x^2+c$ admits a periodic 6-cycle starting at some $a\in\mathbb Q$, but the theorem is contingent on the Birch-Swinnerton-Dyer conjecture being true for the Jacobian $J$ of a certain curve $C$ appearing in Stoll's paper. The reason it's needed is because he needs to know the rank of $J(\mathbb Q)$, but the genus of $C$ is too large for him to do an explicit descent. So he's "just" using the rank part of BSwD. Here's the reference:
MR2465796  Stoll, Michael Rational 6-cycles under iteration of quadratic polynomials. LMS J. Comput. Math. 11 (2008), 367–380.  
A: Implicit in the BSD conjecture are two other basic conjectures about elliptic curves: the Parity Conjecture and the finiteness of the Tate-Shafarevich group.  Most applications I know follow from the rank part, but refined BSD will allow formulas for the order of the Tate-Shafarevich group for instance.
Two specific well-known elementary applications are:


*

*Tunnell's solution to the Congruent Number Problem

*Rodriguez-Villegas and Zagier's solution to which primes are sums of two rational cubes


These are both contingent on BSD for some cases.  Here are a couple other consequences:


*

*There exists an elliptic curve/modular form of analytic rank 4.  (Note the existence of an elliptic curve of analytic rank at least 3 is used to solve the Gauss class number 1 problem.)

*(example) Let $E$ be an elliptic curve of conductor 17, and $-d$ a negative prime discriminant.  Then the $-d$-th quadratic twist of $E$ has infinitely many rational points if and only if the Legendre symbol $(-d/17) = +1$.


It would also provide an analytic way to attack Goldfeld's conjecture on average ranks of elliptic curves.
