Elliptic curve

In this context, an elliptic curve is a plane curve defined by an equation of the form

We know that this equation has its roots at exactly the same x values as

At s=1 (the conductor product can be discarded as it is finite), the L-function becomes

Much like the Riemann hypothesis, the truth of the BSD conjecture would have multiple consequences, including the following two:

provided that the variety it defines is non-singular. If characteristic were not an obstruction, each equation would reduce to the previous ones by a suitable linear change of variables.

The properties of the Hasse–Weil zeta function and the Birch and Swinnerton-Dyer conjecture can also be extended to this more general situation.

then a pair of fundamental periods of E can be calculated very rapidly by

Serge Lang, in the introduction to the book cited below, stated that "It is possible to write endlessly on elliptic curves. (This is not a threat.)" The following short list is thus at best a guide to the vast expository literature available on the theoretical, algorithmic, and cryptographic aspects of elliptic curves.