The Hasse—Weil L-Function of an Elliptic Curve

Abstract

At the end of the last chapter, we used reduction modulo p to find some useful information about the elliptic curves E _n : y² = x³ − n²x and the congruent number problem. We considered E _n as a curve over the prime field ${{\mathbb{F}}_{p}}$ where $p\nmid 2n$ used the easily proved equality $\# {{E}_{n}}({{\mathbb{F}}_{p}}) = p + 1$ when p = 3 (mod 4); and, by making use of infinitely many such p, were able to conclude that the only rational points of finite order on E _n are the four obvious points of order two. This then reduced the congruent number problem to the determination of whether r, the rank of ${{E}_{n}}(\mathbb{Q})$, is zero or greater than zero