In this chapter, we see how to use elliptic curves to build a type of generating function called an L-function. The hope is that important properties of the elliptic curve will be captured by computable properties of the L-function.
The concept and notation for an L-function seem to go back to Lejeune Dirichlet, in his famous paper from 1837, “Beweis eines Satzes über die arithmetische Progression.” In this paper, he proved that if you start with two positive integers a and b that share no common prime factor, then the sequence
contains infinitely many prime numbers. We won’t describe the proof ...