The Point Counting Function on Elliptic Curves

Abstract

Suppose a and m are two coprime integers. Then the arithmetic sequence a, a+m, a+2m, ... contains infinitely many primes. Moreover, their asymptotic density among all primes is known: it is the Euler phi function in m. This is Dirichlet’s theorem on arithmetic progressions. A generalisation of this result is Chebotarev’s density theorem, which can be used to answer the following question. Given two degree three polynomials in two variables with coefficients in the rational numbers satisfying a certain mild regularity condition (i.e. an elliptic curve). Of both polynomials, we count the number of zeroes modulo p for every prime p. Suppose the number of zeroes is equal for a set of primes of asymptotic density 1. Are the number of zeroes then necessarily equal modulo p for every prime p? The answer turns out to be: almost, except possibly for some finite set of “bad” primes. This is the main part of the thesis, which is in fact a special case of a much more general theorem by Serre about schemes. The above situation is the special case when the scheme is an elliptic curve, and for that case the thesis contains a proof. A proof of Chebotarev’s density theorem is also given, assuming some results about L-series and representations of finite groups.