On the representation of the number of integral points of an elliptic curve modulo a prime number

TL;DR

This paper explores how to represent the count of integral points on elliptic curves modulo a prime p, providing explicit formulas and expressing exponential sums involving cubic polynomials in non-exponential terms.

Contribution

It introduces a method to explicitly express exponential sums with cubic polynomials and derives formulas involving the Riemann Zeta function for counting points on elliptic curves.

Findings

Explicit formulas for exponential sums with cubic polynomials

Representation of point counts on elliptic curves modulo p

Formulas involving the Riemann Zeta function

Abstract

In this paper we shall investigate the problem of the representation of the number of integral points of an elliptic curve modulo a prime number p. We present a way of expressing an exponential sum which involves polynomials of third degree, in explicit non-exponential terms. In the process, we present explicit formulas for the calculation of some series involving the Riemann Zeta function.