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

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