Some Identities Involving Character Sums and their Applications

Abstract

§1. The identities. It is well known [8], that if the two elliptic curves y²=f(x) and y²= g(x), defined over the integers Z are isogenous, then for all primes p # 2,3 and not dividing the discriminant of either, the character sums ∑ (f(x)/p) and ∑(g(x)/p) are equal, where (./p) is the Legendre symbol.