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Auxiliary Polynomials in Number Theory by David Masser

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6

Elliptic curves – Stepanov’s Method

We make another jump to count points on elliptic curves E over finite fields. A classical result of Hasse, from around 1930, is that over Fq their number differs from q + 1 by at most image (here the odd-looking q + 1 comes from the zero of the group law). A modern proof (see for example Silverman (1992) chapter V) uses the separability of an endomorphism coming from the Frobenius map. Actually what we prove here, for simplicity with primes q = p, has nothing to do with elliptic curves.

Theorem 6.1Let C in Fp[X] be a cubic polynomial. Then the number N of (x, y) in with y2 = C(x) satisfies

A completely trivial ...

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