**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 F_{q} their number differs from *q* + 1 by at most (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 6.1***Let C in* F_{p}[*X*] *be a cubic polynomial. Then the number N of* (*x*, *y*) *in* *with y*^{2} = *C*(*x*) *satisfies*

A completely trivial ...