How many points has an affine algebraic set in residue classes modulo n ?

Abstract

We show that for every uniform family of affine algebraic sets, there is a formula in the parameters of the family and in $n$, expressing the cardinality of the set in the ring $\mathbb{Z}/n\mathbb{Z}$. In particular, we construct such a formula for the elliptic curves in Weierstrass normal form. These formulas are arithmetic terms: fixed finite compositions of additions, multiplications, subtractions, divisions with remainder and integer exponentiations.