How many points does an affine algebraic set have in residue classes modulo n ?
Mihai Prunescu
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 Z/n 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.
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