Frobenius Maps of Abelian Varieties and Finding Roots of Unity in Finite Fields

Abstract

We give a generalization to Abelian varieties over finite fields of the algorithm of Schoof for elliptic curves. Schoof showed that for an elliptic curve E over ${{\mathbf {F}}_q}$, given by a Weierstrass equation, one can compute the number of ${{\mathbf {F}}_q}$-rational points of E in time $O({(\log q)^9})$. Our result is the following. Let A be an Abelian variety over ${{\mathbf {F}}_q}$. Then one can compute the characteristic polynomial of the Frobenius endomorphism of A in time $O({(\log q)^\Delta })$, where $\Delta$ and the implied constant depend only on the dimension of the embedding space of A, the number of equations defining A and the addition law, and their degrees. The method, generalizing that of Schoof, is to use the machinery developed by Weil to prove the Riemann hypothesis for Abelian varieties. By means of this theory, the calculation is reduced to ideal-theoretic computations in a ring of polynomials in several variables over ${{\mathbf {F}}_q}$. As applications we show how to count the rational points on the reductions modulo primes p of a fixed curve over Q in time polynomial in log p; we show also that, for a fixed prime l, we can compute the lth roots of unity mod p, when they exist, in polynomial time in log p. This generalizes Schoof’s application of his algorithm to find square roots of a fixed integer x mod p.