On the distribution of orders of Frobenius action on $\ell$-torsion of abelian surfaces

Abstract

The computation of the order of Frobenius action on the -torsion is a part of Schoof-Elkies-Atkin algorithm for point counting on an elliptic curve E over a finite field Fq. The idea of Schoof's algorithm is to compute the trace of Frobenius t modulo primes and restore it by the Chinese remainder theorem. Atkin's improvement consists of computing the order r of the Frobenius action on E[ ] and of restricting the number t (mod ) to enumerate by using the formula t 2 ≡ q(ζ + ζ −1 ) 2 (mod ). Here ζ is a primitive r-th root of unity. In this paper, we generalize Atkin's formula to the general case of abelian variety of dimension g. Classically, finding of the order r involves expensive computation of modular polynomials. We study the distribution of the Frobenius orders in case of abelian surfaces and q ≡ 1 (mod ) in order to replace these expensive computations by probabilistic algorithms.