On the distribution of orders of Frobenius action on ℓ-torsion of abelian surfaces
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The computation of the order of Frobenius action on the ℓ-torsion is a part of Schoof-Elkies-Atkin algorithm for point counting on an elliptic curve E over a finite field F_q. The idea of Schoof's algorithm is to compute the trace of Frobenius t modulo primes ℓ and restore it by the Chinese remainder theorem. Atkin's improvement consists of computing the order r of the Frobenius action on E[ℓ] and of restricting the number t ℓ to enumerate by using the formula t^2 ≡ q (ζ + ζ^-1)^2 ℓ. 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 ℓ in order to replace these expensive computations by probabilistic algorithms.
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