On the distribution of orders of Frobenius action on l-torsion of abelian surfaces

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On the distribution of orders of Frobenius action on l-torsion of abelian surfaces	2 MB	Adobe Acrobat PDF	Read	Download
DOI Доступ к ресурсу на сайте издателя			10.17223/20710410/48/3

Title: On the distribution of orders of Frobenius action on l-torsion of abelian surfaces
Creator: Kolesnikov, N. S.
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Novoselov, S. A.
Date: 2020
DOI: 10.17223/20710410/48/3
Description: 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 I 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 F) to enumerate by using the formula t2 = q(Z + Z-1)2 (mod £). Here Z 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 F) in order to replace these expensive computations by probabilistic algorithms.