Geodesic
On the distribution of orders of Frobenius action on $\ell$-torsion of abelian surfaces
Prikladnaâ diskretnaâ matematika, no. 2 (2020), pp. 22-33

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The computation of the order of Frobenius action on the $\ell$-torsion is a part of Schoof — Elkies — Atkin algorithm for point counting on an elliptic curve $E$ over a finite field $\mathbb{F}_q$. The idea of Schoof's algorithm is to compute the trace of Frobenius $t$ modulo primes $\ell$ and restore it by the Chinese remainder theorem. Atkin's improvement consists of computing the order $r$ of the Frobenius action on $E[\ell]$ and of restricting the number $t \pmod{\ell}$ to enumerate by using the formula $t^2 \equiv q (\zeta + \zeta^{-1})^2 \pmod{\ell}$. Here $\zeta$ 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 $q \equiv 1 \pmod{\ell}$ in order to replace these expensive computations by probabilistic algorithms.
Keywords: abelian varieties, finite fields
Mots-clés : Frobenius action, $\ell$-torsion. 
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     author = {N. S. Kolesnikov and S. A. Novoselov},
     title = {On the distribution of orders of {Frobenius} action on $\ell$-torsion of abelian surfaces},
     journal = {Prikladna\^a diskretna\^a matematika},
     pages = {22--33},
     publisher = {mathdoc},
     number = {2},
     year = {2020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/PDM_2020_2_a2/}
}
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N. S. Kolesnikov; S. A. Novoselov. On the distribution of orders of Frobenius action on $\ell$-torsion of abelian surfaces. Prikladnaâ diskretnaâ matematika, no. 2 (2020), pp. 22-33. http://geodesic.mathdoc.fr/item/PDM_2020_2_a2/