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 Prikl. Diskr. Mat.: Year: Volume: Issue: Page: Find

 Prikl. Diskr. Mat., 2020, Number 48, Pages 22–33 (Mi pdm702)

Theoretical Backgrounds of Applied Discrete Mathematics

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

N. S. Kolesnikov, S. A. Novoselov

Immanuel Kant Baltic Federal University, Kaliningrad, Russia

Abstract: 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, Frobenius action, $\ell$-torsion.

 Funding Agency Grant Number Russian Foundation for Basic Research 18-31-00244 The reported study was funded by RFBR according to the research project 18-31-00244.

DOI: https://doi.org/10.17223/20710410/48/3

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UDC: 512.742
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Citation: N. S. Kolesnikov, S. A. Novoselov, “On the distribution of orders of Frobenius action on $\ell$-torsion of abelian surfaces”, Prikl. Diskr. Mat., 2020, no. 48, 22–33

Citation in format AMSBIB
\Bibitem{KolNov20} \by N.~S.~Kolesnikov, S.~A.~Novoselov \paper On the distribution of orders of Frobenius action on $\ell$-torsion of abelian surfaces \jour Prikl. Diskr. Mat. \yr 2020 \issue 48 \pages 22--33 \mathnet{http://mi.mathnet.ru/pdm702} \crossref{https://doi.org/10.17223/20710410/48/3} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000541666400003}