论文标题
关于在$ \ ell $ torsion ob abelian表面上的Frobenius Action订单的分布
On the distribution of orders of Frobenius action on $\ell$-torsion of abelian surfaces
论文作者
论文摘要
$ \ ell $ torsion上Frobenius Action的订单计算是Schoof-elkies-atkin算法的一部分，用于点上椭圆曲线$ e $上的点上的有限字段$ \ mathbb {f} _q $。 Schoof算法的想法是计算Frobenius $ t $ modulo Primes $ \ ell $的痕迹，并根据中国剩余定理恢复它。阿特金（Atkin）的改进包括在$ e [\ ell] $上计算Frobenius Action的$ r $，并通过使用公式$ t^2 \ equiv q（ζ+ζ+ζ^{ - 1}^2 \ pmod^2 \ pmod^\ pmod Q \ ell} $。这里$ζ$是统一的原始$ r $ th。在本文中，我们将Atkin的公式推广到Abelian Varts of Dimension $ g $的一般情况。从经典上讲，查找$ r $的订单涉及对模块化多项式的昂贵计算。我们研究Frobenius订单的分布，如果Abelian表面和$ Q \ equiv 1 \ pmod {\ ell} $，以用概率算法替换这些昂贵的计算。
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 $t^2 \equiv q (ζ+ ζ^{-1})^2 \pmod{\ell}$. 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 $q \equiv 1 \pmod{\ell}$ in order to replace these expensive computations by probabilistic algorithms.
