We present a probabilistic Las Vegas algorithm for computing the local zeta function of a genus-$g$ hyperelliptic curve defined over ${\mathbb{F}}_q$ with explicit real multiplication (RM) by an order $\mathbb{Z}\left[\eta \right]$ in a degree-$g$ totally real number field.

It is based on the approaches by Schoof and Pila in a more favourable case where we can split the $\ell$-torsion into $g$ kernels of endomorphisms, as introduced by Gaudry, Kohel, and Smith in genus 2. To deal with these kernels in any genus, we adapt a technique that the author, Gaudry, and Spaenlehauer introduced to model the $\ell$-torsion by structured polynomial systems. Applying this technique to the kernels, the systems we obtain are much smaller and so is the complexity of solving them.

Our main result is that there exists a constant $c>0$ such that, for any fixed $g$, this algorithm has expected time and space complexity $O\left({(logq)}^c\right)$ as $q$ grows and the characteristic is large enough. We prove that $c\le 9$ and we also conjecture that the result still holds for $c=7$.

我们提出了一种概率性拉斯维加斯算法，用于计算某类的本地zeta函数-$G$ 定义的超椭圆曲线 ${\mathbb{F}}_q$ 与一个订单的显式实数乘法（RM） $\mathbb{ž}\left[\eta \right]$ 在一定程度上$G$ 完全实数字段。

它基于Schoof和Pila的方法，在更有利的情况下，我们可以将 $\ell$扭转成 $G$ 高斯，科埃尔和史密斯在第2属中介绍的同种异形的内核。为了处理任何属中的这些内核，我们采用了作者，高迪和Spaenlehauer引入的建模技术。 $\ell$结构多项式系统的扭转。将这种技术应用于内核，我们获得的系统要小得多，因此解决它们的复杂性也要小得多。

我们的主要结果是存在一个常数 $C>0$ 这样，对于任何固定 $G$，此算法具有预期的时空复杂度 $Ø（{（日志q）}^C）$ 如 $q$增长并且特性足够大。我们证明$C\le 9$ 而且我们也猜想结果仍然成立 $C=7$。