We explore algorithmic aspects of a simply transitive commutative group action coming from the class field theory of imaginary hyperelliptic function fields. Namely, the Jacobian of an imaginary hyperelliptic curve defined over acts on a subset of isomorphism classes of Drinfeld modules. We describe an algorithm to compute the group action efficiently. This is a function field analog of the Couveignes-Rostovtsev-Stolbunov group action. We report on an explicit computation done with our proof-of-concept C++/NTL implementation; it took a fraction of a second on a standard computer. We prove that the problem of inverting the group action reduces to the problem of finding isogenies of fixed -degree between Drinfeld -modules, which is solvable in polynomial time thanks to an algorithm by Wesolowski. We give asymptotic complexity bounds for all algorithms presented in this paper.

中文翻译：

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从虚超椭圆函数域的类域论计算群作用

我们探索来自虚超椭圆函数域的类域论的简单传递交换群作用的算法方面。即，定义的假想超椭圆曲线的雅可比行列式作用于 Drinfeld 模的同构类的子集。我们描述了一种有效计算群体行为的算法。这是 Couveignes-Rostovtsev-Stolbunov 群作用的函数场模拟。我们报告了使用我们的概念验证 C++/NTL 实现完成的显式计算；在标准计算机上只需要几分之一秒的时间。我们证明，反转群作用的问题可以简化为寻找 Drinfeld 模块之间固定度同源性的问题，这得益于 Wesolowski 的算法，可以在多项式时间内解决。我们给出本文中提出的所有算法的渐近复杂度界限。