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arXiv · 2203.06970

Computing a Group Action from the Class Field Theory of Imaginary Hyperelliptic Function Fields

Antoine Leudière·Pierre-Jean Spaenlehauer
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Abstract

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 $\mathbb F_q$ 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 $\tau$-degree between Drinfeld $\mathbb F_q[X]$-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.

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Antoine Leudière, Pierre-Jean Spaenlehauer. 2022-03-14. Computing a Group Action from the Class Field Theory of Imaginary Hyperelliptic Function Fields. https://arxiv.org/abs/2203.06970

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