Computing a group action from the class field theory of imaginary hyperelliptic function fields (Q6543075)

In the classical case, the class group \(\mathrm{Cl}({\mathbb Q} (\sqrt{-D}))\) of an imaginary quadratic number field acts on the set of isomorphism classes of elliptic curves having complex multiplication by \({\mathbb Q}(\sqrt{-D})\). The main objective of this paper is the computation of the comparable group action in class field theory for function fields. The action is expressed in terms of isogenies of Drinfeld modules.\N\NThe authors grow the algorithmic toolbox for isogenies of Drinfeld modules. The focus is on the relationship with class field theory of imaginary hyperelliptic function fields, building on the work of \textit{P. Caranay} et al. [Contemp. Math. 754, 283--313 (2020; Zbl 1469.11178)], of \textit{P. Caranay} [Computing isogeny volcanoes of rank two Drinfeld modules. University of Calgary (PhD Thesis) (2018)] and of \textit{Y. Musleh} and \textit{É. Schost} [ISSAC 2019, 307--314 (2019; Zbl 1467.11057)].\N\NThe main result is the design of an efficient algorithm to compute the group action in the class field theory of hyperelliptic function fields. The algorithm is easy to describe and implement: it relies on computing the right greatest common divisor of two Ore polynomials. The authors provide an asymptotic complexity bound.\N\NNext, it is proved that the inverse problem, that is, given \(x,y\), compute a group element \(g\) such that \(g\cdot x=y\), reduces to the problem of computing isogenies of Drinfeld modules by providing an algorithm computing the ideal in the coordinate ring of the hyperelliptic curve which corresponds to a given isogeny.\N\NFinally, asymptotic complexity bounds are given, and it is presented a concrete calculation of the group action is presented, using a C++/NTL implementation.