Cyclic covers of an algebraic curve from an adelic viewpoint
Luis Manuel Navas Vicente, Francisco J. Plaza Martin
TL;DR
This work develops a purely algebraic program to classify finite Galois covers of a curve $X$ by studying Galois ring extensions of its geometric adele ring $\mathbb{A}_{X}$ via the Harrison framework. Central to the approach are the Harrison groups $\mathbb{H}(R,G)$ and a $p$-Kummerian Kummer theory that links data on $\Sigma$ (the function field) with data on $\mathbb{A}_{X}$, yielding a fundamental exact sequence and a ramification-refined cube that connect algebraic and geometric ramification. The authors show how to recover classical geometric information—ramification, rotation data, and enumeration of $p$-cyclic covers—through purely algebraic constructions, and they connect with Borevich–Cornalba’s and Cornalba’s correspondences to classify and count covers via divisors and line bundles. In particular, they establish a robust algebraic analogue of the Grunwald–Wang problem for $p$-cyclic extensions and provide an algebraic notion of rotation data that matches the analytic notion over $\Bbbk=\mathbb{C}$, highlighting the method’s power and potential for broader abelian settings.
Abstract
We propose an algebraic method for the classification of branched Galois covers of a curve $X$ focused on studying Galois ring extensions of its geometric adele ring $\A_{X}$. As an application, we deal with cyclic covers; namely, we determine when a given cyclic ring extension of $\A_{X}$ comes from a corresponding cover of curves $Y \to X$, which is reminiscent of a Grunwald-Wang problem, and also determine when two covers yield isomorphic ring extensions, which is known in the literature as an equivalence problem. This completely algebraic method permits us to recover ramification, certain analytic data such as rotation numbers, and enumeration formulas for covers.