Cyclic covers of an algebraic curve from an Adelic viewpoint

Luis Manuel Navas Vicente; Francisco J. Plaza Martín

doi:10.56754/0719-0646.2802.261

DOI:

https://doi.org/10.56754/0719-0646.2802.261

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 \(\mathbb{A}_{X}\). As an application, we deal with cyclic covers; namely, we determine when a given cyclic ring extension of \(\mathbb{A}_{X}\) comes from a corresponding cover of curves \(Y\) → \(X\), 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.

Keywords

Coverings of curves , algebraic curves and function fields , Galois theory of commutative rings , geometric adele ring , Kummer theory

Mathematics Subject Classification:

14H30 , 13B05 , 14H05 , 11R56

Luis Manuel Navas Vicente Departamento de Matemáticas and IUFFyM, Universidad de Salamanca, Plaza de la Merced 1-4, 37008 Salamanca. Spain. https://orcid.org/0000-0002-5742-8679
Francisco J. Plaza Martín Departamento de Matemáticas and IUFFyM, Universidad de Salamanca, Plaza de la Merced 1-4, 37008 Salamanca. Spain. https://orcid.org/0000-0002-5532-7567

Pages: 261-293
Date Published: 2026-05-14
Vol. 28 No. 2 (2026)

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