Cyclic covers of an algebraic curve from an adelic viewpoint

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.

💡 Research Summary

The paper develops a purely algebraic framework for classifying finite Galois covers of a smooth projective curve (X) over an algebraically closed field (k). The authors replace the traditional analytic tools (Riemann surfaces, complex uniformization, etc.) with the study of Galois extensions of the geometric adele ring (\mathcal{A}_X). A finite Galois cover (Y\to X) with group (G) naturally yields a (G)-Galois extension of rings (\mathcal{A}_Y/\mathcal{A}_X). The set of isomorphism classes of such extensions is the Harrison set (H(R,G)) (Chase‑Harrison‑Rosenberg), which for abelian (G) carries a natural group structure.

Focusing on cyclic groups of prime order (p\neq\operatorname{char}k), the authors exploit a generalized Kummer exact sequence for any (p)-Kummerian ring (R): \