Kummer theory over the geometric adeles of an algebraic curve
Our goal is to give a purely algebraic characterization of finite abelian Galois covers of a complete, irreducible, non-singular curve $X$ over an algebraically closed field $\k$. To achieve this, we make use of the Galois theory of commutative rings, in particular the Kummer theory of the ring of geometric adeles $\A_{X}$. After we establish the triviality of the Picard group $\Pic(\A_{X})$, the general Kummer sequence for Kummerian rings leads to a characterization of $p$-cyclic extensions of $\A_{X}$ in terms of the closed points of $X$. This is an example of a general local-global principle which we use throughout, allowing us to avoid needing the full spectrum of $\A_{X}$. We prove the existence of primitive elements in $p$-cyclic extensions of $\A_{X}$, which yields explicit invariants lying in $\bigoplus_{x \in X} \Zp$ (summing over closed points) classifying them. From a group-theoretical point of view, we give a complete characterization of which $p$-cyclic subgroups of the full automorphism group of a given $p$-cyclic extension of $\A_{X}$ endow it with a Galois structure. The result is a stratification by the algebraic ramification of the extension modulo a notion of conjugation or twisting of Galois structures, yielding other invariants, in the form of finite tuples over ramified points, which are related to the previous ones in terms of the local Kummer symbols. With these results in hand, a forthcoming paper will identify, inside the set of $p$-cyclic extensions of $\A_{X}$, those arising from extensions of the function field of the curve $X$, eventually leading to the algebraic characterization of abelian covers of $X$.