Characterization of subfields of adelic algebras by a product formula
Luis Manuel Navas Vicente, Francisco J. Plaza Martin
TL;DR
The paper develops a topological framework for geometric adeles on curves and uses it to characterize when subfields of an adelic algebra $\mathbb{A}_{X}\lbrace\mathbbm{p}\rbrace = \mathbb{A}_{X}[T]/(\mathbbm{p}(T))$ arise from covers $Y\to X$. Central results prove that the natural map $\mathbb{A}_{Y} \to \mathbb{A}_{X}\lbrace\mathbbm{p}\rbrace$ is injective if and only if the embedded field $\Omega/\Sigma$ is discrete, and equivalently, that $\Omega$ satisfies an additive product-formula via a content function $V$ on $\mathbb{A}_{X}\lbrace\mathbbm{p}\rbrace^{*}$. The analysis hinges on expressing $\mathbb{A}_{X}\lbrace\mathbbm{p}\rbrace$ as a restricted direct product of local algebras $K_{x}\lbrace\mathbbm{p}_{x}\rbrace$, with integral closures $\mathfrak{B}_{x}$, and equipping it with a natural linear topology defined by $\mathfrak{n}_{x}$-adic neighborhoods, ultimately proving freeness over $\mathbb{A}_{X}$ and a precise discrete-subfield criterion. The additive content-based product formula generalizes classical Artin–Whaples and Iwasawa-type characterizations of global fields to the geometric adelic setting, establishing a bridge between discrete subfields and global reciprocity-type structures.
Abstract
We consider projective, irreducible, non-singular curves over an algebraically closed field $\k$. A cover $Y \to X$ of such curves corresponds to an extension $Ω/Σ$ of their function fields and yields an isomorphism $\A_{Y} \simeq \A_{X} \otimes_Σ Ω$ of their geometric adele rings. The primitive element theorem shows that $\A_{Y}$ is a quotient of $\A_{X}[T]$ by a polynomial. In general, we may look at quotient algebras $\AXp{\p} = \A_{X}[T]/(\p(T))$ where $\p(T) \in \A_{X}[T]$ is monic and separable over $\A_{X}$, and try to characterize the field extensions $Ω/Σ$ lying in $\AXp{\p}$ which arise from covers as above. We achieve this topologically, namely, as those $Ω$ which embed discretely in $\AXp{\p}$, and in terms of an additive analog of the product formula for global fields, a result which is reminiscent of classical work of Artin-Whaples and Iwasawa. The technical machinery requires studying which topology on $\AXp{\p}$ is natural for this problem. Local compactness no longer holds, but instead we have linear topologies defined by commensurability of $\k$-subspaces which coincide with the restricted direct product topology with respect to integral closures. The content function is given as an index measuring the discrepancy in commensurable subspaces.