Restricted tensor product of local test-function spaces

Definition

Let $F$ be a number field, $F_v$ its completions, and $\mathcal S(F_v)$ the local Schwartz–Bruhat spaces. Put $F_\infty=\prod_{v\mid\infty}F_v$ , and let $\mathcal S(F_\infty)$ be the completed nuclear tensor product of the archimedean local Schwartz spaces. For each finite place set $e_v=1_{\mathcal O_v}$ . The restricted tensor product of the local test-function spaces is

\mathcal S(F_\infty)\,\widehat\otimes\! \bigotimes_{v\nmid\infty}'\mathcal S(F_v) =\underset{S}{\mathop{\mathrm{colim}}}\, \left(\mathcal S(F_\infty)\widehat\otimes \bigotimes_{v\in S}\mathcal S(F_v)\right) \otimes\left(\bigotimes_{\substack{v\nmid\infty\\v\notin S}} \mathbb C e_v\right),

where $S$ ranges over finite sets of finite places. The vectors $e_v$ are part of the definition.

Canonical realization

An elementary tensor determines a function on the restricted product $\mathbb A_F=\prod_v'F_v$ by

x=(x_v)_v\longmapsto\prod_v f_v(x_v).

Only finitely many factors differ from $1_{\mathcal O_v}$ , and an adele lies in $\mathcal O_v$ at almost every finite place, so this product is well defined. Finite sums of completely separated local functions realize a dense algebraic subspace of the adelic test-function space. They need not exhaust $\mathcal S(F_\infty)$ when there is more than one archimedean place.

Topology

For a fixed finite $S$ , the unrestricted factors form a completed nuclear test-function tensor product, while factors outside $S$ remain fixed. The adelic Schwartz topology is the standard locally convex inductive-limit (LF) topology over these finite sets. Completely separated elementary tensors are dense in each archimedean completion and hence in the resulting adelic space Weil, Chapter II, “Adeles”.

Dependence on distinguished vectors

A restricted tensor product is not determined by the spaces $\mathcal S(F_v)$ alone; the vectors $e_v$ are part of its data. For the additive adeles, $1_{\mathcal O_v}$ is the standard unramified choice. Replacing finitely many $e_v$ gives the same restricted tensor-product space up to its evident canonical identification, whereas changing infinitely many can produce a different restricted product.

References

André Weil, Basic Number Theory, 2nd ed., Springer, 1973. DOI record. Relevant: Chapter II, “Adeles,” restricted products and adelic test functions.
Daniel Bump, Automorphic Forms and Representations, Cambridge University Press, 1997. DOI record. Relevant: Chapter 3, restricted tensor products in adelic harmonic analysis.