Schwartz–Bruhat space on the adeles - Imported Knowlpedia Content

Definition

Let $F$ be a number field and $\mathbb A_F$ its ring of adeles. The Schwartz–Bruhat space on the adeles is

\mathcal S(\mathbb A_F) =\mathcal S(F_\infty)\,\widehat\otimes\! \bigotimes_{v\nmid\infty}'\mathcal S(F_v),

where $\mathcal S(F_\infty)$ is the Schwartz space of the finite-dimensional real vector space $F_\infty=\prod_{v\mid\infty}F_v$ . The finite-place factor is the restricted tensor product of the local Schwartz–Bruhat spaces relative to $1_{\mathcal O_v}$ . Finite sums of products of local functions form a dense algebraic subspace; at several archimedean places they need not exhaust the completed space $\mathcal S(F_\infty)$ .

Fourier transform

Choose a nontrivial additive character $\psi:\mathbb A_F/F\to\mathbb T$ and compatible local Haar measures. The adelic Fourier transform factors on elementary tensors:

\widehat{\bigotimes_v f_v}=\bigotimes_v\widehat f_v.

For the standard unramified data, $\widehat{1_{\mathcal O_v}}=1_{\mathcal O_v}$ at almost every finite place, so the transform preserves the restricted tensor product. With self-dual measures it is an automorphism of $\mathcal S(\mathbb A_F)$ and satisfies Fourier inversion Weil, Chapter II.

Basic examples

For $F=\mathbb Q$ , there is one archimedean factor, and a typical elementary function is

f_\infty\otimes\bigotimes_p f_p,

where $f_\infty\in\mathcal S(\mathbb R)$ , each $f_p$ is locally constant and compactly supported on $\mathbb Q_p$ , and $f_p=1_{\mathbb Z_p}$ for all but finitely many primes. The constant function $1$ on $\mathbb A_{\mathbb Q}$ is not in the space because its archimedean factor is not rapidly decreasing.

Role in global harmonic analysis

The space $\mathcal S(\mathbb A_F)$ supplies the test functions in adelic Poisson summation and Tate's zeta integrals. The restricted tensor-product description makes a global integral factor into local integrals when both the function and measure are factorizable. This factorization is algebraic for elementary tensors and extends by linearity and continuity to the completed archimedean factor and the standard locally convex LF topology.

Conventions and scope

For a global function field there are no archimedean factors, and the same definition uses locally constant compactly supported functions at every place. Some sources denote this space by $\mathcal S(\mathbb A)$ or $\mathcal D(\mathbb A)$ . It is not the ordinary Euclidean Schwartz space on a finite-dimensional real vector space.

References

André Weil, Basic Number Theory, 2nd ed., Springer, 1973. DOI record. Relevant: Chapter II, adelic Schwartz functions and Fourier analysis.
John Tate, “Fourier Analysis in Number Fields and Hecke's Zeta-Functions,” in J. W. S. Cassels and A. Fröhlich, eds., Algebraic Number Theory, Academic Press, 1967, 305–347. Author-hosted scan. Relevant: §§2.2–2.4, adelic test functions, Fourier transform, and zeta integrals.