Arthur–Selberg trace formula

Builds on Selberg trace formula Adele ring Haar measure Trace class operator Conjugacy class Automorphic form

Statement

Let $F$ be a number field with adele ring $\mathbb A$ , and let $G$ be a connected reductive group over $F$ . Let $G(\mathbb A)^1$ be the common kernel of the maps $x \mapsto \abs{\chi(x)}$ for the rational characters $\chi$ of $G$ , and assume that $G(F) \backslash G(\mathbb A)^1$ is compact. Let $R$ be the right regular representation of $G(\mathbb A)^1$ on $L^2(G(F) \backslash G(\mathbb A)^1)$ . For a smooth compactly supported function $f$ on $G(\mathbb A)^1$ , let $R(f) = \int f(y) R(y) \, dy$ , with a fixed Haar measure. Then $R(f)$ is a trace class operator. Its trace is given by $\sum_{\pi} m(\pi) \operatorname{tr} \pi(f) = \sum_{\gamma} \operatorname{vol}\big(G_\gamma(F) \backslash G_\gamma(\mathbb A)^1\big) \int_{G_\gamma(\mathbb A) \backslash G(\mathbb A)^1} f(x^{-1} \gamma x) \, dx.$ Here $\pi$ runs over the irreducible unitary representations of $G(\mathbb A)^1$ , and $m(\pi)$ is the multiplicity of $\pi$ in $R$ . The index $\gamma$ runs over representatives of the conjugacy classes of $G(F)$ , and $G_\gamma$ is the centralizer of $\gamma$ . Both sides equal $\operatorname{tr} R(f)$ .

Remarks

This compact-quotient case is the direct generalisation of the Selberg trace formula. The proof writes $R(f)$ as an integral operator with kernel $K(x, y) = \sum_{\gamma \in G(F)} f(x^{-1} \gamma y)$ and integrates the kernel over the diagonal. Grouping the terms by conjugacy class gives the geometric side; the discrete decomposition of $L^2$ gives the spectral side. For general $G$ , such as $\mathrm{GL}_n$ with $n \ge 2$ , the quotient is not compact and $R(f)$ is not trace class. Arthur's trace formula then replaces both sides by truncated distributions $J_{\mathrm{geom}}(f) = J_{\mathrm{spec}}(f)$ , which contain extra terms from the continuous spectrum and from non-semisimple conjugacy classes. The spectral side encodes automorphic forms, the geometric side encodes orbital integrals, and comparing the formulas for two groups is the main tool for proving cases of Langlands functoriality.

Going deeper

For a student

The Selberg trace formula is the case of a cocompact Fuchsian group, seen through the real place only. Jacquet and Langlands used a trace formula comparison to relate automorphic forms on $\mathrm{GL}_2$ and on a quaternion algebra. The stabilization of the trace formula, completed with the fundamental lemma, allowed Arthur to classify the automorphic representations of classical groups.

Reading path

To understand this page from the ground up, read these 25 pages in order. The foundations come first.

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Adele ring concept
Automorphic form concept
Complex numbers object
Conjugacy class concept
Eigenvalue concept
Fuchsian group concept
General linear group object
Geodesic concept
Haar measure concept
Identity matrix object
Laplace–Beltrami operator concept
Matrix definition
Möbius transformation concept
Riemann surface concept
Trace class operator concept
Vector space definition
Square matrix definition
Linear map definition
Multilinear map definition
Alternating multilinear map definition
Determinant definition
Special linear group object
Upper half-plane object
Selberg trace formula statement
Arthur–Selberg trace formula statement

Tag	`BCETMS8K3X`
URI	`m2://BCETMS8K3X`
Kind	statement, theorem
Writing	W0 stub
MSC	11F72, 11F70, 22E55
Status	This copy of the site has no build data. Run `m2 status BCETMS8K3X`.
About	Automorphic form
Uses	Selberg trace formula, Adele ring, Haar measure, Trace class operator, Conjugacy class, Automorphic form
Same as	Wikidata Q4800826 Wikipedia Arthur–Selberg trace formula