Arthur-Selberg trace formula

For a connected reductive group $G$ over a global field $F$ and a suitable adelic test function $f$ , the Arthur–Selberg trace formula is an identity

J_{\mathrm{geom}}(f)=J_{\mathrm{spec}}(f)

between a geometric expansion indexed by conjugacy data in $G(F)$ and a spectral expansion indexed by automorphic representations and Levi subgroups.

Kernel heuristic

Right convolution by $f$ has the formal automorphic kernel

K_f(x,y)=\sum_{\gamma\in G(F)}f(x^{-1}\gamma y).

If the automorphic quotient were compact, integrating $K_f(x,x)$ would give both a sum of orbital integrals and a sum of traces of automorphic representations. For a general reductive group, the quotient and the continuous spectrum make this naive integral diverge.

Arthur's truncation operator removes the divergent constant-term contributions and produces the actual formula.

Geometric side

The fine geometric expansion is a sum over Levi subgroups and conjugacy classes of coefficients times weighted orbital integrals. Elliptic regular terms resemble centralizer volumes multiplied by products of ordinary local orbital integrals. Unipotent and singular terms require separate distributions.

Spectral side

The spectral expansion contains the discrete automorphic spectrum of Levi subgroups together with normalized intertwining operators and integrals representing the continuous spectrum. Under simplifying support hypotheses it can reduce to a trace on the discrete spectrum, but that is not the general formula.

Invariant and stable forms

Arthur first reorganizes the truncated identity into an invariant trace formula. Stabilization then expresses its unstable terms through stable trace formulas of endoscopic groups. These are successive refinements, not synonyms for the initial coarse formula.

Uses

Comparing trace formulas proves instances of functoriality, constructs or classifies automorphic representations, computes cohomological traces, and links orbital geometry to spectral multiplicities.

References

James Arthur, “An introduction to the trace formula,” 2005. Clay Mathematics Proceedings.
James Arthur, “A trace formula for reductive groups I,” Duke Mathematical Journal 45 (1978), 911–952. DOI.