Math 2.0

Number theory Theorem

Selberg trace formula

Builds on Fuchsian groupUpper half-planeRiemann surfaceLaplace–Beltrami operatorEigenvalueGeodesicConjugacy class

Statement

Let Γ⊂PSL2(ℝ)\Gamma \subset \mathrm{PSL}_2(\R) be a torsion-free Fuchsian group such that X=Γ\ℍX = \Gamma \backslash \mathbb H is compact, where ℍ\mathbb H is the upper half-plane with the hyperbolic metric (dx2+dy2)/y2(dx^2 + dy^2)/y^2. Let 0=λ0<λ1≤λ2≤⋯0 = \lambda_0 < \lambda_1 \le \lambda_2 \le \cdots be the eigenvalues of the Laplace–Beltrami operator Δ=−y2(∂x2+∂y2)\Delta = -y^2 (\partial_x^2 + \partial_y^2) on XX, repeated with multiplicity, and write λj=14+rj2\lambda_j = \tfrac14 + r_j^2 with rj∈ℂr_j \in \C. Let hh be an even function, holomorphic on a strip ∣Im⁡r∣≤12+ε\abs{\operatorname{Im} r} \le \tfrac12 + \varepsilon with ε>0\varepsilon > 0, such that ∣h(r)∣≤C(1+∣r∣)−2−ε\abs{h(r)} \le C (1 + \abs{r})^{-2-\varepsilon} there. Let g(u)=12π∫−∞∞h(r)e−iru drg(u) = \frac{1}{2\pi} \int_{-\infty}^{\infty} h(r) e^{-iru} \, dr. Then ∑j=0∞h(rj)=area⁡(X)4π∫−∞∞r h(r)tanh⁡(πr) dr+∑γ∑k=1∞ℓ(γ)2sinh⁡(kℓ(γ)/2) g(kℓ(γ)),\sum_{j=0}^{\infty} h(r_j) = \frac{\operatorname{area}(X)}{4\pi} \int_{-\infty}^{\infty} r \, h(r) \tanh(\pi r) \, dr + \sum_{\gamma} \sum_{k=1}^{\infty} \frac{\ell(\gamma)}{2 \sinh(k \ell(\gamma)/2)} \, g(k \ell(\gamma)), where γ\gamma runs over the primitive oriented closed geodesics of XX and ℓ(γ)\ell(\gamma) is the length of γ\gamma. All sums and integrals converge absolutely.

Remarks

The closed geodesics of XX correspond to the nontrivial conjugacy classes of Γ\Gamma, and kℓ(γ)k \ell(\gamma) is the length of the kk-th power. The proof computes the trace of an integral operator on L2(X)L^2(X) in two ways. The kernel ∑σ∈Γk(z,σw)\sum_{\sigma \in \Gamma} k(z, \sigma w) is built from a point-pair invariant kk whose spectral transform is hh. Expanding in eigenfunctions gives the left side; integrating the kernel over the diagonal and grouping σ\sigma by conjugacy classes gives the right side, with the identity giving the area term. Example: h(r)=e−t(r2+1/4)h(r) = e^{-t(r^2 + 1/4)} with t>0t > 0 gives the trace of the heat kernel ∑je−tλj\sum_j e^{-t \lambda_j}, and the area term behaves like area⁡(X)/(4πt)\operatorname{area}(X)/(4\pi t) as t→0t \to 0. This yields Weyl's law: the number of λj≤T\lambda_j \le T is asymptotic to area⁡(X) T/(4π)\operatorname{area}(X) \, T / (4\pi). For non-compact XX of finite area, such as SL2(ℤ)\ℍ\mathrm{SL}_2(\Z) \backslash \mathbb H, the formula needs extra terms from the cusps and from the continuous spectrum.

Going deeper

For a student

The formula is a non-abelian analogue of the Poisson summation formula, which links a lattice in ℝ\R with its dual lattice. It gives the prime geodesic theorem, the analytic properties of the Selberg zeta function, and Weyl's law for XX. Applied to congruence subgroups with Hecke operators inserted, it links Maass forms with arithmetic, and it is the prototype of the Arthur–Selberg trace formula.

Reading path

To understand this page from the ground up, read these 20 pages in order. The foundations come first.

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Complex numbers, object, no evidence levelComplex numbersobjectConjugacy class, concept, no evidence levelConjugacy classconceptEigenvalue, concept, no evidence levelEigenvalueconceptFuchsian group, concept, no evidence levelFuchsian groupconceptGeneral linear group, object, no evidence levelGeneral linear groupobjectGeodesic, concept, no evidence levelGeodesicconceptIdentity matrix, object, no evidence levelIdentity matrixobjectLaplace–Beltrami operator, concept, no evidence levelLaplace–Beltrami oper…conceptMatrix, definition, no evidence levelMatrixdefinitionMöbius transformation, concept, no evidence levelMöbius transformationconceptRiemann surface, concept, no evidence levelRiemann surfaceconceptVector space, definition, no evidence levelVector spacedefinitionSquare matrix, definition, no evidence levelSquare matrixdefinitionLinear map, definition, no evidence levelLinear mapdefinitionMultilinear map, definition, no evidence levelMultilinear mapdefinitionAlternating multilinear map, definition, no evidence levelAlternating multiline…definitionDeterminant, definition, no evidence levelDeterminantdefinitionSpecial linear group, object, no evidence levelSpecial linear groupobjectUpper half-plane, object, no evidence levelUpper half-planeobjectSelberg trace formula, statement, no evidence levelSelberg trace formulastatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Complex numbers object
  2. Conjugacy class concept
  3. Eigenvalue concept
  4. Fuchsian group concept
  5. General linear group object
  6. Geodesic concept
  7. Identity matrix object
  8. Laplace–Beltrami operator concept
  9. Matrix definition
  10. Möbius transformation concept
  11. Riemann surface concept
  12. Vector space definition
  13. Square matrix definition
  14. Linear map definition
  15. Multilinear map definition
  16. Alternating multilinear map definition
  17. Determinant definition
  18. Special linear group object
  19. Upper half-plane object
  20. Selberg trace formula statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Selberg_trace_formula
About this page
TagBDAWKPBNQP
URIm2://BDAWKPBNQP
Kindstatement, theorem
WritingW0 stub
MSC11F72, 58J50, 30F35
StatusThis copy of the site has no build data. Run m2 status BDAWKPBNQP.
AboutLaplace–Beltrami operator, Fuchsian group
UsesFuchsian group, Upper half-plane, Riemann surface, Laplace–Beltrami operator, Eigenvalue, Geodesic, Conjugacy class
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