nLab
Selberg zeta function

Context

Theta functions

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in physics

Riemannian geometry

Basic definitions

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Further concepts

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Theorems

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Applications

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under construction

Contents

Idea

Motivated by the resemblance of the Selberg trace formula to Weil’s formula for the sum of zeros of the Riemann zeta function, (Selberg 56) defined for any compact hyperbolic Riemann surface a zeta function-like expression, the Selberg zeta function of a Riemann surface. (e.g. Bump, below theorem 19). There is also a Selberg zeta function “of odd type” for odd-dimensional manifolds (Millson 78, Bunke-Olbrich 94a, prop. 4.5).

Definition

For even-dimensional manifolds

(…)

For odd-dimensional manifolds

Let

Then the quotient

XΓ\G/K X \coloneqq \Gamma \backslash G / K

is a hyperbolic manifold of odd dimension with fundamental group being π 1(X)Γ\pi_1(X) \simeq \Gamma. Accordingly the representation χ\chi is equivalently a flat vector bundle on XX.

Write Conj(Γ)Conj(\Gamma) for the set of conjugacy classes of Γ\Gamma and write

Prim(Γ)Conj(Γ) Prim(\Gamma) \hookrightarrow Conj(\Gamma)

for the subset of elements [g][g] for which n Γ(g)=1n_\Gamma(g) = 1. Regarded as elements of the fundamental group as above, these elements correspond to paths which are prime geodesics in XX.

Definition

The Selberg zeta function ζ χ\zeta_\chi of this data is defined for Re(s)>ρ(n1)/2Re(s)\gt \rho \coloneqq (n-1)/2 to be the infinite product

ζ χ(s)=[g]Prim(Γ)[g]1k=0det(1e (ρ+s)l(g)S k(Ad(g) n 1)σ(m)χ(g)) \zeta_\chi(s) = \underset{{[g] \in Prim(\Gamma)} \atop {[g] \neq 1}}{\prod} \; \underoverset{k = 0}{\infty}{\prod} \det\left( 1 - e^{-(\rho + s)l(g)} S^k(Ad(g)^{-1}_{\mathbf{n}}) \sigma(m) \; \chi(g) \right)

(…) (BunkeOlbrich 94a, def. 4.1)

Properties

Analogy with Artin L-function

That the Selberg zeta function is equivalently an Euler product of characteristic polynomials is due to (Gangolli 77, (2.72), Fried 86, prop. 5).

That it is in particular the Euler product of characteristic polynomials of the determinants of the monodromies of the flat connection corresponding to the given group representation (similar to the Ruelle zeta function) is (Bunke-Olbrich 94, prop. 6.3) for the even-dimensional case and (Bunke-Olbrich 94a, def. 4.1) for the odd-dimensional case. (Or rather, the Ruelle zeta function (Bunke-Olbrich 94a, def. 5.1)).

This is analogous to the standard definition of an Artin L-function if one interprets a) a Frobenius map Frob pFrob_p (as discussed there) as an element of the arithmetic fundamental group of an arithmetic curve and b) a Galois representation as a flat connection.

So under this analogy the Selberg zeta function for hyperbolic 3-manifolds as well as the Artin L-function for a number field both are like an infinite product over primes (prime geodesics in one case, prime ideals in the other, see also at Spec(Z) – As a 3-dimensional space containing knots) of determinants of monodromies of the given flat connection.

See at Artin L-function – Analogy with Selberg zeta function for more. This analogy has been highlighted in (Brown 09, Morishita 12, remark 12.7).

Relation to the eta-function

Under suitable conditions, the Selberg zeta function of odd type is an exponential of the eta function of a suitable Dirac operator

ζ S(0)=exp(iπη D(0)) \zeta_S(0) = \exp\left(i \pi \eta_D(0)\right)

(Millson 78, Bunke-Olbrich 94a, prop. 4.5, Park 01, theorem 1.2, Guillarmou-Moroianu-Park 09).

Relation to analytic torsion

The Ruelle zeta function at 0 gives a power of analytic torsion

(Fried 86, Bunke-Olbrich 94a, theorem 5.5.)

Relation to prime geodesic asymptotics

The Selberg zeta function controls the asymptotics of prime geodesics via the prime geodesic theorem in direct analogy to how the Riemann zeta function controls the asymptotics of prime numbers via the prime number theorem.

context/ θ\theta ζ\zeta (= of θ(0,)\theta(0,-)) L zL_{\mathbf{z}} (= of θ(z,)\theta(\mathbf{z},-)) η\eta
/ θ(z,τ)=Tr(exp(τ(D z) 2))\theta(\mathbf{z},\mathbf{\tau}) = Tr(\exp(-\mathbf{\tau} \cdot (D_\mathbf{z})^2)) as function of τ\mathbf{\tau} of Σ\Sigma (hence of ) and / z\mathbf{z}analytically continued of ζ(s)=Tr reg(1(D 0) 2) s= 0 τ s1θ(0,τ)dτ\zeta(s) = Tr_{reg}\left(\frac{1}{(D_{0})^2}\right)^s = \int_{0}^\infty \tau^{s-1} \,\theta(0,\tau)\, d\tauanalytically continued of in z\mathbf{z}: L z(s)Tr reg(1(D z) 2) s= 0 τ s1θ(z,τ)dτL_{\mathbf{z}}(s) \coloneqq Tr_{reg}\left(\frac{1}{(D_{\mathbf{z}})^2}\right)^s = \int_{0}^\infty \tau^{s-1} \,\theta(\mathbf{z},\tau)\, d\tauanalytically continued of in z\mathbf{z} η z(s)=Tr reg(sgn(D z)|D z|) s\eta_{\mathbf{z}}(s) = Tr_{reg} \left(\frac{sgn(D_{\mathbf{z}})}{ { \vert D_{\mathbf{z}} } \vert }\right)^s pvL z(1)=Tr reg(1(D z) 2)pv\, L_{\mathbf{z}}(1) = Tr_{reg}\left(\frac{1}{(D_{\mathbf{z}})^2}\right) / fermionic pvη z(1)=Tr reg(D z(D z) 2)pv\, \eta_{\mathbf{z}}(1)= Tr_{reg} \left( \frac{D_{\mathbf{z}}}{(D_{\mathbf{z}})^2} \right) / 12L z (0)=Z H=12lndet reg(D z 2)-\frac{1}{2}L_{\mathbf{z}}^\prime(0) = Z_H = \frac{1}{2}\ln\;det_{reg}(D_{\mathbf{z}}^2)
(),
θ(z,τ)\theta(\mathbf{z},\mathbf{\tau}) of over J(Σ τ)J(\Sigma_{\mathbf{\tau}}) in terms of covering coordinates z\mathbf{z} on gJ(Σ τ)\mathbb{C}^g \to J(\Sigma_{\mathbf{\tau}})
for a (for ) and (in )
for a , (being the L zL_{\mathbf{z}} for z=0\mathbf{z} = 0 the ) L zL_{\mathbf{z}} of a z\mathbf{z}, expressible “in coordinates” (by ) as a finite-order (for 1-dimensional representations) and generally (via ) by an (for higher dimensional reps) \cdot
for \mathbb{Q} (z=0\mathbf{z} = 0)/ (z=χ\mathbf{z} = \chi a ) (being the L zL_{\mathbf{z}} for z=0\mathbf{z} = 0) of a z\mathbf{z} , expressible “in coordinates” (via ) as a (for 1-dimensional Galois representations) and generally (via ) as an

References

The original article is

  • Atle Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, Journal of the Indian Mathematical Society 20 (1956) 47-87.

Review includes

Expression of the Selberg/Ruelle zeta function as an Euler product of characteristic polynomials is due to

  • Ramesh Gangolli, Zeta functions of Selberg’s type for compact space forms of symmetric spaces of rank one, Illinois J. Math. Volume 21, Issue 1 (1977), 1-41. (Euclid)

  • David Fried, The zeta functions of Ruelle and Selberg. I, Annales scientifiques de l’École Normale Supérieure, Sér. 4, 19 no. 4 (1986), p. 491-517 (Numdam)

The analogy with the Artin L-function is highlighted in

  • Darin Brown, Lifting properties of prime geodesics, Rocky Mountain J. Math. Volume 39, Number 2 (2009), 437-454 (euclid)

  • Masanori Morishita, section 12.1 of Knots and Primes: An Introduction to Arithmetic Topology, 2012 (web)

Discussion of the relation between, on the one hand, zeta function of Laplace operators/eta funcstions of Dirac operators and, on the other hand, Selberg zeta functions includes

  • Eric D'HokerDuong Phong, Communications in Mathematical Physics, Volume 104, Number 4 (1986), 537-545 (Euclid)

  • Peter Sarnak, Determinants of Laplacians, Communications in Mathematical Physics, Volume 110, Number 1 (1987), 113-120. (Euclid)

  • Ulrich Bunke, Martin Olbrich, Andreas Juhl, The wave kernel for the Laplacian on the classical locally symmetric spaces of rank one, theta functions, trace formulas and the Selberg zeta function, Annals of Global Analysis and Geometry February 1994, Volume 12, Issue 1, pp 357-405

  • Ulrich Bunke, Martin Olbrich, Theta and zeta functions for locally symmetric spaces of rank one (arXiv:dg-ga/9407013)

and for odd-dimensional spaces also in

Last revised on July 18, 2015 at 04:24:36. See the history of this page for a list of all contributions to it.