nLab Selberg zeta function

under construction

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

• $n \in \mathbb{N}$, $n \geq 1$;

• $G = SO(n,1)$ the Lorentz group or $G = Spin(n,1)$ its spin group;

• $K = SO(n)$ the special orthogonal group or $K = Spin(n)$ the spin group;

• $K \hookrightarrow G$ the canonical inclusion, exhibiting a maximal compact subgroup;

• $G = K A N$ an Iwasawa decomposition;

• $\Gamma \hookrightarrow G$ a discrete subgroup;

• $E$ a complex finite dimensional vector space

• $\chi \colon \Gamma \to U(E)$ a unitary representation of $\Gamma$.

Then the quotient

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

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

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

$$Prim(\Gamma) \hookrightarrow Conj(\Gamma)$$

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

Definition

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

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

(…) (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 $Frob_p$ (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

$$\zeta_S(0) = \exp\left(i \pi \eta_D(0)\right)$$

(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.)

• analytic torsion – relation to Selberg zeta function

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.

Related concepts

• Ruelle zeta function

context/function field analogy	theta function $\theta$	zeta function $\zeta$ (= Mellin transform of $\theta(0,-)$)	L-function $L_{\mathbf{z}}$ (= Mellin transform of $\theta(\mathbf{z},-)$)	eta function $\eta$	special values of L-functions
physics/2d CFT	partition function $\theta(\mathbf{z},\mathbf{\tau}) = Tr(\exp(-\mathbf{\tau} \cdot (D_\mathbf{z})^2))$ as function of complex structure $\mathbf{\tau}$ of worldsheet $\Sigma$ (hence polarization of phase space) and background gauge field/source $\mathbf{z}$	analytically continued trace of Feynman propagator $\zeta(s) = Tr_{reg}\left(\frac{1}{(D_{0})^2}\right)^s = \int_{0}^\infty \tau^{s-1} \,\theta(0,\tau)\, d\tau$	analytically continued trace of Feynman propagator in background gauge field $\mathbf{z}$: $L_{\mathbf{z}}(s) \coloneqq Tr_{reg}\left(\frac{1}{(D_{\mathbf{z}})^2}\right)^s = \int_{0}^\infty \tau^{s-1} \,\theta(\mathbf{z},\tau)\, d\tau$	analytically continued trace of Dirac propagator in background gauge field $\mathbf{z}$ $\eta_{\mathbf{z}}(s) = Tr_{reg} \left(\frac{sgn(D_{\mathbf{z}})}{ { \vert D_{\mathbf{z}} } \vert }\right)^s$	regularized 1-loop vacuum amplitude $pv\, L_{\mathbf{z}}(1) = Tr_{reg}\left(\frac{1}{(D_{\mathbf{z}})^2}\right)$ / regularized fermionic 1-loop vacuum amplitude $pv\, \eta_{\mathbf{z}}(1)= Tr_{reg} \left( \frac{D_{\mathbf{z}}}{(D_{\mathbf{z}})^2} \right)$ / vacuum energy $-\frac{1}{2}L_{\mathbf{z}}^\prime(0) = Z_H = \frac{1}{2}\ln\;det_{reg}(D_{\mathbf{z}}^2)$
Riemannian geometry (analysis)		zeta function of an elliptic differential operator	zeta function of an elliptic differential operator	eta function of a self-adjoint operator	functional determinant, analytic torsion
complex analytic geometry	section $\theta(\mathbf{z},\mathbf{\tau})$ of line bundle over Jacobian variety $J(\Sigma_{\mathbf{\tau}})$ in terms of covering coordinates $\mathbf{z}$ on $\mathbb{C}^g \to J(\Sigma_{\mathbf{\tau}})$	zeta function of a Riemann surface	Selberg zeta function		Dedekind eta function
arithmetic geometry for a function field		Goss zeta function (for arithmetic curves) and Weil zeta function (in higher dimensional arithmetic geometry)
arithmetic geometry for a number field	Hecke theta function, automorphic form	Dedekind zeta function (being the Artin L-function $L_{\mathbf{z}}$ for $\mathbf{z} = 0$ the trivial Galois representation)	Artin L-function $L_{\mathbf{z}}$ of a Galois representation $\mathbf{z}$, expressible “in coordinates” (by Artin reciprocity) as a finite-order Hecke L-function (for 1-dimensional representations) and generally (via Langlands correspondence) by an automorphic L-function (for higher dimensional reps)		class number $\cdot$ regulator
arithmetic geometry for $\mathbb{Q}$	Jacobi theta function ($\mathbf{z} = 0$)/ Dirichlet theta function ($\mathbf{z} = \chi$ a Dirichlet character)	Riemann zeta function (being the Dirichlet L-function $L_{\mathbf{z}}$ for Dirichlet character $\mathbf{z} = 0$)	Artin L-function of a Galois representation $\mathbf{z}$ , expressible “in coordinates” (via Artin reciprocity) as a Dirichlet L-function (for 1-dimensional Galois representations) and generally (via Langlands correspondence) as an automorphic L-function

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

• Wikipedia, Selberg zeta function

• Matthew Watkins, citation collection on Selberg trace formula and zeta functions

• Bump, below theorem 19 in Spectral theory of $\Gamma \backslash SL(2,\mathbb{R})$ (pdf)

• Selberg and Ruelle zeta functions for compact hyperbolic manifolds (pdf)

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

• David Fried, Analytic torsion and closed geodesics on hyperbolic manifolds, Invent. math. 84, 523-540 (1986) (pdf)

• John Millson, Closed geodesic and the $\eta$-invariant, Ann. of Math., 108, (1978) 1-39 (jstor)

• Ulrich Bunke, Martin Olbrich, Theta and zeta functions for odd-dimensional locally symmetric spaces of rank one (arXiv:dg-ga/9407012)

• Ulrich Bunke, Martin Olbrich $\Gamma$-Cohomology and the Selbeg zeta function (arXiv:dg-ga/9411004)

• Ulrich Bunke, Martin Olbrich, Group cohomology and the singularities of the Selberg zeta function associated to a Kleinian group (arXiv:dg-ga/9603003)

• Ulrich Bunke, Martin Olbrich, Selberg zeta and theta functions: a differential operator approach, Akademie Verlag 1995

• Jinsung Park, Eta invariants and regularized determinants for odd dimensional hyperbolic manifolds with cusps (arXiv:0111175)

• Joshua Friedman, The Selberg trace formula and Selberg zeta-function for cofinite Kleinian groups with finite-dimensional unitary representations (arXiv:math/0410067)

• Joshua Friedman, Regularized determinants of the Laplacian for cofinite Kleinian groups with finite-dimensional unitary representations, Communications in Mathematical Physics (arXiv:math/0605288)

• Colin Guillarmou, Sergiu Moroianu, Jinsung Park, Eta invariant and Selberg Zeta function of odd type over convex co-compact hyperbolic manifolds (arXiv:0901.4082)