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Werner Mueller

Publications and source records attributed to Werner Mueller.

30 records · Page 2Linked to original sources
arXiv2011

On the torsion in the cohomology of arithmetic hyperbolic 3-manifolds

In this paper we consider the cohomology of a closed arithmetic hyperbolic 3-manifold with coefficients in the local system defined by the even symmetric powers of the standard representation of SL(2,C). The cohomology is defined over the integers and is a finite abelian group. We show that the order of the 2nd cohomology grows exponentially as the local system grows. We also consider the twisted Ruelle zeta function of a closed arithmetic hyperbolic 3-manifold and we express the leading coefficient of its Laurent expansion at the origin in terms of the orders of the torsion subgroups of the cohomology.

math.NT↗
arXiv2010

The asymptotics of the Ray-Singer analytic torsion of hyperbolic 3-manifolds

In this paper we consider the analytic torsion of a closed hyperbolic 3-manifold associated with the m-th symmetric power of the standard representation of SL(2,C) and we study its asymptotic behavior if m tends to infinity. The leading coefficient of the asymptotic formula is given by the volume of the hyperbolic 3-manifold. It follows that the Reidemeister torsion associated with the symmetric powers determines the volume of a closed hyperbolic 3-manifold.

math.SP↗
arXiv2009

The Selberg trace formula for non-unitary representations of the lattice

For a compact locally symmetric space X, we establish a version of the Selberg trace fromula for a non-unitary representation of the fundamental group of X. On the spectral side appears the spectrum of the "flat Laplacian", acting in the space of sections of the associated flat bundle. In general, this is a non-self-adjoint operator with the same principal symbol as the Laplace operator associated to any choice of a Hermitian fiber metric in the flat bundle.

math.SP↗
arXiv2007

Spectral asymptotics for arithmetic quotients of SL(n,R)/SO(n)

In this paper we study the asymptotic distribution of the cuspidal spectrum of arithmetic quotients of the symmetric space S=SL(n,R)/SO(n). In particular, we obtain Weyl's law with an estimation on the remainder term. This extends results of Duistermaat-Kolk-Varadarajan on spectral asymptotics for compact locally symmetric spaces to this non-compact setting.

math.RT↗
arXiv2007

Weyl's law in the theory of automorphic forms

For a compact Riemannian manifold, Weyl's law describes the asymptotic behavior of the counting function of the eigenvalues of the associated Laplace operator. In this paper we discuss Weyl's law in the context of automorphic forms. The underlying manifolds are locally symmetric spaces of finite volume. In the non-compact case Weyl's law is closely related to the problem of existence of cusp forms.

math.SP↗
arXiv2007

A spectral interpretation of the zeros of the constant term of certain Eisenstein series

In this paper we consider the constant term $ϕ_K(y,s)$ of the non-normalized Eisenstein series attached to $\PSL(2,\cO_K)$, where $K$ is either $\Q$ or an imaginary quadratic field of class number one. The main purpose of this paper is to show that for every $a\ge 1$ the zeros of the Dirichlet series $ϕ_K(a,s)$ admit a spectral interpretation in terms of eigenvalues of a natural self-adjoint operator $Δ_a$. This implies that, except for at most two real zeros, all zeros of $ϕ_K(a,s)$ are on the critical line, and all zeros are simple. For $K=\Q$ this is due to Lagarias and Suzuki and Ki.

math.NT↗
arXiv2007

Scattering theory for the Laplacian on manifolds with bounded curvature

In this paper we study the behaviour of the continuous spectrum of the Laplacian on a complete Riemannian manifold of bounded curvature under perturbations of the metric. The perturbations that we consider are such that its covariant derivatives up to some order decay with some rate in the geodesic distance from a fixed point. Especially we impose no conditions on the injectivity radius. One of the main results are conditions on the rate of decay, depending on geometric properties of the underlying manifold, that guarantee the existence and completeness of the wave operators.

math.SP↗
arXiv2004

Regularized determinants of Laplace type operators, analytic surgery and relative determinants

We study the asymptotic behaviour of regularized determinants of certain Laplace type operators with respect to singular deformations of the underlying manifold which are obtained by stretching a tubular neighborhood of an embedded separating hypersurface to a cylinder of infinite length. Using the asymptotic expansions, we derive adiabatic splitting formulas for regularized determinants of Laplace type operators.

math.DG↗
arXiv2004

Weak Weyl's law for congruence subgroups

Let $G$ be a connected and simply connected semisimple algebraic group over $\Bbb Q$ and let $Γ\subset G(\Bbb Q)$ be an arithmetic subgroup. Let $K_\infty\subset G(\Bbb R)$ be a maximal compact subgroup and let $d$ be the dimension of the symmetric space $G({\mathbb R})/K_\infty$. Let $σ$ be an irreducible unitary representation of $K_\infty$. We prove that for every $Γ$ there exists a normal subgroup $Γ_1\subset Γ$ of finite index such that the quotient of the counting function of the $Γ_1$-cuspidal spectrum of weight $σ$ and $T^{d/2}$ has a positive lower bound as $T\to\infty$.

math.RT↗
arXiv2003

Weyl's law for the cuspidal spectrum of SL(n)

Let $Γ$ be a principal congruence subgroup of $SL_n(Z)$ and let $σ$ be an irreducible representation of SO(n). Let $N(T,σ)$ be the counting function of the eigenvalues of the Casimir operator acting in the space of cusp forms for $Γ$ which transform under SO(n) according to $σ$. We prove that the counting function $N(T,σ)$ satisfies Weyl's law as $T\to\infty$. Especially this implies that there exist infinitely many cusp forms for the full modular group $SL_n(Z)$.

math.RT↗
arXiv1999

Extremal Kaehler metrics and Ray-Singer analytic torsion

Let (X,[ω]) be a compact Kaehler manifold with a fixed Kaehler class [ω]. Let K_ωbe the set of all Kaehler metrics on X whose Kaehler class equals [ω]. In this paper we investigate the critical points of the functional Q(g)= |v|_g T_0(X,g)^{1/2} for g \in K_ω, where v is a fixed nonzero vector of the determinant line λ(X) associated to H^*(X) and T_0(X,g) is the Ray-Singer analytic torsion. For a polarized algebraic manifold (X,L) we consider a twisted version Q_L(g) of this functional and assume that c_1(L)=[ω]. Then the critical points of Q_L are exactly the metrics g\in K_ωof constant scalar curvature. In particular, if c_1(X)=0 or if c_1(X)<0 and 1/(2π)[ω] = -c_1(X), then K_ωcontains a unique Kaehler-Einstein metric g_{KE} and Q_L attains its absolut maximum at g_{KE}.

math.DG↗