Automorphic forms and rational homology 3--spheres

Abstract

We investigate a question of Cooper adjacent to the Virtual Haken Conjecture. Assuming certain conjectures in number theory, we show that there exist hyperbolic rational homology 3-spheres with arbitrarily large injectivity radius. These examples come from a tower of abelian covers of an explicit arithmetic 3-manifold. The conjectures we must assume are the Generalized Riemann Hypothesis and a mild strengthening of results of Taylor et al on part of the Langlands Program for GL_2 of an imaginary quadratic field. The proof of this theorem involves ruling out the existence of an irreducible two dimensional Galois representation (rho) of Gal(Qbar/Q(sqrt(-2))) satisfying certain prescribed conditions. In contrast to similar questions of this form, (rho) is allowed to have arbitrary ramification at some prime of Z[sqrt(-2)]. Finally, we investigate the congruence covers of twist-knot orbifolds. Our experimental evidence suggests that these topologically similar orbifolds have rather different behavior depending on whether or not they are arithmetic. In particular, the congruence covers of the nonarithmetic orbifolds have a paucity of homology.

arXiv commentThis is the version published by Geometry & Topology on 2 April 2006

References in corpus (4)

• Ricci flow with surgery on three-manifolds
• The virtual Haken conjecture: Experiments and examples
• Pro-p groups and towers of rational homology spheres
• Compatibility of local and global Langlands correspondences

Cited by in corpus (14)

• A torsion Jacquet--Langlands correspondence
• Bounds for eigenforms on arithmetic hyperbolic 3-manifolds
• Pro-p groups and towers of rational homology spheres
• An Eisenstein ideal for imaginary quadratic fields and the Bloch-Kato conjecture for Hecke characters
• Injectivity radii of hyperbolic integer homology 3-spheres
• Problems In Groups, Geometry, and Three-Manifolds
• The cohomology of lattices in SL(2,C)
• Growth of homology torsion in finite coverings and hyperbolic volume
• On compact hyperbolic manifolds of Euler characteristic two
• Bounds for Multiplicities of Unitary Representations of Cohomological Type in Spaces of Cusp Forms
• On geometric aspects of diffuse groups
• Normal Subgroups of Profinite Groups of Non-negative Deficiency
• Fundamental groups and group presentations with bounded relator lengths
• A deformation problem for Galois representations over imaginary quadratic fields