A Caltech Library Service

Automorphic forms and rational homology 3–spheres

Calegari, Frank and Dunfield, Nathan M. (2006) Automorphic forms and rational homology 3–spheres. Geometry and Topology, 10 (8). pp. 295-329. ISSN 1465-3060. doi:10.2140/gt.2006.10.295.

PDF - Published Version
See Usage Policy.


Use this Persistent URL to link to this item:


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 GL2 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/Qsqrt-2) satisfying certain prescribed ramification conditions. In contrast to similar questions of this form, rho is allowed to have arbitrary ramification at some prime pi of Z[sqrt -2]. In the next paper in this volume, Boston and Ellenberg apply pro–p techniques to our examples and show that our result is true unconditionally. Here, we give additional examples where their techniques apply, including some non-arithmetic examples. 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 non-arithmetic orbifolds have a paucity of homology.

Item Type:Article
Related URLs:
URLURL TypeDescription Paper
Additional Information:Proposed: Walter Neumann; Seconded: David Gabai, Tomasz Mrowka Received: 18 August 2005; Revised: 28 February 2006 Calegari was partially supported as a 5–year fellow of the American Institute of Mathematics. Dunfield was partially supported by U.S. National Science Foundation grant #DMS-0405491, as well as a Sloan Fellowship. The authors thank Nigel Boston, Kevin Buzzard, Jordan Ellenberg, Oliver Goodman, Damian Heard, Craig Hodgson, Alex Lubotzky, Hee Oh, Dinakar Ramakrishnan, Alan Reid, and Richard Taylor for helpful conversations and correspondence.
Issue or Number:8
Record Number:CaltechAUTHORS:CALgt06b
Persistent URL:
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:2836
Deposited By: Archive Administrator
Deposited On:01 May 2006
Last Modified:08 Nov 2021 19:51

Repository Staff Only: item control page