Automorphic forms and rational homology 3--spheres

By Frank Calegari, Nathan M. Dunfield. 2005

51 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/022dc20d1bc6c87e822a7272073cb1daf28c0401

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Citations in the corpus

The asymptotic growth of torsion homology for arithmetic groups (2010, methodology): “Numerical experiments ([23], and unpublished data computed by Calegari-Dunfield in connection with [11]) as well as the work of Calegari-Mazur [15] suggest that arithmetic hyperbolic 3-manifolds should have a lot of torsion in their homology.”
A torsion Jacquet--Langlands correspondence (2012, methodology): “The orbifold YB was the main object of study in [16, 11], where it was showed that the π-adic tower ΓB(π) had vanishing Q-cohomology for”
Arithmetic Fuchsian Groups of Genus Zero (2006, background): “As in [7], the relevance of these conjectures to the cocompact setting follows from the Jacquet-Langlands correspondence (see [18] and also [7] Theorem 3.”
Bounds for multiplicities of unitary representations of cohomological type in spaces of cusp forms (2007, background): “The main result of Calegari–Dunfield [2] shows that there exists at least one (F,Γ, p) for which H(1) cusp(Yk,C) = 0 for all k.”