Volume 10, issue 1 (2006)

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Automorphic forms and rational homology 3–spheres

Frank Calegari and Nathan M Dunfield

Geometry & Topology 10 (2006) 295–329
 arXiv: math.GT/0508271
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 $Gal\left(\overline{ℚ}∕ℚ\left(\sqrt{-2}\right)\right)$ 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 $ℤ\left[\sqrt{-2}\right]$.

In the next paper in this volume, Boston and Ellenberg apply pro–$\phantom{\rule{0.3em}{0ex}}p$ 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.

Keywords
virtual Haken Conjecture, Cooper's question, rational homology sphere, injectivity radius, automorphic forms, Galois representations
Mathematical Subject Classification 2000
Primary: 57M27, 11F80, 11F75