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 ρ of
satisfying certain prescribed
ramification conditions. In contrast to similar questions of this
form, ρ is allowed to have arbitrary ramification at some prime
π of Z[√(-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