Automorphic forms and rational homology 3--spheres
  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. 
Frank Calegari, Nathan M Dunfield