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Fix a prime p, and a modular residual representation ρ : GQ → GL2(Fp). Suppose f is a normalized cuspidal Hecke eigenform of some level N and weight k that gives rise to ρ, and let Kf denote the… (More)

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… (More)

- Frank Calegari
- 2016

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This note summarizes the theory of p-adically completed cohomology. This construction was first introduced in paper [4] (although insufficient attention was given there to the integral aspects of the… (More)

Let G∞ be a semisimple real Lie group with unitary dual Ĝ∞. The goal of this note is to produce new upper bounds for the multiplicities with which representations π ∈ Ĝ∞ of cohomological type appear… (More)

Let p be a prime, and let N be a positive integer coprime to p. Let Mk(Γ1(N);Qp) denote the weight k modular forms of level Γ1(N) defined over Qp. In recent years, work of Coleman and others (for… (More)

We study torsion in the homology of arithmetic groups and give evidence that it plays a role in the Langlands program. We prove, among other results, a numerical form of a Jacquet–Langlands… (More)

Gouvêa and Mazur made a precise conjecture about slopes of modular forms. Weaker versions of this conjecture were established by Coleman and Wan. In this Note, we exhibit examples contradicting the… (More)

We prove that the completed homology groups of $$\mathrm {GL}_n(\mathbf {Z})$$GLn(Z) in fixed degree stabilize as N goes to infinity. We also prove that the action of Hecke operators on stable… (More)