Frank Calegari

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 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.

Modularity lifting beyond the Taylor–Wiles method

We prove new modularity lifting theorems for p-adic Galois representations in situations where the methods of Wiles and Taylor–Wiles do not apply. Previous generalizations of these methods have been restricted to situations where the automorphic forms in question contribute to a single degree of cohomology. In practice, this imposes several restrictions—one must be in a Shimura variety setting and the automorphic forms must be of regular weight at infinity. In this paper, we essentially show how to remove these restrictions. Our most general result is a modularity lifting theorem which, on the automorphic side, applies to automorphic forms on the group

Cyclotomic Integers, Fusion Categories, and Subfactors

\mathrm {GL}(n)

Even Galois representations and the Fontaine–Mazur conjecture

GL(n) over a general number field; it is contingent on a conjecture which, in particular, predicts the existence of Galois representations associated to torsion classes in the cohomology of the associated locally symmetric space. We show that if this conjecture holds, then our main theorem implies the following: if E is an elliptic curve over an arbitrary number field, then E is potentially automorphic and satisfies the Sato–Tate conjecture. In addition, we also prove some unconditional results. For example, in the setting of

p

\mathrm {GL}(2)

p

GL(2) over