Related papers: On Ihara's lemma for degree one and two cohomology…

An analogue over imaginary quadratic fields of a result in algebraic number theory known as Ihara's lemma is established. More precisely, we show that for a prime ideal P of the ring of integers of an imaginary quadratic field F, the kernel…

We construct an Euler system associated to regular algebraic, essentially conjugate self-dual cuspidal automorphic representations of GL(3) over imaginary quadratic fields, using the cohomology of Shimura varieties for GU(2, 1).

We prove a power saving over the trivial bound for the number of cohomological cuspidal automorphic representations of fixed level and growing weight on $GL_3/\mathbb{Q}$, by adapting the methods of our earlier paper on $GL_2$.

We use category-theoretic techniques to provide two proofs showing that for a higher-rank graph $\Lambda$, its cubical (co-)homology and categorical (co-)homology groups are isomorphic in all degrees, thus answering a question of Kumjian,…

We prove a rank-two potential automorphy theorem for mod $l$ representations satisfying an ordinary condition. Combined with an ordinary automorphy lifting theorem, we prove a rank-two, $p \ne l$ case of local-global compatibility for…

We prove various results on the cohomology of arithmetic lattices arising from quaternion algebras over a number field with at least one complex place, including a strong restriction on the allowable weights of cuspidal cohomological…

We prove the existence of a cuspidal automorphic representation $\pi$ for $GL_{79}/\mathbf{Q}$ of level one and weight zero. We construct $\pi$ using symmetric power functoriality and a change of weight theorem, using Galois deformation…

We prove a new upper bound for the dimension of the space of cohomological automorphic forms of fixed level and growing parallel weight on $\mathrm{GL}_2$ over a number field which is not totally real, improving the one obtained by…

Given the L-series of a half-integral weight cusp form, we construct a cohomology class with coefficients in a finite dimensional vector space in a way that parallels the Eichler cohomology in the integral weight case. We also define a lift…

We construct the compatible system of $l$-adic representations associated to a regular algebraic cuspidal automorphic representation of $GL_n$ over a CM (or totally real) field and check local-global compatibility for the $l$-adic…

We study the motivic cohomology of the special fiber of quaternionic Shimura varieties at a prime of good reduction. We exhibit classes in these motivic cohomology groups and use this to give an explicit geometric realization of level…

In this article, we establish an asymptotic estimate on the number of cuspidal automorphic representations of ${\rm GL}_4(\mathbb A_{\mathbb Q})$ which contribute to the cuspidal cohomology of ${\rm GL}_4$ and are obtained from symmetric…

We investigate the correspondence between holomorphic automorphic forms on the upper half-plane with complex weight and parabolic cocycles. For integral weights at least 2 this correspondence is given by the Eichler integral. Knopp…

By adapting the work of Kudla and Millson we obtain a lifting of cuspidal cohomology classes for the symmetric space associated to GO(V) for an indefinite rational quadratic space V of even dimension to holomorphic Siegel modular forms on…

The main purpose of this article is to give the integral cohomology of classical principal congruence subgroups in SL(2,Z) as well as their analogues in the third braid group with local coefficients in symmetric powers of the natural…

We present a level raising result for families of p-adic automorphic forms for a definite quaternion algebra D over the rational numbers. The main theorem is an analogue of a theorem for classical automorphic forms due to Diamond and…

Clozel, Harris, and Taylor proposed a conjectural generalized Ihara's lemma for definite unitary groups. In this paper, we prove their conjecture with banal coefficients under some conditions. As an application, we prove a level-raising…

We extend methods of Greenberg and the author to compute in the cohomology of a Shimura curve defined over a totally real field with arbitrary class number. Via the Jacquet-Langlands correspondence, we thereby compute systems of Hecke…

In this paper we study the cohomology of the de Rham complex of sheaves of reflexive differential forms on a normal complex space. First, we prove that the complex is exact in degree one under suitable conditions on the underlying…

We prove Ihara's lemma for the mod $l$ cohomology of Shimura curves, localised at a maximal ideal of the Hecke algebra, under a large image hypothesis on the associated Galois representation. This was proved by Diamond and Taylor, for…