On a conjecture of Ash

Banach Center Publications, 2016

The paper contains an expanded version of the talk delivered by the first author during the conference ALANT3 in Będlewo in June 2014. We survey recent results on independence of systems of Galois representations attached to-adic cohomology of schemes. Some other topics ranging from the Mumford-Tate conjecture and the Geyer-Jarden conjecture to applications of geometric class field theory are also considered. In addition, we have highlighted a variety of open questions which can lead to interesting research in near future.

Proceedings of The National Academy of Sciences, 1982

E. Hecike initiated the application of representation theory to the study of cusp forms. He showed that, for p a prime congruent to 3 mod 4, the difference ofmultiplicities ofcertain conjugate representations of SL4(F) on cusp forms of degree 1, level p, and weight 22 is given by the class number h(-p) of the field Q(Vj;). We apply the holomorphic Lefschetz theorem to actions on the Igusa compactification ofthe Siegel moduli space of degree 2 to compute the values of characters of the representations of SpN(FV) on certain spaces of cusp forms of degree 2 and level p at parabolic elements ofthis group. Our results imply that here too, the difference in multiplicities of conjugate representations of Sp4(F,) is a multiple of h(-p).

International Journal of Number Theory, 2009

In 1992, Ash and McConnell presented computational evidence of a connection between three-dimensional Galois representations and certain arithmetic cohomology classes. For some examples, they were unable to determine the attached representation. For several Hecke eigenclasses (including one for which Ash and McConnell did not find the Galois representation), we find a Galois representation which appears to be attached and show strong evidence for the uniqueness of this representation. The techniques that we use to find defining polynomials for the Galois representations include a targeted Hunter search, class field theory and elliptic curves.

Inventiones Mathematicae, 1993

2014

Hecke Eigenvalues and Arithmetic Cohomology William Leonard Cocke Department of Mathematics, BYU Master of Science We provide algorithms and documention to compute the cohomology of congruence subgroups of SL3(Z) using the well-rounded retract and the Voronoi decomposition. We define the Sharbly complex and how one acts on a k-sharbly by the Hecke operators. Since the norm of a sharbly is not preserved by the Hecke operators we also examine the reduction techniques described by Gunnells and present our implementation of said techniques for n = 3.

Noncommutative Geometry and Number Theory

We survey some of the known results on the relation between the homology of the full Hecke algebra of a reductive p-adic group G, and the representation theory of G. Let us denote by C ∞ c (G) the full Hecke algebra of G and by HP * (C ∞ c (G)) its periodic cyclic homology groups. LetĜ denote the admissible dual of G. One of the main points of this paper is that the groups HP * (C ∞ c (G)) are, on the one hand, directly related to the topology ofĜ and, on the other hand, the groups HP * (C ∞ c (G)) are explicitly computable in terms of G (essentially, in terms of the conjugacy classes of G and the cohomology of their stabilizers). The relation between HP * (C ∞ c (G)) and the topology ofĜ is established as part of a more general principle relating HP * (A) to the topology of Prim(A), the primitive ideal spectrum of A, for any finite typee algebra A. We provide several new examples illustrating in detail this principle. We also prove in this paper a few new results, mostly in order to better explain and tie together the results that are presented here. For example, we compute the Hochschild homology of O(X) ⋊ Γ, the crossed product of the ring of regular functions on a smooth, complex algebraic variety X by a finite group Γ. We also outline a very tentative program to use these results to construct and classify the cuspidal representations of G. At the end of the paper, we also recall the definitions of Hochschild and cyclic homology.

Let ρ 1 and ρ 2 be a pair of residual, odd, absolutely irreducible two-dimensional Galois representations of a totally real number field F. In this article we propose a conjecture asserting existence of "safe" chains of compatible systems of Galois representations linking ρ 1 to ρ 2. Such conjecture implies the generalized Serre's conjecture and is equivalent to Serre's conjecture under a modular version of it. We prove a weak version of the modular variant using the connectedness of certain Hecke algebras, and we comment on possible applications of these results to establish some cases of Langlands functoriality.

2016

Let ρ 1 and ρ 2 be a pair of residual, odd, absolutely irreducible two-dimensional Galois representations of a totally real number field F. In this article we propose a conjecture asserting existence of "safe" chains of compatible systems of Galois representations linking ρ 1 to ρ 2. Such conjecture implies the generalized Serre's conjecture and is equivalent to Serre's conjecture under a modular version of it. We prove a weak version of the modular variant using the connectedness of certain Hecke algebras, and we comment on possible applications of these results to establish some cases of Langlands functoriality.

1986

1996

In this paper, Hecke eigenvalues of several automorphic forms for congruence subgroupsof SL(3; Z) are listed. To compute such tables, we describe an algorithm which combinestechniques developed by Ash, Grayson and Green with the Lenstra--Lenstra--Lov'asz algorithm.