Hecke operators and Hilbert modular forms

Let F be a real quadratic field with ring of integers O and with class number 1. Let Γ be a congruence subgroup of GL2 (O)GL2() . We describe a technique to compute the action of the Hecke operators on the cohomology H3 (G; \mathbb C)H3(;C) . For F real quadratic this cohomology group contains the cuspidal cohomology corresponding to cuspidal Hilbert modular forms of parallel weight 2. Hence this technique gives a way to compute the Hecke action on these Hilbert modular forms

PERFECT FORMS OVER TOTALLY REAL NUMBER FIELDS

A rational positive-definite quadratic form is perfect if it can be reconstructed from the knowledge of its minimal nonzero value m and the finite set of integral vectors v such that f(v) = m. This concept was introduced by Vorono¨ý and later generalized by Koecher to arbitrary number fields. One knows that up to a natural “change of variables” equivalence, there are only finitely many perfect forms, and given an initial perfect form one knows how to explicitly compute all perfect forms up to equivalence. In this paper we investigate perfect forms over totally real number fields. Our main result explains how to find an initial perfect form for any such field. We also compute the inequivalent binary perfect forms over real quadratic fields Q(pd) with d 66

Hecke operators and Hilbert modular forms

Let F be a real quadratic field with ring of integers O and with class number 1. Let Γ be a congruence subgroup of GL2 (O)GL2() . We describe a technique to compute the action of the Hecke operators on the cohomology H3 (G; \mathbb C)H3(;C) . For F real quadratic this cohomology group contains the cuspidal cohomology corresponding to cuspidal Hilbert modular forms of parallel weight 2. Hence this technique gives a way to compute the Hecke action on these Hilbert modular forms

Hecke operators and Hilbert modular forms Hecke operators and Hilbert modular forms

Abstract. Let F be a real quadratic field with ring of integers Ø and with class number 1. Let Γ be a congruence subgroup of GL2(Ø). We describe a technique to compute the action of the Hecke operators on the cohomology H 3 (Γ ; C). For F real quadratic this cohomology group contains the cuspidal cohomology corresponding to cuspidal Hilbert modular forms of parallel weight 2. Hence this technique gives a way to compute the Hecke action on these Hilbert modular forms

Hyperbolic tessellations and generators of for imaginary quadratic fields

We develop methods for constructing explicit generators, modulo torsion, of the K3 -groups of imaginary quadratic number fields. These methods are based on either tessellations of hyperbolic 3 -space or on direct calculations in suitable pre-Bloch groups and lead to the very first proven examples of explicit generators, modulo torsion, of any infinite K3 -group of a number field. As part of this approach, we make several improvements to the theory of Bloch groups for K3 of any field, predict the precise power of 2 that should occur in the Lichtenbaum conjecture at −1 and prove that this prediction is valid for all abelian number fields

On the topological computation of K_4 of the Gaussian and Eisenstein integers

In this paper we use topological tools to investigate the structure of the algebraic K-groups K4(R) for R=Z[i] and R=Z[ρ] where i:=−1−−−√ and ρ:=(1+−3−−−√)/2. We exploit the close connection between homology groups of GLn(R) for n≤5 and those of related classifying spaces, then compute the former using Voronoi’s reduction theory of positive definite quadratic and Hermitian forms to produce a very large finite cell complex on which GLn(R) acts. Our main result is that K4(Z[i]) and K4(Z[ρ]) have no p-torsion for p≥5