• Corpus ID: 118715458

Modular symbols over number fields

@inproceedings{Arans2010ModularSO,
  title={Modular symbols over number fields},
  author={M. Aran{\'e}s},
  year={2010}
}
Let K be a number field, R its ring of integers. For some classes of fields, spaces of cusp forms of weight 2 for GL(2;K) have been computed using methods based on modular symbols. J.E. Cremona [9] began the programme of extending the classical methods over Q to the case of imaginary quadratic fields. This work was continued by some of his Ph.D. students [35, 6, 22], and results have been obtained for some imaginary quadratic fields with small class number. More recently, P. Gunnells and D… 

Figures and Tables from this paper

Modular Symbols Modulo Eisenstein Ideals for Bianchi Spaces
The goal of this thesis is two-fold. First, it gives an efficient method for calculating the action of Hecke operators in terms of “Manin” symbols, otherwise know as “Msymbols,” in the first homology
Bianchi modular forms over an imaginary quadratic
In this paper, we present the outcome of vast computer calculations, locating several of the very rare instances of level one cuspidal Bianchi modular forms that are not lifts of elliptic modular
The homological torsion of PSL_2 of the imaginary quadratic integers
Denote by Q(sqrt{-m}), with m a square-free positive integer, an imaginary quadratic number field, and by A its ring of integers. The Bianchi groups are the groups SL_2(A). We reveal a correspondence
Title On Level One Cuspidal Bianchi Modular Forms
In this paper, we present the outcome of vast computer calculations, locating several of the very rare instances of level one cuspidal Bianchi modular forms that are not lifts of elliptic modular
Congruence Subgroups, Cusps and Manin Symbols over Number Fields
We develop an explicit theory of congruence subgroups, their cusps, and Manin symbols for arbitrary number fields. While our motivation is in the application to the theory of modular symbols over
Bianchi’s additional symmetries
In a 2012 note in Comptes Rendus Math{e}matique, the author did try to answer a question of Jean-Pierre Serre; it has recently been announced that the scope of that answer needs an adjustment, and
Bounds on entries in Bianchi group generators
Upper and lower bounds are given for the maximum Euclidean curvature among faces in Bianchi’s fundamental polyhedron for PSL2(O) in the upper-half space model of hyperbolic space, where O is an
Arithmetic Aspects of Bianchi Groups
We discuss several arithmetic aspects of Bianchi groups, especially from a computational point of view. In particular, we consider computing the homology of Bianchi groups together with the Hecke
On level one cuspidal Bianchi modular forms
In this paper, we present the outcome of vast computer calculations, locating several of the very rare instances of level one cuspidal Bianchi modular forms that are not lifts of elliptic modular

References

SHOWING 1-10 OF 33 REFERENCES
Algorithms for Modular Elliptic Curves
TLDR
This book presents a thorough treatment of many algorithms concerning the arithmetic of elliptic curves with remarks on computer implementation and an extensive set of tables giving the results of the author's implementations of the algorithms.
Modular symbols for 1 ( N ) and elliptic curves with everywhere good reduction
The modular symbols method developed by the author in [4] for the computation of cusp forms for Γ0(N) and related elliptic curves is here extended to Γ1(N). Two applications are given: the
(Co)homologies and K-theory of Bianchi groups using computational geometric models
This thesis consists of the study of the geometry of a certain class of arithmetic groups, by means of a proper action on a contractible space. We will explicitly compute their group homology, and
Hecke Operators and Hilbert Modular Forms
TLDR
A technique to compute the action of the Hecke operators on the cuspidal cohomology H3(Γ;C) of GL2(O) for F real quadratic, which contains cuspid Hilbert modularforms of parallel weight 2.
Applications of a computer implementation of Poincaré’s theorem on fundamental polyhedra
PoincarCs Theorem asserts that a group F of isometries of hyperbolic space H is discrete if its generators act suitably on the boundary of some polyhedron in H, and when this happens a presentation
Modular Symbols for Q-Rank One Groups and Voronı Reduction
Abstract LetGbe a reductive algebraic group of Q -rank one associated to a self-adjoint homogeneous cone defined over Q , and letΓ⊂Gbe a torsion-free arithmetic subgroup. Letdbe the cohomological
A course in computational algebraic number theory
  • H. Cohen
  • Computer Science, Mathematics
    Graduate texts in mathematics
  • 1993
TLDR
The first seven chapters guide readers to the heart of current research in computational algebraic number theory, including recent algorithms for computing class groups and units, as well as elliptic curve computations, while the last three chapters survey factoring and primality testing methods.
Quaternionic Manin symbols, Brandt matrices, and Hilbert modular forms
TLDR
A more efficient way to compute Hilbert modular forms over totally real number fields, especially quadratic fields, and a theory of quaternionic M-symbols whose definition bears some resemblance to the classical M-Symbols, except for their combinatorial nature is developed.
The Normalizer Of Certain Modular Subgroups
  • M. Newman
  • Mathematics
    Canadian Journal of Mathematics
  • 1956
Introduction. Let G denote the multiplicative group of matrices where a, b, c, d are integers and ad — bc = 1. G is one of the well-known modular groups. Let G0(n) denote the subgroup of G
Hyperbolic Tessellations Associated to Bianchi Groups
TLDR
This paper computes the structure of these polytopes for a range of imaginary quadratic fields using the model of positive definite binary Hermitian forms over F.
...
1
2
3
4
...