Modular curves
Congruence subgroups of PSL(2,ℝ)
Arithmetic Fuchsian groups
Modular forms
Modular symbols
Brandt modules
Supersingular divisors on modular curves
Modular Abelian varieties
Hilbert modular forms
Modular forms over imaginery quadratic fields
Admissible representations of GL2(Qp)
A modular curve in Magma is an affine curve given by an equation in a standard form. These equations are stored in precomputed databases. The modular curves currently available are defined by "modular equations" of three kinds, each of which is a bivariate polynomial relation between the j-invariant and another standard function on X0(N). These give singular plane affine models for X0(N). One application for which these models are suited is to compute isogenies of degree N between elliptic curves.
Features:
Creation of a modular curve of specified level from a database. Possible model types are Atkin, Canonical and Classical.
Database of modular equations: Atkin, Canonical and Classical.
Parametrization of the isogenies of an elliptic curve by points on some X0(N).
The j-invariant of a modular curve as a function on the curve.
Automorphisms: Atkin–Lehner involutions.
Hilbert and Weber class polynomials.
The group GL+2(ℝ) of 2 by 2 matrices defined over ℝ with positive determinant acts on the upper half complex plane ℍ = {x∈C | Im(x) > 0}} by fractional linear transformation:
| a | b |
| c | d |
Calculations on the upper half complex plane ℍ: distances, angles and geodesics
Action of PSL2(ℝ) on ℍ: fixed points and stabilizers
Construction of the congruence subgroups listed above
Construction of cusps, cusp widths, and elliptic points of congruence subgroups
Computation of a fundamental domain for the action of a congruence subgroup, described by the vertices, and Farey symbols
Equivalence of points under the action of a congruence subgroup
Computation of generators of congruence subgroups, and coset representatives
Graphics: postscript output of pictures of fundamental domains, points and geodesics, and polygons with geodesic edges (all on the upper half complex plane)
Since V2.8, Magma has included packages for modular forms and modular symbols. These were originally developed by William Stein, and are continually being developed further and improved by the Magma group. The modular forms package is, to a large extent, built on top of the modular symbols package. However, it also contains several independent features, notably Eisenstein series, half-integral weight forms and weight 1 forms.
Construction of spaces of modular forms of weight k ≥ 1/2 on Γ0(N) or Γ1(N) (or with specified character)
Decomposition into Eisenstein, cuspidal, and new subspaces
Computation of dimensions (by formulae)
Computation of bases of these spaces, expressing basis elements as q-expansions with desired number of terms
Arithmetic operations for modular forms
Computation of Hecke operators and Atkin-Lehner operators
Decomposition into invariant subspaces with respect to these operators
Characteristic polynomials of Hecke operators
Determination of all newforms of given level (with Fourier coefficients given in suitable number fields)
Determination of all reductions of newforms modulo a given prime
Modular symbols provide explicit representations of homology groups associated to modular curves, which are suited to efficient computation. Computing the Hecke action on these groups yields q-expansions of modular forms. Other computations yield arithmetic information about the Jacobians of modular curves and their irreducible factors. This package contains many routines of this kind, which are used by the other packages, and which may also be called directly. (There is some duplication, and in some instances, the routines in this package are "lower level".)
Many of the algorithms implemented here are described in William Stein's thesis [WAS00] or in [WAS07]. Additional references are given in the Magma Handbook.
Construction of spaces of modular symbols of given character, level, and weight
Computation of Hecke operators and Atkin-Lehner operators
Decomposition into invariant subspaces under these operators
Computation of degeneracy maps, and standard subspaces
Determination of twists of minimal level, for an eigenform
Computation of certain invariants of modular abelian varieties
The intersection pairing on the integral homology of modular curves
Special values of L-functions and computation of complex period lattices
Modular abelian varieties over ℚ are the ℚ-irreducible quotients of Jacobians of modular curves X0(N) or X1(N). In Magma, modular abelian varieties are viewed as explicit quotients or subvarieties of these modular Jacobians. The implementation is built on the modular symbols package. (The algorithms do not involve explicit equations for the Jacobians as varieties, which would be impractical).
Construction of modular abelian varieties associated to Hecke-invariant subspaces of modular forms.
Finite direct sums and quotients may be formed.
Explicit computation of the group Hom(A,B) or the ring End(A), as a subgroup of homology, for modular abelian varieties A, B over ℚ.
Computation of kernels, cokernels, and images of homomorphisms of abelian varieties.
Intersections of subvarieties.
Computation of discriminants of subgroups of endomorphism rings, such as Hecke algebras.
Upper and lower bounds on the order of the K-rational torsion subgroup of A.
The determination of whether or not two modular abelian varieties are isomorphic (in some cases).
Characteristic polynomial of Frobenius.
Tamagawa numbers and component group orders (in some cases).
Computation with torsion points as elements of rational homology.
Computation of all inner and CM twists (not provably correct).
Computation of "building blocks".
Brandt modules provide an alternative approach to computing modular forms: they admit a natural Hecke action, and can be identified with spaces of classical modular forms having the same Hecke action. Brandt modules are defined in terms of ideals of Eichler orders in quaternion algebras.
Note that this package is only for Brandt modules over ℚ. A separate implementation for arbitrary totally real fields, using a more efficient algorithm, is at the core of the package for Hilbert modular forms (described below).
Features:
Construction of the Brandt module on the left ideal class of an Eichler order in a definite quaternion algebra over ℚ.
Elementary invariants: level, discriminant, conductor, etc.
Calculation of dimensions by standard formulae.
Arithmetic operations for module elements.
Construction of Hecke and Atkin Lehner operators on Brandt modules.
Decomposition of a Brandt module into invariant subspaces.
The Eisenstein subspace and the cuspidal subspace of a Brandt module.
Operations on subspaces: Orthogonal complement, intersection.
Properties of subspaces: Eisenstein, cuspidal, decomposable.
Inner product of elements with respect to the canonical pairing on their parent.
q-expansions associated with a pair of elements of a Brandt module.
Another construction of interesting Hecke-modules is via the Hecke action on divisors on the supersingular points on X0(N) in characteristic p. More precisely, the module considered is the free abelian group on the supersingular elliptic curves in characteristic p enhanced with level N structure. It is computed using the "method of graphs" of Mestre and Oesterlé and the Brandt modules algorithm.
Computation of Hecke operators and Atkin-Lehner involutions on modules of supersingular divisors.
Decomposition of a Brandt module into invariant subspaces with respect to these operators.
The monodromy pairing.
Since V2.15, Magma has included a package for computing Hecke operators on spaces of Hilbert modular forms, over arbitrary totally real fields and for arbitrary level.
Two separate algorithms are implemented (both rely on the Jacquet-Langlands correspondence, but they make use of different kinds of quaternion algebras). The algorithm due to Dembele is available for all weights greater than or equal to 2; it is an efficient approach to Brandt module computations. The implementation of the core calculations is fast, and this method has been used for fields of degree up to 10. The algorithm due to Greenberg and Voight is available for parallel weight 2, and makes use of Voight's algorithm for fundamental domains of Fuchsian groups associated to Shimura curves.
Construction of cuspidal spaces of Hilbert modular forms over any totally real field, of given level and weight
The quaternion order which underlies the computations for a particular space may also be specified
Computation of dimensions (by "formulae", or by computing the space)
Computation of Hecke operators
Computation of new subspaces
Decomposition into invariant subspaces with respect to the Hecke action
Since V2.16, Magma has included a package for modular forms of weight 2 over arbitrary imaginary quadratic fields. In the current version, one can compute Hecke operators for principal ideals on these spaces, and determine the newforms.
The algorithm, developed by Gunnells and Yasaki, is based on "Sharbly complexes", and involves computations with the Voronoi polyhedron of the given imaginary field.
Construction of cuspidal spaces of modular forms over any imaginary quadratic field, of given level and weight 2
Computation of dimensions
Computation of Hecke operators
Computation of new subspaces
This package, included since V2.16, provides functionality for working with the local components of an automorphic representation associated to a modular form. An eigenform in a classical space of cusp forms determines an automorphic representation, which is made up of "local components" at each prime p. The local component at p is an admissible representation on GL2(ℚp).
The package has two main parts. Starting with a cuspidal newform, the admissible representation (or data describing it) can be computed. Furthermore, via the local Langlands correspondence, there exists a related Galois representation on the absolute Galois group of ℚp. The interesting part of this representation can be computed from the admissible representation.
Construction of the admissible representation on GL2(ℚp) arising from a given cuspidal eigenform, of any weight and level
Conductor and central character of an admissible representation
Determination of a twist with minimal level, of a given representation
Recognition of whether a representation is principal series, or supercuspidal
In the case of principal series, the "principal series parameters" can be determined (as two characters on ℤp)
In the supercuspidal case, a "cuspidal inducing datum" can be obtained (as a representation on a suitable subgroup of GL2(ℚp))
Computation of the associated local Galois representation: its restriction to inertia is returned as a representation on a finite group Gal(L/ℚp) where L is some finite extension of ℚp.