This package deals with cuspidal spaces of weight 2 modular forms on Γ0(N), over any imaginary quadratic field. In the current version, one can compute Hecke operators for principal ideals on these spaces, and determine the newforms. The package will be developed further in the future.
We will refer to modular forms over imaginary quadratic fields as Bianchi modular forms. These are defined similarly to classical modular forms, with the modular group SL(2, ZZ) is replaced by SL(2, OF), where OF is the ring of integers in an imaginary quadratic field F. This group acts on HH3, 3-dimensional hyperbolic space, and Bianchi modular forms (of weight k ≥2) are functions on HH3 that satisfy a natural automorphy relation under this action.
For an ideal N of OF, the congruence subgroup Γ0(N) of GL(2, OF) is defined as the subgroup of matrices that are upper triangular modulo N. The space of Bianchi modular forms on F of level N is defined as the space of functions satisfying the automorphy relation on the subgroup Γ0(N).
For precise definitions see [EGM98].
There exist several previous implementations of weight 2 Bianchi modular forms, each for specific fields F. These projects are described in [Cre84] and references cited there for Euclidean fields, [Whi90] for fields of class number one, [Byg99] for QQ(Sqrt( - 5)) and [Lin05] for QQ(Sqrt( - 23)) and QQ(Sqrt( - 31)).
A theorem of Franke states that for algebraic group (G) defined over QQ and arithmetic subgroup Γ,
H * (Γ;E) simeq H * (tenfrak(g), K;A(Γ, G) tensor E)
where A(Γ, G) denotes the space of automorphic forms. Thus we can think of the cohomology H * (Γ;E) as a concrete realization of certain automorphic forms.
Ash, Gunnells, and Lee-Szczarba [LS78], [Ash94], [Gun00] define a homology complex S * (Γ) known as the sharbly complex. A theorem of Borel-Serre [BS73] gives
Hν - k(Γ;CC) simeq Hk(S_ * (Γ)),
where ν = vcd(Γ) is the virtual cohomological dimension of Γ.
There is a natural Hecke action on this complex which agrees with the Hecke action on the automorphic forms.
The space of positive definite binary Hermitian forms over F form an open cone in a real vector space. There is a natural decomposition of this cone into polyhedral cones corresponding to the facets of the Vor polyhedron [Gun99], [Koe60], [Ash77]. These facets are in 1-1 correspondence with perfect forms over F. The polyhedral cones give rise to ideal polytopes in HH3, 3-dimensional hyperbolic space.
The structure of the polytopes allows us to find a finite (modulo Γ) spanning set for the sharbly complex. This is the analogue of unimodular symbols in the classical case. The modular symbol algorithm, for describing the action of Hecke operators, can now be replaced by a 0-sharbly reduction algorithm. An advantage of this is, compared with a straightforward generalization of the usual modular symbols algorithm, is that the number field does not need to be Euclidean.
Given an imaginary quadratic field F, we compute the structure of the Vor polyhedron by computing a complete set of GL2(OO)-class representatives of perfect forms. Given a level fn ⊆OO defining the level of the congruence subgroup, the polyhedron is used to compute H2(Γ0(fn)). Given a prime ideal fp ⊂OO, the 0-sharbly reduction algorithm is used to compute the action of the Hecke operator Tfp on H2(Γ0(fn)).
The algorithm described in [Gun99] is an efficient algorithm for computing the Vor polyhedron and provides a replacement for the modular symbol algorithm for computing the action of Hecke operators.
One advantage of this algorithm is that the most expensive steps come in the precomputation phase (computations depending only on the imaginary quadratic field). This means that for a given field, forms of many different levels can be computed based on the same precomputation.
The cohomology, regarded as a module for the algebra generated by the Hecke operators Tfp, decomposes into an Eisenstein piece and a cuspidal piece. The cuspidal piece is the same as the cuspidal subspace of modular forms (regarding both as Hecke modules). The Eisenstein part is recognisable by looking at Hecke eigenvalues of the eigenforms contained in it. By this means, we strip away the Eisenstein part; the space returned is the cuspidal subspace only.
Spaces of Bianchi modular forms in Magma are objects of type ModFrmBianchi.
This type shares many features with the type ModFrmHil of Hilbert modular forms. Many of the functions for Hilbert modular forms (see Chapter HILBERT MODULAR FORMS) can also be applied to Bianchi spaces, such as NewSubspace, NewformDecomposition, etc.
Technically, ModFrmBianchi is implemented as a subtype of ModFrmHil. Some intrinsics may not be displayed separately for the two types in Magma, for instance when one enters NewSubspace; at the Magma prompt.
To see some information printed during computation about what the program is doing, use SetVerbose("Bianchi",n), where n is 0 (silent, by default), 1 (for concise information), 2, 3 or 4 (which may display bulky data).