Brandt Modules (New) [HB 89]

Brandt modules provide a representation in terms of quaternion ideals of certain cohomology subgroups associated to Shimura curves X^D₀(N) which generalize the classical modular curves X₀(N). The Brandt module datatype is that of a Hecke module - a free module of finite rank with the action of a ring of Hecke operators - which is equipped with a canonical basis (identified with left quaternion ideal classes) and an inner product which is adjoint with respect to the Hecke operators.

Features:

• Construction of a Brandt module on the left ideal class of a definite order in a quaternion algebra over $\mbox{\bf Q}$.
• Arithmetic operations with module elements.
• Inner product of elements with respect to the canonical pairing on their parent.
• Elementary invariants of a Brandt module: Level, discriminant, conductor etc.
• Decomposition of a Brandt module under the action of Atkin-Lehner and Hecke operators.
• Eisenstein subspace, cuspidal subspace.
• Operations on subspaces: Orthogonal complement, intersection.
• Properties of subspaces: Eisenstein, cuspidal, decomposable.
• Construction of Hecke and Atkin Lehner operators.
• q-expansions associated with a pair of elements of a Brandt module.
• Determination of the dimension of a Brandt module of given level obtained using standard formulae.