Hecke Modules of Brandt

A new quaternion algebra package, allowing ideal enumeration, provides the basis for the construction of Brandt modules. This uses the arithmetic of quaternions as a model for the equivalent construction of Mestre and Oesterlé via supersingular elliptic curves. The Brandt module provide the only known effective means for computing component groups of the Néron model of Jacobians of Shimura curves and modular curves.

Features:

• Construction of Brandt module from the left ideals of an order in a quaternion algebra, together with Hecke operators given by the Brandt matrices.
• Decomposition of the Brandt module under the Hecke algebra.
• Natural inner product structure on the Brandt module.