Brandt Modules Over F_q[t]

This section concerns the case of quaternion orders whose base ring is F_q[t]. The definitions and constructions are similar to the case of quaternion orders over the integers. The implementation follows the new implementation over the integers (avoiding working explicitly in terms of ideals in Eichler orders), and makes use of techniques for quadratic forms over F_q[t] (developed by Markus Kirschmer).

The following intrinsics are provided. Where no description is given, the arguments and return values are similar to the corresponding intrinsics over the integers.

QuaternionOrder(M) : ModBrdt -> AlgQuatOrd

Level(M) : ModBrdt -> RngElt

Discriminant(M) : ModBrdt -> RngElt

Conductor(M) : ModBrdt -> RngElt

Ideals(M) : ModBrdt -> []

InnerProductMatrix(M) : ModBrdt -> AlgMatElt

HeckeOperator(M, n) : ModBrdtNew, RngElt -> Mtrx

BrandtModuleDimension(D, N) : RngElt, RngElt -> RngIntElt

BrandtModuleDimensionOfNewSubspace(D, N) : RngElt, RngElt -> RngIntElt

BrandtModule(M, N) : AlgQuatOrd, RngElt -> ModBrdt

This constructs the Brandt module attached to an Eichler order of level N in the maximal order M.

HeckeEigenvectors(M) : ModBrdt -> [ ModBrdt ]

This returns the common eigenvectors for the Hecke operators on the Brandt module M, as elements of M.

HeckeEigenvalue(f, p) : ModBrdtElt, RngElt -> RngElt

For a Hecke eigenform f in a Brandt module, this returns the eigenvalue for the Hecke operator at the prime p.