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Brandt modules provide a representation in terms of quaternion ideals
of certain cohomology subgroups associated to Shimura curves XD0(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. The machinery of modular symbols, Brandt modules, and,
in a future release, a module of singular elliptic curves, form the
computational machinery underlying modular forms in Magma.
Brandt modules were implemented by David Kohel, motivated by the
article of Mestre and Oesterl'e [Mes86] on the method
of graphs for supersingular elliptic curves, the article of
Pizer [Piz80] on computing spaces of modular forms using
quaternion arithmetic, and grew out of research in the author's
thesis [Koh96] on endomorphism ring structure of elliptic
curves over finite fields. The Brandt module machinery is
described in the article [Koh01] and has been used, together with
modular symbols, in the computation of component groups of quotients
of the Jacobians J0(N) of classical modular curves [KS00].
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