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Endomorphism Rings over Number Fields

The problem of computing $ {\mathrm{End}}(A/\mathbb{Q})$ has been solved by Stein (and is implemented in MAGMA). Stein has not published the algorithm, though he wrote it up.

Problem 8.2.1   Implement in SAGE Stein's algorithm to compute $ {\mathrm{End}}(A/\mathbb{Q})$ .

Problem 8.2.2   Design an algorithm to compute $ {\mathrm{End}}(A/\overline{\mathbb{Q}})$ . This would use Ribet's theory in [Rib92] combined with some ideas related to splitting of quaternion algebras (see [GL01]).

Problem 8.2.3   Implement in SAGE a basic framework for computing with modular abelian varieties. Stein implemented such a package in MAGMA. Much of the solution could be simply a ``design document'' and some basic classes.

There is also a paper of Calegari-Stein (unpublished) that goes into great detail in one case.

Problem 8.2.4   Help finish Calegari-Stein.

William Stein 2006-10-20