Modular Abelian Varieties (New) [HB 104]

This is a new package that offers some nontrivial functionality for modular abelian varieties, which we view as explicitly given quotients or subvarieties of modular Jacobians. No explicit algebraic defining equations are used in these algorithms, so computations with abelian varieties of large dimension are feasible. Some highlights of the package include:

• Construction of quite general modular abelian varieties, in the sense that arbitrary finite direct sums and quotients may be formed.
• Explicit computation of the group Hom(A, B) or the ring End(A), as a subgroup of homology, for modular abelian varieties A, B over Q.
• Computation of kernels, cokernels, and images of homomorphisms of abelian varieties.
• Intersections of subvarieties.
• Computation of discriminants of subgroups of endomorphism rings, such as Hecke algebras.
• A divisor and a multiple of the order of the K-rational torsion subgroup of A.
• The determination of whether or not two modular abelian varieties are isomorphic (in some cases).
• Characteristic polynomial of Frobenius.
• Tamagawa numbers and component group orders (in some cases).
• Computation of all inner and CM twists (not provably correct).
• Computation with torsion points as elements of rational homology.