Header : /home/was/papers/thesis/RCS/algorithms.tex, v1.142000/05/1500 : 13 : 18wasExp Log : algorithms.tex, v Revision 1.14 2000/05/15 00:13:18 was quick fix! Revision 1.13 2000/05/10 20:31:51 was done. Revision 1.12 2000/05/10 09:30:12 was indexing and cleaning up. Revision 1.11 2000/05/10 03:45:57 was Wrote an introduction. Revision 1.10 2000/05/09 22:47:19 was fixed modular degree notation m_{i}nsteadofδ Revision 1.9 2000/05/09 19:51:48 was Fixed sectionThe Cap problem Revision 1.8 2000/05/09 19:49:45 was More details on the modular degree. Revision 1.7 2000/05/08 18:47:46 was added lario ref Revision 1.6 2000/05/08 03:27:43 was Fixed errors pointed out by Hendrik. Applications of modular symbols chap:computing In the previous chapter we introduced several spaces

of modular symbols, and observations such as “Manin’s trick” suggested that we could compute with them. The duality between modular symbols and modular forms hints that modular symbols might be useful in computing information about modular forms. In the present chapter, we gather together the fruits of our investigation into this connection.

Sections sec:computingmk–sec:decomposemodsym of this chapter give a method to compute the irreducible components of the spaces _{k}(N,) of modular symbols. In Section sec:intersection we observe that computing intersections of certain abelian varieties can be reduced to linear algebra over by viewing the abelian varieties as complex tori and considering the appropriate diagrams. In Sections sec:ratperiod, we continue this trend by pointing out that many invariants of the complex torus attached to a modular form can be computed without computing any approximate period lattices. In Section sec:torsionsubgroup, we discuss well-known methods for computing both an upper and lower bound on the order of the torsion subgroup of certain abelian varieties. Section sec:moddeg presents an algorithm for computing the modular degree of the complex torus associated to a newform.

In Section sec:ratpartformula we aim squarely at the problem of gathering data related to the Birch and Swinnerton-Dyer conjectureBSD conjecture!and L(A, j)/Ω_{j} and its generalizations, where we give a formula for the rational numbers |L(A_{f}, j)/Ω_{j}| attached to a newform. In Section sec:maninconstant we compare the ratio computed in the previous section to the one considered in the Birch and Swinnerton-Dyer conjecture;BSD conjecture the two numbers differ by a ManinManin constant, which we bound. Finally, in Section sec:analytic we give algorithms for approximating the period lattice and related numerical quantities associated to a newform of arbitrary weight.

Computing the space of modular symbolsModular symbols!computingsec:computingmkdefinition Let W be a subspace of a finite-dimensional vector space V. To compute the quotient V/W means to give a matrix representing the projection V V/W, with respect to some basis for V and some basis B for V/W, along with a lift to V of each element of B.