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Computing weight \(2\) modular forms of level \(p^2\) (with an appendix by B. Gross). (English) Zbl 1093.11027

The authors describe an effectively computable Hecke stable subspace \(V_0\) of the space \(V\) of modular forms of weight 2 and level \(p^2\), where \(p\equiv3\bmod4\) is a prime, with \(V_0\) containing the space of modular forms with CM by the ring of integers in \({\mathbb Q}(\sqrt{- p})\). The space \(V_0\) is constructed using Brandt matrices associated to ideal classes of an order (of index \(p\) in the maximal order) in the quaternion algebra over \({\mathbb Q}\) ramified at \(p\) and at \(\infty\). The total space \(V\) has dimension that grows proportionally to \(p^2\), while \(V_0\) has dimension that grows proportionally to \(p\). Thus introducing \(V_0\) allows one to compute effectively with a Hecke stable subspace of smaller dimension than \(V\).
Three tables are also presented, one consisting of dimensions and the other two matching various subspaces of \(V_0\) to their corresponding abelian varieties. An appendix by B. Gross proves that the space of CM modular forms of weight 2 and level \(p^2\) injects into the space \(V_0\); it makes use of the local and global Jacquet-Langlands correspondence.

MSC:

11F11 Holomorphic modular forms of integral weight
11E20 General ternary and quaternary quadratic forms; forms of more than two variables
11Y99 Computational number theory
11F25 Hecke-Petersson operators, differential operators (one variable)

Software:

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References:

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