Identities between modular graph forms

Congruences between Modular Forms

1. Basics 1 1.

Higher congruences between modular forms

It is well-known that two modular forms on the same congruence subgroup and of the same weight, with coefficients in the integer ring of a number field, are congruent modulo a prime ideal in this integer ring, if the first B coefficients of the forms are congruent modulo this prime ideal, where B is an effective bound depending only on the congruence subgroup and the weight of the forms. In thi...

Higher Congruences Between Modular Forms

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18 Congruences between Modular Forms

Jacobi identities, modular lattices, and modular towers

We give first a simple proof of a generalized Jacobi identity for n-dimensional odd diagonal lattices which specializes to the classical Jacobi identity for the lattice Z2. For Z +√ Z, it recovers a one-parameter family of Jacobi identities discovered recently by Chan, Chua and Solé, used to deduce two quadratically converging algorithms for computing π corresponding to elliptic functions for t...