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+ℓZ, 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 the cubic and septic bases. Next, motivated by strongly modular lattices for the ten special levels ℓ, where σ1(ℓ)∣24, we derive quadratic iterations in these ten special levels generalizing the cubic and septic cases. This also gives a uniform proof of the equations used by N.D. Elkies for 13 of his explicit modular towers. They correspond exactly to the case where all eta terms occur to the same power in his list. This provides a link between strongly modular lattices and modular towers.