2016 Volume 70 Issue 2 Pages 281-314
Erokhin showed that the Siegel theta series associated with the even unimodular 32-dimensional extremal lattices of degree up to three is unique. Later, Salvati Manni showed that the difference of the Siegel theta series of degree four associated with the two even unimodular 32-dimensional extremal lattices is a constant multiple of the square J2 of the Schottky modular form J, which is a Siegel cusp form of degree four and weight eight. In the present paper we show that the Fourier coefficients of the Siegel theta series associated with the even unimodular 32-dimensional extremal lattices of degrees two and three can be computed explicitly, and the Fourier coefficients of the Siegel theta series of degree four for those lattices are computed almost explicitly.
