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Erokhin showed that the Siegel theta series associated with the even unimodular 32-dimensional extremal lattices of degree up to 3 is unique. Later Salvati Manni showed that the difference of the Siegel theta series of degree 4 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 4 and weight 8. 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 2 and 3 can be computed explicitly, and the Fourier coefficients of the Siegel theta series of degree 4 for those lattices are computed almost explicitly.

A numerical study of Siegel theta series of various degrees for the 48-dimensional even unimodular extremal lattices