A Numerical Study of Siegel Theta Series of Various Degrees for the 32-Dimensional Even Unimodular Extremal Lattices

Abstract

Abstract. 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 J 2 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.

Distinguishing Siegel theta series of degree 4 for the 32-dimensional even unimodular extremal lattices

“…In this article we compute the Fourier coefficient a(T 40 , L 32 ) for each of the five extremal even unimodular lattices L 32 that are constructed from the five self-dual doubly even binary codes C of length 32. This settles one of the questions left unanswered in [11].…”

“…Suppose that ρ(δ) = u ∩ v.Then (12) holds, which implies (11). From the last condition we have (x, y) = 2.…”

“…Using the result in the previous subsection and Proposition 4.3 in [11] we obtain the following table. …”

“…In our previous paper [11] we described a method for computing the Fourier coefficients of Siegel theta series of degree up to 3 associated with the 32-dimensional even unimodular extremal lattices L 32 . The computation is independent of the lattices chosen in this family.…”

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

“…In [11] we improved the method, which is initiated in [14], to compute the Fourier coe‰cients of Siegel theta series of various degrees for the class of 32-dimensional even unimodular extremal lattices. Siegel theta series of degree 4 associated with the 32-dimensional even unimodular extremal lattices depend on the individual lattices.…”