E8 Lattice - Theta Function

Theta Function

One can associate to any (positive-definite) lattice Λ a theta function given by

The theta function of a lattice is then a holomorphic function on the upper half-plane. Furthermore, the theta function of an even unimodular lattice of rank n is actually a modular form of weight n/2. The theta function of an integral lattice is often written as a power series in so that the coefficient of qn gives the number of lattice vectors of norm n.

Up to normalization, there is a unique modular form of weight 4: the Eisenstein series G4(τ). The theta function for the E8 lattice must then be proportional to G4(τ). The normalization can be fixed by noting that there is a unique vector of norm 0. This gives

where σ3(n) is the divisor function. It follows that the number of E8 lattice vectors of norm 2n is 240 times the sum of the cubes of the divisors of n. The first few terms of this series are given by (sequence A004009 in OEIS):

The E8 theta function may be written in terms of the Jacobi theta functions as follows:

where

\theta_2(q) = \sum_{n=-\infty}^{\infty}q^{(n+\frac{1}{2})^2}\qquad \theta_3(q) = \sum_{n=-\infty}^{\infty}q^{n^2}\qquad \theta_4(q) = \sum_{n=-\infty}^{\infty}(-1)^n q^{n^2}.

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