Siegel theta series of degree 2 of extremal lattices
Let \(\Lambda\) be an even unimodular lattice of rank N (N must satisfy \(N\equiv 0 (mod 8))\). Furthermore let \(\Lambda\) be an extremal lattice, i.e. \(\min_{x\in \Lambda,x\neq 0}Q(x)=[N/24]+1\), where Q(x) is the quadratic form of the lattice \(\Lambda\). In J. Number Theory 28, 119-131 (1988; Zbl 0638.10031), the reviewer raised the question whether the Siegel theta-series of degree 2 distinguish the equivalence classes of even unimodular extremal lattices of rank 40. In the present paper, this question is solved negatively. The author shows that there is more than one class of even unimodular positive extremal lattices of rank 40 with the same Siegel theta-series of degree 2, using (i) Igusa's structure theorem of Siegel modular forms of degree 2, (ii) a relation between the coefficients of Jacobi theta-series attached to the lattices in question and (iii) the pigeon hole principle. He also gives a sketch that the same statement holds for the case \(N=64\).
even unimodular extremal lattices of rank 40
scientific article; zbMATH DE number 4148178