Slope of cusp forms and theta series (Q1581873)

It is proved that the theta series of degree 3 for extremal even unimodular lattices of rank 32 and 48 are unique, whereas for ranks 56 and 72 their difference is a multiple of a certain cusp form. For rank 40, if two extremal theta series are equal in degree 2, then in degree 3 their difference is a multiple of a certain cusp form. For ranks 32 and 48 and degree 4 the difference of two theta series is a multiple of a power of Schottky's polynomial. The proofs are based on results about the vanishing of cusp forms on the hyperelliptic locus [\textit{J.-I. Igusa}, Am. J. Math. 89, 817-855 (1967; Zbl 0159.50401)] and unpublished results of Weissauer.