Slope of Cusp Forms and Theta Series Original Research Article

The aim of this paper is to improve the results of [o] about the theta series associated to the even extremal unimodular quadratic form. We prove that, in degree 3, the associated theta series is still unique for extremal even unimodular lattices of rank 32 and 48. For the ranks 40, 56, 72 we analyze the situation. We discuss also consequences in degree 4. To obtain these results we use some known results about the vanishing of the cusp forms on the hyperelliptic locus Ig. We give also some improvements in this subject, namely we show that the estimate about the slope of cusp forms vanishing on Ig is sharp.