Slope of Cusp Forms and Theta Series

Abstract

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 I g . We give also some improvements in this subject, namely we show that the estimate about the slope of cusp forms vanishing on I g is sharp.

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

“…In the last section we show that the Siegel theta series of degree 4 associated with the five extremal even unimodular lattices are distinct. This result settles a question raised by Salvati Manni [14].…”

“…The number of pairs ρ(δ 2 ), ρ(δ 3 ) which satisfy the conditions (13) 2 and (14) (13) and (14). In this case we obtain…”

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

“…We use the symbol (T 3 * , {a/2, b/2, c/2}, 2) to denote the matrix ⎛ ⎜ ⎜ ⎝ 30 , {a/2, b/2, c/2}, 2)U , the minimal value of the non-zero diagonal entries of the resulting matrix is two. We note that it is easy to find all possible ordered triples a, b, c satisfying the condition (1) only.…”

“…To eliminate the triples a, b, c not satisfying the condition (2) we use the theta series of one complex variable associated with the quadratic form defined by (T, {a/2, b/2, c/2}, 2). If (T, {a/2, b/2, c/2}, 2) integrally represents 1, then (T 30 , {a/2, b/2, c/2}, 2) does not satisfy the condition (2). At this stage we do not know whether two triples that lead to the quaternary forms with the identical discriminant belong to the same set (T 30 , {a/2, b/2, c/2}, 2).…”

“…Much later Salvati Manni [30] showed that the difference of the Siegel theta series of degree four associated with 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. We remark that Salvati Manni implies Erokhin, since Siegel's operator sends Siegel theta series of higher degree to theta series of lower degree, and sends cusp forms to zero.…”