On the Genus Version of the Basis Problem II: The Case of Oldforms

Abstract

A famous result of Waldspurger [22] asserts that for a given square-free level N > 1 we can choose a genus $\mathfrak{G}$ of positive definite even integral quadratic forms of rank m = 2k with determinant a perfect square such that all newforms of level N can be written as linear combinations of theta series attached to lattices from that genus (m > 4). Later on it was shown [4] that the statement above is true for all genera of lattices of precise level N provided that $p^{m} \nmid det(\mathfrak{G})$ for all prime divisors of N. Like Waldspurger most other authors working on the basis problem only treat the case of newforms. This seemed to be sufficient, because the oldforms can then be obtained from theta series attached to lattices of lower levels or scaled versions of them (provided that they exist!). Already for prime level and weight congruent 2 mod 4 there is a problem because even unimodular lattices do not exist. The purpose of the present paper is to complete the result of Waldspurger and others by a precise description, which cusp forms beyond the newforms are linear combinations of theta series attached to lattices in a given genus. Due to Siegel’s theorem it is reasonable to restrict attention to cusp forms. The answer is simply “ALL” as long as the genus in question is not maximal or adjoint to maximal at any prime divisor of the level N. It seems that only in these cases the arithmetic of these lattices is “strong enough” to have some influence on the kind of modular forms presented. Locally this is quite natural because (in the case adjoint to the maximal case) the p-modular Jordan-component is anisotropic mod p.