On the local density formula and the Gross-Keating invariant with an Appendix 'The local density of a binary quadratic form' by T. Ikeda and H. Katsurada

Abstract

Ikeda and Katsurada have developed the theory of the Gross-Keating invariant of a quadratic form in their recent papers Ikeda and Katsurada (Am J Math 140:1521-1565, 2018, Explicit formula of the Siegel series of a quadratic form over a non-archimedean local field, 2017. arXiv:1602.06617). In particular, they prove that the local factors of the Fourier coefficients of the Siegel-Eisenstein series are completely determined by the Gross-Keating invariants with extra datums, called the extended GK datums, in Ikeda and Katsurada (Explicit formula of the Siegel series of a quadratic form over a non-archimedean local field, 2017. arXiv:1602.06617). On the other hand, such a local factor is a special case of the local density for a pair of two quadratic forms. Thus we propose a general question if the local density can be expressed in terms of a certain series of extended GK datums. In this paper, we prove that the answer to this question is affirmative, for the local density of a single quadratic form defined over an unramified finite extension of Z(2) and over a finite extension of Z(p) with p odd. In the appendix, Ikeda and Katsurada compute the local density formula of a single binary quadratic form defined over any finite extension of Z(2).