An explicit formula for the Siegel series of a quadratic form over a non-archimedean local field

Abstract Let F be a non-archimedean local field of characteristic 0, and 𝔬 {{\mathfrak{o}}} the ring of integers in F . We give an explicit formula for the Siegel series of a half-integral matrix over 𝔬 {{\mathfrak{o}}} . This formula expresses the Siegel series of a half-integral matrix B explicitly in terms of the Gross–Keating invariant of B and its related invariants.