Local densities of 2-adic quadratic forms

Abstract

In this paper, we give an explicit from formula for the local density number of representing a two by two 2-integral matrix T by a quadratic 2-integral lattice L over Z 2 . The non-dyadic case was dealt in a previous paper. The special case when L is a (maximal) lattice in the space of trace zero elements in a quaternion algebra over Q 2 yields a clean and interesting formula, which matches up perfectly with the non-dyadic case in terms of the Gross–Keating invariants. This work is used to compare the central derivative of a genus two Eisenstein series with certain generating function of arithmetic 0-cycles on certain Shimura curve, in a joint work with Kudla and Rapoport.