An Explicit Formula for Local Densities of Quadratic Forms

Let S and T be two positive definite integral matrices of rank m and n respectively. It is an ancient but still very challenging problem to determine how many times S can represent T , i.e., the number of integral matrices X with XSX = T . However, Siegel proved in his celebrated paper ([Si1]) that certain weighted averages of these numbers over the genus of S can be expressed as the Euler product of pure local data—confluent hypergeometric functions for p = ∞ and local densities αp(T, S) for p < ∞ (see (1.1) for definition). Siegel himself extended this result to indefinite forms ([Si2-3]) in early fifties. A. Weil reinterpreted Siegel’s results in terms of representation and extended his results to other classical groups in 1965 ([We]). Roughly speaking, the Siegel-Weil formula says that the theta integral associated to a vector space (quadratic or Hermitian) is the special value of some Eisenstein series at certain point when both the theta integral and Eisenstein series (at the point) are both absolutely convergent. Recently, Kudla and Rallis pushed the results to non-convergent regions ([KR1-3]). From the point view of representation theory, the local density can be viewed as the special value of a local Whittaker function, which is the local factor of the Fourier coefficients of the Eisenstein series. For a lot of arithmetic applications, it is very important to have an exact formula for the local densities. For example, in his work on central derivative of Eisenstein series ([Ku1]), Kudla needed to compare the local density of certain ternary form with intersection number on some formal group. The explicit formula for local density was also used in Gross and Keating’s work in the intersection section of modular correspondence ([GK]). However, explicit formulas of local densities are known hard to obtain and are complicated in general. Siegel himself obtained an explicit formula for n = 1 or m = n assuming S is unimodular and p 6= 2. Under the