Quaternions And Kudla’s Matching Principle

The representation number of a quadratic space is very important in number theory. In this paper, we only consider the problems related to quaternion algebra over Q.Let D be a square free integer, and let B=B(D) be the unique quaternion algebra of discriminant D over Q with the reduced norm det. For a positive integer N prime to D, let OD(N) be an Eichler order in B of conductor N. When B is definite, for the lattice L=(Op(N), det) it is a very interesting and hard question to compute the representation number (for a positive integer m) On the other hand, the average over the genus gen(L), which we denote by is a product of so-called local densities, thanks to Siegel’s seminal work in1930’s [Si]. These densities are computable (see example [Yal]).The number TD,N associated with the coeffieient of certain modular form. Using Kudla’s matching principle ([Ku2, Section4], see also in Chapter1) and Siegel-Weil formula [KR1], we will prove some interesting identities between average representation numbers (associated to definite quaternion algebras). We will also prove the identities with degrees of Hecke correspondences on Shimura curves (associated to indefinite quaternion algebras).There is no Siegel-Weil formula for the quaternion algebra M2(Q) until now. We get some special case in Chapter4, and using these results and Kudla’s matching we could relate the number TD,N to the computation in M2(Q).