...This chapter computes the local heights and compares them with the derivatives computed before. It checks the theorem place by place and takes into account all the assumptions on the Schwartz function. According to the reduction of the Shimura curve, the situation is divided to the following four...

...This chapter introduces two classes of degenerate Schwartz functions which significantly simplify the computations and arguments of both the analytic kernel and the geometric kernel functions. It first restates the kernel identity in terms of un-normalized kernel functions before stating...

.... The chapter then considers the v-part for non-archimedean v. An explicit formula is given in the unramified case, and an approximation is presented in the ramified case assuming the Schwartz function is degenerate. An explicit result of the v-part...

...This chapter describes the decomposition of the geometric kernel. It considers the assumptions on the Schwartz function and decomposes the height series into local heights using arithmetic models. The intersections with the Hodge bundles are zero, and a decomposition to a sum of local heights...

...This chapter proves the theorem that asserts the modularity of the generating series and the theorem dealing with abelian varieties parametrized by Shimura curves. Before presenting the proofs, the chapter considers the new space of Schwartz functions and constructs theta series and Eisenstein...