Abstract
There are two parts in the thesis. Part One (Chapter 1 to 4) is on arithmetic of definite Shimura curves over function fields and automorphic forms. We construct certain theta series from definite quaternion algebras over function fields which generate the space of harmonic automorphic forms. From the special points on definite Shimura curves and those theta series we deduce the critical central value of $L$-series of automoprhic forms. As applications, an analogue of Waldspurger's formula and critical central values of Hasse-Weil $L$-function of certain elliptic curves over function fields are obtained. Part Two (Chapter 5) is a published paper: On the independence of Heegner points over function fields. We prove the independence of Heegner points for different "imaginary" quadratic function fields and get a subgroup of elliptic curves with arbitrary large rank.