Modular forms, quaternion algebras, and special values of L-functions

Let f be a cusp form of weight 2 and level N. Let K be an imaginary quadratic field of discriminant, --D, and A an ideal class of K. We obtain precise formulas for the special values of the L-functions associated to the Rankin convolution of f and a theta series associated to the ideal class A , in terms of the Petersson scalar product of f with the theta series associated to an Eichler order in a positive definite quaternion algebra. Our work is an extension of the work done by Gross [7]. The central tools used in this thesis are Rankin's method and a reformulation of Gross of work of Waldspurger concerning central critical values