Central values of L-functions of twisted modular forms of composite level

In Chapter 1 we provide some background information about modular forms, describe the correspondence between half-integral and integer weight modular forms and explain how the coefficients of half-integral weight modular forms encode the central values of L-functions of twisted modular forms. <br>In Chapter 2 we study the rank of the modular curve X_0(49) over quadratic extensions. Assuming the Birch and Swinnerton-Dyer Conjecture, we show that the rank over quadratic number fields is positive if and only if the number of integer solutions of two explicit ternary quadratic forms is the same.<br>In Chapter 3 we study the product of two central values of L-functions of a twisted modular. We show that it suffices to compute a local polynomial at a finite number of points to decide whether the product is zero. <br>