On congruences for the coefficients
نویسندگان
چکیده
1997 2 Kevin Lee James On con gruences for the coefficients of modular forms and some applications (Under the direction of Andrew Granville) In this dissertation, we will study two different conjectures about elliptic curves and modular forms. First, we will exploit the theory developed by Shimura and Waldspurger to address Goldfeld's conjecture which states that the density of rank zero curves in a family of quadratic twists of an elliptic curve should be 1/2. In particular, we will find lower bounds for the density of rank zero curves in several families of quadratic twists. Next, we will use a beautiful theorem of Frey to verify that the 3-part of the Birch and Swinnerton-Dyer conjecture holds for four different families of elliptic curves. More precisely, we will verify for four different elliptic
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