Relating Special Values of L-functions with orders of Tate-Shafarevich groups

Relating Special Values of L-functions with orders of Tate-Shafarevich groups Relating Special Values of L-functions with orders of Tate-Shafarevich groups L-functions of modular forms are believed to encode sizes of TateShafarevich groups (Sha) and other arithmetic invariants. Two concrete conjectures are the Birch and Swinnerton-Dyer conjectures (one of the Millenium Problems), and the Tamagawa Number C