Tate-Shafarevich groups 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 Conjecture of Bloch and Kato. The PI has shed light on Sha by 1) proving part of the Birch and Swinnerton-Dyer conjectures, 2) building the foundation for a generalization of this theory to modular forms, and 3) developing the Iwasawa theory to estimate the size of Sha in towers of number fields.
|Effective start/end date||9/1/19 → 8/31/24|
- Simons Foundation: $42,000.00
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