Project description
Tackling one of the most important open questions in mathematics from a new angle
The Birch and Swinnerton-Dyer (BSD) conjecture is one of the seven unsolved problems in mathematics identified as the Millennium Prize Problems in 2000. Relating to rational solutions to equations defining elliptic curves, much progress has been made for a wide class of elliptic curves over the rationals. The EU-funded ShimBSD project is working towards expanding descriptions significantly. Researchers plan to prove new cases of the BSD conjecture and other conjectures beyond the realm of the last critical breakthroughs in the 1990s using an innovative approach.
Objective
"One of the most famous open problems in mathematics is the Birch–Swinnerton-Dyer (BSD) conjecture, which predicts that the size of the set of rational points on an elliptic curve is determined by the order of vanishing at s = 1 of its Hasse–Weil L-function. Building a crucial breakthrough due to Kolyvagin in the 1990's—the discovery of the first example of an ""Euler system""—the BSD conjecture has now been proved for a wide class of elliptic curves over the rationals: those where the order of vanishing of the L-function (the ""analytic rank"") is 0 or 1, which conjecturally accounts for 100% of elliptic curves.
However, the case of elliptic curves over the rationals is only the tip of an iceberg. Versions of the BSD conjecture are also expected to hold for elliptic curves over number fields, and more generally for abelian varieties of any dimension (with elliptic curves being the case of dimension 1). Even more generally, the Bloch–Kato conjecture predicts that for any L-function arising from geometry, its order of vanishing at any integer point encodes geometric information. However, these conjectures are far beyond the reach of Kolyvagin's Euler system.
The aim of my proposal is to prove new cases of the BSD conjecture and the Bloch–Kato conjecture, using new Euler systems arising from the geometry of unitary and symplectic Shimura varieties. In particular, I will prove the rank 0 case of the BSD conjecture for abelian surfaces over the rationals, elliptic curves over imaginary quadratic fields, and abelian three-folds with complex multiplication, assuming appropriate modularity results hold for these objects (which are known in many cases)."
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Funding Scheme
ERC-COG - Consolidator GrantHost institution
3900 BRIG
Switzerland