Description du projet
Aborder sous un angle nouveau l’une des plus importantes questions ouvertes en mathématiques
La conjecture de Birch et Swinnerton-Dyer (BSD) est l’un des sept problèmes non résolus en mathématiques désignés en 2000 comme les problèmes du prix du millénaire. Concernant les solutions rationnelles aux équations définissant les courbes elliptiques, de nombreux progrès ont été réalisés pour une importante classe de courbes elliptiques sur les rationnels. Le projet ShimBSD, financé par l’UE, entend considérablement élargir les descriptions. Les chercheurs envisagent de prouver, par une approche innovante, de nouveaux cas de la conjecture BSD et d’autres conjectures au-delà du domaine des dernières percées critiques des années 1990.
Objectif
"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)."
Champ scientifique
Mots‑clés
Programme(s)
Régime de financement
ERC-COG - Consolidator GrantInstitution d’accueil
3900 BRIG
Suisse