Innovating Works

ShimBSD

Financiado
Shimura varieties and the Birch Swinnerton Dyer conjecture
"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 o... "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)." ver más
30/04/2026
Mimotec SA
2M€
Duración del proyecto: 61 meses Fecha Inicio: 2021-03-24
Fecha Fin: 2026-04-30

Línea de financiación: concedida

El organismo H2020 notifico la concesión del proyecto el día 2021-03-24
Línea de financiación objetivo El proyecto se financió a través de la siguiente ayuda:
ERC-2020-COG: ERC CONSOLIDATOR GRANTS
Cerrada hace 4 años
Presupuesto El presupuesto total del proyecto asciende a 2M€
Líder del proyecto
Mimotec SA No se ha especificado una descripción o un objeto social para esta compañía.