Projektbeschreibung
Eine der wichtigsten offenen Fragen der Mathematik aus neuem Blickwinkel betrachten
Der Beweis der Vermutung von Birch und Swinnerton-Dyer ist eines der sieben ungelösten Probleme der Mathematik, die im Jahr 2000 in die Liste der Millennium-Probleme aufgenommen wurden. In Hinsicht auf rationale Lösungen für elliptische Kurven definierende Gleichungen konnten für eine breite Klasse elliptischer Kurven über den rationalen Zahlen große Fortschritte erzielt werden. Das EU-finanzierte Projekt ShimBSD arbeitet daran, diese Beschreibungen maßgeblich zu erweitern. Die Forschenden planen, anhand eines innovativen Ansatzes neue Fälle der Vermutung von Birch und Swinnerton-Dyer und anderer Vermutungen zu beweisen, die über die zuletzt in den 1990er Jahren erzielten kritischen Durchbrüche hinausgehen.
Ziel
"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)."
Wissenschaftliches Gebiet
Schlüsselbegriffe
Programm/Programme
Thema/Themen
Finanzierungsplan
ERC-COG - Consolidator GrantGastgebende Einrichtung
3900 BRIG
Schweiz