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Euler systems and the Birch--Swinnerton-Dyer conjecture

Objective

The Birch--Swinnerton-Dyer conjecture, one of the Millennium Prize Problems, is one of the central unsolved problems in mathematics. It predicts a relation between the arithmetic of an elliptic curve and the properties of the L-function of the elliptic curve. Some special cases of the conjecture were proven by Kolyvagin; the main ingredient in his proof is an algebraic construction called an Euler system. Even though Euler systems are extremely powerful tools, so far only five examples are known to exist. I propose to construct several new examples of Euler systems, in order to prove new cases of the Birch--Swinnerton-Dyer conjecture. In particular, I believe the following theorem to be within reach:

Let A be either a modular elliptic curve over a (real or imaginary) quadratic number field, or a modular abelian surface over the rational numbers. If the L-value L(A, 1) is non-zero, then the Mordell--Weil group of A is finite (i.e. the Birch--Swinnerton-Dyer conjecture holds for A).

Field of science

  • /natural sciences/mathematics/pure mathematics/arithmetic
  • /natural sciences/mathematics

Call for proposal

ERC-2014-CoG
See other projects for this call

Funding Scheme

ERC-COG - Consolidator Grant

Host institution

UNIVERSITY COLLEGE LONDON
Address
Gower Street
WC1E 6BT London
United Kingdom
Activity type
Higher or Secondary Education Establishments
EU contribution
€ 1 070 473

Beneficiaries (1)

UNIVERSITY COLLEGE LONDON
United Kingdom
EU contribution
€ 1 070 473
Address
Gower Street
WC1E 6BT London
Activity type
Higher or Secondary Education Establishments