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Euler systems SIGNED

Euler systems and the Birch--Swinnerton-Dyer conjecture

Total Cost €

EC-Contrib. €

Partnership

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Outcomes and
results

Euler systems project word cloud

Explore the words cloud of the Euler systems project. It provides you a very rough idea of what is the project "Euler systems" about.

unsolved construct zero relation powerful proof finite mordell euler holds exist weil prove surface proven construction quadratic prize dyer conjecture curve elliptic imaginary arithmetic swinnerton central special algebraic examples believe predicts mathematics though birch kolyvagin theorem modular tools function rational millennium group either abelian ingredient extremely

Project "Euler systems" data sheet

The following table provides information about the project.

Coordinator	UNIVERSITY COLLEGE LONDON Organization address address: GOWER STREET city: LONDON postcode: WC1E 6BT website: n.a. contact info title: n.a. name: n.a. surname: n.a. function: n.a. email: n.a. telephone: n.a. fax: n.a.
Coordinator Country	United Kingdom [UK]
Total cost	1˙070˙473 €
EC max contribution	1˙070˙473 € (100%)
Programme	1. H2020-EU.1.1. (EXCELLENT SCIENCE - European Research Council (ERC))
Code Call	ERC-2014-CoG
Funding Scheme	ERC-COG
Starting year	2015
Duration (year-month-day)	from 2015-07-01 to 2020-06-30

Partnership

Take a look of project's partnership.

#	participants	country	role	EC contrib. [€]
1	UNIVERSITY COLLEGE LONDON UNIVERSITY COLLEGE LONDON Organization address address: GOWER STREET city: LONDON postcode: WC1E 6BT website: n.a. contact info title: n.a. name: n.a. surname: n.a. function: n.a. email: n.a. telephone: n.a. fax: n.a.	UK (LONDON)	coordinator	1˙070˙473.00

Map

Project 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).

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The information about "EULER SYSTEMS" are provided by the European Opendata Portal: CORDIS opendata.