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Root numbers and parity of ranks of elliptic curves

Identifieur interne : 000263 ( Main/Exploration ); précédent : 000262; suivant : 000264

Root numbers and parity of ranks of elliptic curves

Auteurs : Tim Dokchitser [Royaume-Uni] ; Vladimir Dokchitser [Royaume-Uni]

Source :

Journal für die reine und angewandte Mathematik (Crelles Journal) [ 0075-4102 ] ; 2011-09.

RBID : ISTEX:35B3B0263FD154132798BAC3C6D3AEFF4F97F640

English descriptors

KwdEn :
- Abelian, Burnside ring, Conjecture, Continuity argument, Corollary, Dokchitser, Eld, Elliptic, Elliptic curve, Elliptic curves, Galois, Galois extension, Galois group, Global root number, Good reduction, Hilbert, Hilbert symbol, Isogeny, Local root numbers, Local terms, Multiplicative, Multiplicative reduction, Nite, Ord2, Ord3, Ordp, Other hand, Other words, Parity, Parity conjecture, Product formula, Quadratic, Quadratic extension, Quadratic twist, Quadratic twists, Real number, Root number, Root numbers, Same parity, Selmer, Selmer group, Selmer groups, Selmer rank, Semistable, Theorem theorem, Vladimir, Vladimir dokchitser.
Teeft :
- Abelian, Burnside ring, Conjecture, Continuity argument, Corollary, Dokchitser, Eld, Elliptic, Elliptic curve, Elliptic curves, Galois, Galois extension, Galois group, Global root number, Good reduction, Hilbert, Hilbert symbol, Isogeny, Local root numbers, Local terms, Multiplicative, Multiplicative reduction, Nite, Ord2, Ord3, Ordp, Other hand, Other words, Parity, Parity conjecture, Product formula, Quadratic, Quadratic extension, Quadratic twist, Quadratic twists, Real number, Root number, Root numbers, Same parity, Selmer, Selmer group, Selmer groups, Selmer rank, Semistable, Theorem theorem, Vladimir, Vladimir dokchitser.

Abstract

The purpose of the paper is to complete several global and local results concerning parity of ranks of elliptic curves. Primarily, we show that the Shafarevich–Tate conjecture implies the parity conjecture for all elliptic curves over number fields, we give a formula for local and global root numbers of elliptic curves and complete the proof of a conjecture of Kramer and Tunnell in characteristic 0. The method is to settle the outstanding local formulae by deforming from local fields to totally real number fields and then using global parity results.

Url:

https://api.istex.fr/document/35B3B0263FD154132798BAC3C6D3AEFF4F97F640/fulltext/pdf

DOI: 10.1515/crelle.2011.060

Affiliations:

Royaume-Uni

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Le document en format XML

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Root numbers and parity of ranks of elliptic curves

Root numbers and parity of ranks of elliptic curves

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English descriptors

Abstract

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