Root numbers and parity of ranks of elliptic curves
Identifieur interne : 000B17 ( Istex/Corpus ); précédent : 000B16; suivant : 000B18Root numbers and parity of ranks of elliptic curves
Auteurs : Tim Dokchitser ; Vladimir DokchitserSource :
- Journal für die reine und angewandte Mathematik (Crelles Journal) [ 0075-4102 ] ; 2011-09.
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:
DOI: 10.1515/crelle.2011.060
Links to Exploration step
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<issue>658</issue>
<fpage>39</fpage>
<lpage>64</lpage>
<history><date date-type="received"><day>02</day>
<month>11</month>
<year>2009</year>
<string-date>Eingegangen 2. November 2009.</string-date>
</date>
<date date-type="rev-recd"><day>25</day>
<month>04</month>
<year>2010</year>
<string-date>in revidierter Fassung 25. April 2010.</string-date>
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<related-article related-article-type="pdf" xlink:href="crelle.2011.060.pdf"></related-article>
<abstract><title>Abstract</title>
<p>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.</p>
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<titleInfo type="alternative" lang="en" contentType="CDATA"><title>Root numbers and parity of ranks of elliptic curves</title>
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<name type="personal"><namePart type="given">Tim</namePart>
<namePart type="family">Dokchitser</namePart>
<affiliation>Robinson College, Cambridge CB3 9AN, United Kingdom</affiliation>
<affiliation>Robinson College, Cambridge CB3 9AN, United Kingdom</affiliation>
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<name type="personal"><namePart type="given">Vladimir</namePart>
<namePart type="family">Dokchitser</namePart>
<affiliation>Emmanuel College, Cambridge CB2 3AP, United Kingdom</affiliation>
<affiliation>Emmanuel College, Cambridge CB2 3AP, United Kingdom</affiliation>
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<abstract lang="en">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.</abstract>
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