Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-18T08:48:50.470Z Has data issue: false hasContentIssue false
  • English
  • Français

Parity conjectures for elliptic curves over global fields of positive characteristic

Part of: Arithmetic algebraic geometry

Published online by Cambridge University Press:  04 May 2011

Fabien Trihan  and
Christian Wuthrich
Fabien Trihan
Affiliation:
School of Mathematical Sciences, University Nottingham, Nottingham NG7 2RD, UK (email: fabien.trihan@nottingham.ac.uk)
Christian Wuthrich
Affiliation:
School of Mathematical Sciences, University Nottingham, Nottingham NG7 2RD, UK (email: christian.wuthrich@gmail.com)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We prove the p-parity conjecture for elliptic curves over global fields of characteristic p>3. We also present partial results on the ℓ-parity conjecture for primes ℓ≠p.

Keywords

parity conjectureelliptic curvesfunction fieldsBirch and Swinnerton-Dyer conjecture

MSC classification

Secondary: 11G40: $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 11G05: Elliptic curves over global fields 11G07: Elliptic curves over local fields
Type
Research Article
Information
Compositio Mathematica , Volume 147 , Issue 4 , July 2011 , pp. 1105 - 1128
Copyright
Copyright © Foundation Compositio Mathematica 2011

References

[Cas65]Cassels, J. W. S., Arithmetic on curves of genus 1. VIII. On conjectures of Birch and Swinnerton-Dyer, J. Reine Angew. Math. 217 (1965), 180–199.CrossRefGoogle Scholar
[CFKS10]Coates, J., Fukaya, T., Kato, K. and Sujatha, R., Root numbers, Selmer groups, and non-commutative Iwasawa theory, J. Algebraic Geom. 19 (2010), 19–97.CrossRefGoogle Scholar
[Dok05]Dokchitser, V., Root numbers of non-abelian twists of elliptic curves, Proc. Lond. Math. Soc. (3) 91 (2005), 300–324, with an appendix by Tom Fisher.CrossRefGoogle Scholar
[DD08]Dokchitser, T. and Dokchitser, V., Parity of ranks for elliptic curves with a cyclic isogeny, J. Number Theory 128 (2008), 662–679.CrossRefGoogle Scholar
[DD09a]Dokchitser, T. and Dokchitser, V., Regulator constants and the parity conjecture, Invent. Math. 178 (2009), 23–71.CrossRefGoogle Scholar
[DD09b]Dokchitser, T. and Dokchitser, V., Root numbers and parity of ranks for elliptic curves, J. Reine Angew. Math., to appear, Preprint (2009) available at http://arxiv.org/abs/0906.1815.Google Scholar
[DD09c]Dokchitser, T. and Dokchitser, V., Self-duality of Selmer groups, Math. Proc. Cambridge Philos. Soc. 146 (2009), 257–267.CrossRefGoogle Scholar
[DD10]Dokchitser, T. and Dokchitser, V., On the Birch–Swinnerton-Dyer quotients modulo squares, Ann. of Math. (2) 172 (2010), 567–596.CrossRefGoogle Scholar
[Gon09]González-Avilés, C. D., Arithmetic duality theorems for 1-motives over function fields, J. Reine Angew. Math. 632 (2009), 203–231.Google Scholar
[GT07]González-Avilés, C. D. and Tan, K.-S., A generalization of the Cassels–Tate dual exact sequence, Math. Res. Lett. 14 (2007), 295–302.CrossRefGoogle Scholar
[GD70]Grothendieck, A. and Demazure (ed), M., Schémas en groupes. I: Propriétés générales des schémas en groupes, in Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3). Dirigé par M. Demazure et A. Grothendieck, Lecture Notes in Mathematics, vol. 151 (Springer, Berlin, 1970).Google Scholar
[HS09]Harari, D. and Szamuely, T., Corrigenda for: arithmetic duality theorems for 1-motives, J. Reine Angew. Math. 632 (2009), 233–236.Google Scholar
[Hel09]Helfgott, H. A., On the behaviour of root numbers in families of elliptic curves, Preprint (2009), available at http://arxiv.org/abs/math.NT/0408141.Google Scholar
[KT03]Kato, K. and Trihan, F., On the conjectures of Birch and Swinnerton-Dyer in characteristic p>0, Invent. Math. 153 (2003), 537–592.CrossRefGoogle Scholar
[KM85]Katz, N. M. and Mazur, B., Arithmetic moduli of elliptic curves, Annals of Mathematics Studies, vol. 108 (Princeton University Press, Princeton, NJ, 1985).CrossRefGoogle Scholar
[Kim07]Kim, B. D., The parity conjecture for elliptic curves at supersingular reduction primes, Compositio Math. 143 (2007), 47–72.CrossRefGoogle Scholar
[MR07]Mazur, B. and Rubin, K., Finding large Selmer rank via an arithmetic theory of local constants, Ann. of Math. (2) 166 (2007), 579–612.CrossRefGoogle Scholar
[Mil68]Milne, J. S., The Tate–Šafarevič group of a constant abelian variety, Invent. Math. 6 (1968), 91–105.CrossRefGoogle Scholar
[Mil80]Milne, J. S., Étale cohomology, Princeton Mathematical Series, vol. 33 (Princeton University Press, Princeton, NJ, 1980).Google Scholar
[Mil06]Milne, J. S., Arithmetic duality theorems, second edition (BookSurge, LLC, Charleston, SC, 2006).Google Scholar
[Nek01]Nekovář, J., On the parity of ranks of Selmer groups. II, C. R. Math. Acad. Sci. Paris Sér. I 332 (2001), 99–104.CrossRefGoogle Scholar
[Nek09]Nekovář, J., On the parity of ranks of Selmer groups. IV, Compositio Math. 145 (2009), 1351–1359, with an appendix by Jean-Pierre Wintenberger.CrossRefGoogle Scholar
[Nek10]Nekovář, J., Some consequences of a formula of Mazur and Rubin for arithmetic local constants, Preprint (2010).Google Scholar
[Roh94]Rohrlich, D. E., Elliptic curves and the Weil–Deligne group, in Elliptic curves and related topics, CRM Proceedings Lecture Notes, vol. 4 (American Mathematical Society, Providence, RI, 1994), 125–157.CrossRefGoogle Scholar
[Tat95]Tate, J., On the conjectures of Birch and Swinnerton-Dyer and a geometric analog, Séminaire Bourbaki, vol. 9 (Soc. Math. France, Paris, 1995), Exp. No. 306, 415–440.Google Scholar
[TO70]Tate, J. and Oort, F., Group schemes of prime order, Ann. Sci. École Norm. Sup. (4) 3 (1970), 1–21.CrossRefGoogle Scholar
[Ulm91]Ulmer, D. L., p-descent in characteristic p, Duke Math. J. 62 (1991), 237–265.CrossRefGoogle Scholar
[Ulm04]Ulmer, D., Elliptic curves and analogies between number fields and function fields, in Heegner points and Rankin L-series, Mathematical Sciences Research Institute Publications, vol. 49 (Cambridge University Press, Cambridge, 2004), 285–315.CrossRefGoogle Scholar
[Ulm05]Ulmer, D. L., Geometric non-vanishing, Invent. Math. 159 (2005), 133–186.CrossRefGoogle Scholar