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Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Huybrechts, Daniel / Hwang, Jun-Muk / Williamson, Geordie

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Volume 2016, Issue 719


The p-parity conjecture for elliptic curves with a p-isogeny

Kęstutis Česnavičius
Published Online: 2014-06-11 | DOI: https://doi.org/10.1515/crelle-2014-0040


For an elliptic curve E over a number field K, one consequence of the Birch and Swinnerton-Dyer conjecture is the parity conjecture: the global root number matches the parity of the Mordell–Weil rank. Assuming finiteness of Ш(E/K)[p] for a prime p this is equivalent to the p-parity conjecture: the global root number matches the parity of the p-corank of the p-Selmer group. We complete the proof of the p-parity conjecture for elliptic curves that have a p-isogeny for p>3 (the cases p3 were known). Tim and Vladimir Dokchitser have showed this in the case when E has semistable reduction at all places above p by establishing respective cases of a conjectural formula for the local root number. We remove the restrictions on reduction types by proving their formula in the remaining cases. We apply our result to show that the p-parity conjecture holds for every E with complex multiplication defined over K. Consequently, if for such an elliptic curve Ш(E/K)[p] is infinite, it must contain (p/p)2.


  • [1]

    Borel A. and Serre J.-P., Théorèmes de finitude en cohomologie galoisienne, Comment. Math. Helv. 39 (1964), 111–164. Google Scholar

  • [2]

    Bosch S., Lütkebohmert W. and Raynaud M., Néron models, Ergeb. Math. Grenzgeb. (3) 21, Springer-Verlag, Berlin 1990. Google Scholar

  • [3]

    Breuil C., Groupes p-divisibles, groupes finis et modules filtrés, Ann. of Math. (2) 152 (2000), no. 2, 489–549. Google Scholar

  • [4]

    Cassels J. W. S., Arithmetic on curves of genus 1. IV. Proof of the Hauptvermutung, J. reine angew. Math. 211 (1962), 95–112. Google Scholar

  • [5]

    Coates J., Elliptic curves with complex multiplication and Iwasawa theory, Bull. Lond. Math. Soc. 23 (1991), no. 4, 321–350. Google Scholar

  • [6]

    Coates J., Fukaya T., Kato K. and Sujatha R., Root numbers, Selmer groups, and non-commutative Iwasawa theory, J. Algebraic Geom. 19 (2010), no. 1, 19–97. Google Scholar

  • [7]

    Deligne P., Courbes elliptiques: formulaire d’après J. Tate, Modular functions of one variable. IV (Antwerp 1972), Lecture Notes in Math. 476, Springer-Verlag, Berlin (1975), 53–73. Google Scholar

  • [8]

    Dokchitser T. and Dokchitser V., Parity of ranks for elliptic curves with a cyclic isogeny, J. Number Theory 128 (2008), no. 3, 662–679. Google Scholar

  • [9]

    Dokchitser T. and Dokchitser V., Regulator constants and the parity conjecture, Invent. Math. 178 (2009), no. 1, 23–71. Google Scholar

  • [10]

    Dokchitser T. and Dokchitser V., On the Birch–Swinnerton-Dyer quotients modulo squares, Ann. of Math. (2) 172 (2010), no. 1, 567–596. Google Scholar

  • [11]

    Dokchitser T. and Dokchitser V., Root numbers and parity of ranks of elliptic curves, J. reine angew. Math. 658 (2011), 39–64. Google Scholar

  • [12]

    Dokchitser T. and Dokchitser V., Local invariants of isogenous elliptic curves, Trans. Amer. Math. Soc., to appear. Google Scholar

  • [13]

    Illusie L., Grothendieck’s existence theorem in formal geometry, Fundamental algebraic geometry, Math. Surveys Monogr. 123, American Mathematical Society, Providence (2005), 179–233. Google Scholar

  • [14]

    Kim B. D., The parity conjecture for elliptic curves at supersingular reduction primes, Compos. Math. 143 (2007), no. 1, 47–72. Google Scholar

  • [15]

    Kobayashi S., The local root number of elliptic curves with wild ramification, Math. Ann. 323 (2002), no. 3, 609–623. Google Scholar

  • [16]

    Liedtke C. and Schröer S., The Néron model over the Igusa curves, J. Number Theory 130 (2010), no. 10, 2157–2197. Google Scholar

  • [17]

    Mazur B. and Rubin K., Ranks of twists of elliptic curves and Hilbert’s tenth problem, Invent. Math. 181 (2010), no. 3, 541–575. Google Scholar

  • [18]

    Milne J. S., On the arithmetic of abelian varieties, Invent. Math. 17 (1972), 177–190. Google Scholar

  • [19]

    Nekovář J., Selmer complexes, Astérisque 310, Société Mathématique de France, Paris 2007. Google Scholar

  • [20]

    Nekovář J., On the parity of ranks of Selmer groups. IV, Compos. Math. 145 (2009), no. 6, 1351–1359. Google Scholar

  • [21]

    Nekovář J., Some consequences of a formula of Mazur and Rubin for arithmetic local constants, Algebra Number Theory 7 (2013), no. 5, 1101–1120. Google Scholar

  • [22]

    Nekovář J., Compatibility of arithmetic and algebraic local constants (the case lp), preprint 2014, http://www.math.jussieu.fr/~nekovar/pu/loc.pdf.

  • [23]

    Rohrlich D. E., Elliptic curves and the Weil–Deligne group, Elliptic curves and related topics, CRM Proc. Lecture Notes 4, American Mathematical Society, Providence (1994), 125–157. Google Scholar

  • [24]

    Rohrlich D. E., Galois theory, elliptic curves, and root numbers, Compos. Math. 100 (1996), no. 3, 311–349. Google Scholar

  • [25]

    Rubin K., Elliptic curves with complex multiplication and the conjecture of Birch and Swinnerton-Dyer, Arithmetic theory of elliptic curves (Cetraro 1997), Lecture Notes in Math. 1716, Springer-Verlag, Berlin (1999), 167–234. Google Scholar

  • [26]

    Schaefer E. F., Class groups and Selmer groups, J. Number Theory 56 (1996), no. 1, 79–114. Google Scholar

  • [27]

    Serre J.-P., Local class field theory, Algebraic number theory (Brighton 1965), Thompson, Washington (1967), 128–161. Google Scholar

  • [28]

    Serre J.-P., Propriétés galoisiennes des points d’ordre fini des courbes elliptiques, Invent. Math. 15 (1972), no. 4, 259–331. Google Scholar

  • [29]

    Serre J.-P., Linear representations of finite groups, Grad. Texts in Math. 42, Springer-Verlag, New York 1977. Google Scholar

  • [30]

    Serre J.-P., Local fields, Grad. Texts in Math. 67, Springer-Verlag, New York 1979. Google Scholar

  • [31]

    Serre J.-P., Galois cohomology, Springer Monogr. Math., Springer-Verlag, Berlin 2002. Google Scholar

  • [32]

    Tate J. T., The arithmetic of elliptic curves, Invent. Math. 23 (1974), 179–206. Google Scholar

  • [33]

    Tate J., Algorithm for determining the type of a singular fiber in an elliptic pencil, Modular functions of one variable. IV (Antwerp 1972), Lecture Notes in Math. 476, Springer-Verlag, Berlin (1975), 33–52. Google Scholar

  • [34]

    Tate J. and Oort F., Group schemes of prime order, Ann. Sci. Éc. Norm. Supér. (4) 3 (1970), 1–21. Google Scholar

About the article

Received: 2012-09-03

Revised: 2014-03-10

Published Online: 2014-06-11

Published in Print: 2016-10-01

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2016, Issue 719, Pages 45–73, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: https://doi.org/10.1515/crelle-2014-0040.

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