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Regulator constants and the parity conjecture

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Abstract

The p-parity conjecture for twists of elliptic curves relates multiplicities of Artin representations in p ∞-Selmer groups to root numbers. In this paper we prove this conjecture for a class of such twists. For example, if E/ℚ is semistable at 2 and 3, K/ℚ is abelian and K ∞ is its maximal pro-p extension, then the p-parity conjecture holds for twists of E by all orthogonal Artin representations of \(\mathop{\mathrm{Gal}}(K^{\infty}/{\mathbb{Q}})\) . We also give analogous results when K/ℚ is non-abelian, the base field is not ℚ and E is replaced by an abelian variety. The heart of the paper is a study of relations between permutation representations of finite groups, their “regulator constants”, and compatibility between local root numbers and local Tamagawa numbers of abelian varieties in such relations.

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References

  1. Birch, B.J., Stephens, N.M.: The parity of the rank of the Mordell-Weil group. Topology 5, 295–299 (1966)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bosch, S., Liu, Q.: Rational points of the group of components of a Néron model. Manuscr. Math. 98(3), 275–293 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  3. Coates, J., Fukaya, T., Kato, K., Sujatha, R.: Root numbers, Selmer groups and non-commutative Iwasawa theory. J. Algebraic Geom. (2007, to appear)

  4. Deligne, P.: Valeur de fonctions L et périodes d’intégrales. In: Borel, A., Casselman, W. (eds.) Automorphic Forms, Representations and L-Functions, Part 2. Proc. Symp. in Pure Math., vol. 33, pp. 313–346. AMS, Providence (1979)

    Google Scholar 

  5. Dokchitser, T., Dokchitser, V.: Parity of ranks for elliptic curves with a cyclic isogeny. J. Number Theory 128, 662–679 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  6. Dokchitser, T., Dokchitser, V.: On the Birch–Swinnerton-Dyer quotients modulo squares. Ann. Math. (2006, to appear). arXiv:math.NT/0610290

  7. Dokchitser, T., Dokchitser, V.: Self-duality of Selmer groups. Math. Proc. Camb. Philos. Soc. 146, 257–267 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  8. Dokchitser, V.: Root numbers of non-abelian twists of elliptic curves, with an appendix by T. Fisher. Proc. Lond. Math. Soc. 91(3), 300–324 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  9. Faltings, G.: Endlichkeitssätze für abelsche Varietäten über Zahlkörpern. Invent. Math. 73(3), 349–366 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  10. Fröhlich, A., Queyrut, J.: On the functional equation of the Artin L-function for characters of real representations. Invent. Math. 20, 125–138 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  11. Greenberg, R.: On the Birch and Swinnerton-Dyer conjecture. Invent. Math. 72(2), 241–265 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  12. Greenberg, R.: Iwasawa theory, projective modules and modular representations. Preprint (2008)

  13. Grothendieck, A.: Modèles de Néron et monodromie. In: Séminaire de Géométrie 7, Exposé IX. LNM, vol. 288. Springer, Berlin (1973)

    Google Scholar 

  14. Guo, L.: General Selmer groups and critical values of Hecke L-functions. Math. Ann. 297(2), 221–233 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  15. Hachimori, Y., Venjakob, O.: Completely faithful Selmer groups over Kummer extensions. In: Documenta Mathematica, Extra Volume: Kazuya Kato’s Fiftieth Birthday, pp. 443–478 (2003)

  16. Kim, B.D.: The parity theorem of elliptic curves at primes with supersingular reduction. Compos. Math. 143, 47–72 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  17. Kim, B.D.: The parity conjecture over totally real fields for elliptic curves at supersingular reduction primes. Preprint

  18. Kramer, K.: Arithmetic of elliptic curves upon quadratic extension. Trans. Am. Math. Soc. 264(1), 121–135 (1981)

    Article  MATH  Google Scholar 

  19. Kramer, K., Tunnell, J.: Elliptic curves and local ε-factors. Compos. Math. 46, 307–352 (1982)

    MATH  MathSciNet  Google Scholar 

  20. Mazur, B., Rubin, K.: Finding large Selmer ranks via an arithmetic theory of local constants. Ann. Math. 166(2), 579–612 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  21. Mazur, B., Rubin, K.: Growth of Selmer rank in nonabelian extensions of number fields. Duke Math. J. 143, 437–461 (2008)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  23. Milne, J.S.: Arithmetic duality theorems. Perspect. Math. 1 (1986)

  24. Monsky, P.: Generalizing the Birch-Stephens theorem. I: Modular curves. Math. Z. 221, 415–420 (1996)

    MATH  MathSciNet  Google Scholar 

  25. Nekovář, J.: Selmer complexes. Astérisque 310 (2006)

  26. Nekovář, J.: On the parity of ranks of Selmer groups IV, with an appendix by J.-P. Wintenberger. Compos. Math. (2009, to appear)

  27. Nekovář, J.: Growth of Selmer groups of Hilbert modular forms over ring class fields. Ann. E.N.S., Sér. 4, 41(6), 1003–1022 (2008)

    MATH  Google Scholar 

  28. Raynaud, M.: Variétés abéliennes et géométrie rigide. Actes Congr. Int. Nice 1, 473–477 (1970)

    Google Scholar 

  29. Rohrlich, D.: The vanishing of certain Rankin-Selberg convolutions. In: Automorphic Forms and Analytic Number Theory, pp. 123–133. Les publications CRM, Montreal (1990)

    Google Scholar 

  30. Rohrlich, D.: Elliptic curves and the Weil-Deligne group. In: Elliptic Curves and Related Topics. CRM Proc. Lecture Notes, vol. 4, pp. 125–157. Am. Math. Soc., Providence (1994)

    Google Scholar 

  31. Rohrlich, D.: Galois theory, elliptic curves, and root numbers. Compos. Math. 100, 311–349 (1996)

    MATH  MathSciNet  Google Scholar 

  32. Rohrlich, D.: Scarcity and abundance of trivial zeros in division towers. J. Algebraic Geom. 17, 643–675 (2008)

    MATH  MathSciNet  Google Scholar 

  33. Rohrlich, D.: Galois invariance of local root numbers. Preprint (2008)

  34. Sabitova, M.: Root numbers of abelian varieties and representations of the Weil-Deligne group. Ph.D. thesis, Univ. Pennsylvania (2005)

  35. Sabitova, M.: Root numbers of abelian varieties. Trans. Am. Math. Soc. 359(9), 4259–4284 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  36. Serre, J.-P.: Abelian l-adic Representations and Elliptic Curves. Addison-Wesley, Reading (1989)

    Google Scholar 

  37. Serre, J.-P.: Linear Representations of Finite Groups, GTM 42. Springer, Berlin (1977)

    Google Scholar 

  38. Silverman, J.H.: The Arithmetic of Elliptic Curves, GTM 106. Springer, Berlin (1986)

    Google Scholar 

  39. Silverman, J.H.: Advanced Topics in the Arithmetic of Elliptic Curves, GTM 151. Springer, Berlin (1994)

    Google Scholar 

  40. Tate, J.: Number theoretic background. In: Borel, A., Casselman, W. (eds.) Automorphic Forms, Representations and L-Functions, Part 2. Proc. Symp. in Pure Math., vol. 33, pp. 3–26. AMS, Providence (1979)

    Google Scholar 

  41. Tate, J.: On the conjectures of Birch and Swinnerton-Dyer and a geometric analog. Séminaire Bourbaki, 18e année, no. 306 (1965/66)

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Authors and Affiliations

  1. Robinson College, Cambridge, CB3 9AN, UK

    Tim Dokchitser

  2. Gonville & Caius College, Cambridge, CB2 1TA, UK

    Vladimir Dokchitser

Authors
  1. Tim Dokchitser
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  2. Vladimir Dokchitser
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Correspondence to Tim Dokchitser.

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T. Dokchitser is supported by a Royal Society University Research Fellowship.

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Dokchitser, T., Dokchitser, V. Regulator constants and the parity conjecture. Invent. math. 178, 23–71 (2009). https://doi.org/10.1007/s00222-009-0193-7

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  • DOI: https://doi.org/10.1007/s00222-009-0193-7

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