Growth of Selmer groups of Hilbert modular forms over ring class fields

Jan Nekovář

Annales scientifiques de l'École Normale Supérieure (2008)

  • Volume: 41, Issue: 6, page 1003-1022
  • ISSN: 0012-9593

Abstract

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We prove non-trivial lower bounds for the growth of ranks of Selmer groups of Hilbert modular forms over ring class fields and over certain Kummer extensions, by establishing first a suitable parity result.

How to cite

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Nekovář, Jan. "Growth of Selmer groups of Hilbert modular forms over ring class fields." Annales scientifiques de l'École Normale Supérieure 41.6 (2008): 1003-1022. <http://eudml.org/doc/272180>.

@article{Nekovář2008,
abstract = {We prove non-trivial lower bounds for the growth of ranks of Selmer groups of Hilbert modular forms over ring class fields and over certain Kummer extensions, by establishing first a suitable parity result.},
author = {Nekovář, Jan},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {Selmer groups; Hilbert modular forms},
language = {eng},
number = {6},
pages = {1003-1022},
publisher = {Société mathématique de France},
title = {Growth of Selmer groups of Hilbert modular forms over ring class fields},
url = {http://eudml.org/doc/272180},
volume = {41},
year = {2008},
}

TY - JOUR
AU - Nekovář, Jan
TI - Growth of Selmer groups of Hilbert modular forms over ring class fields
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2008
PB - Société mathématique de France
VL - 41
IS - 6
SP - 1003
EP - 1022
AB - We prove non-trivial lower bounds for the growth of ranks of Selmer groups of Hilbert modular forms over ring class fields and over certain Kummer extensions, by establishing first a suitable parity result.
LA - eng
KW - Selmer groups; Hilbert modular forms
UR - http://eudml.org/doc/272180
ER -

References

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  1. [1] E. Aflalo & J. Nekovář, Non-triviality of CM points in ring class field towers, to appear in Israel J. Math.. Zbl1273.11094MR2607546
  2. [2] S. Bloch & K. Kato, L -functions and Tamagawa numbers of motives, in The Grothendieck Festschrift, Vol. I, Progr. Math. 86, Birkhäuser, 1990, 333–400. Zbl0768.14001MR1086888
  3. [3] J. Coates, T. Fukaya, K. Kato & R. Sujatha, Root numbers, Selmer groups and non-commutative Iwasawa theory, preprint. Zbl1213.11135MR2551757
  4. [4] C. Cornut & V. Vatsal, Nontriviality of Rankin-Selberg L -functions and CM points, in L -functions and Galois representations (Durham, July 2004), LMS Lecture Note Series 320, Cambridge Univ. Press, 2007, 121–186. Zbl1153.11025MR2392354
  5. [5] T. Dokchitser & V. Dockchitser, Regulator constants and the parity conjecture, preprint arXiv:0709.2852. Zbl1219.11083
  6. [6] T. Dokchitser & V. Dokchitser, On the Birch-Swinnerton-Dyer quotients modulo squares, preprint arXiv:math/0610290. Zbl1223.11079
  7. [7] V. Dokchitser, Root numbers of non-abelian twists of elliptic curves, Proc. London Math. Soc.91 (2005), 300–324. Zbl1076.11042MR2167089
  8. [8] J.-M. Fontaine & B. Perrin-Riou, Autour des conjectures de Bloch et Kato: cohomologie galoisienne et valeurs de fonctions L , in Motives (Seattle, WA, 1991), Proc. Sympos. Pure Math. 55, Amer. Math. Soc., 1994, 599–706. Zbl0821.14013MR1265546
  9. [9] B. H. Gross & D. Prasad, Test vectors for linear forms, Math. Ann.291 (1991), 343–355. Zbl0768.22004MR1129372
  10. [10] H. Jacquet, Automorphic forms on GL ( 2 ) . Part II, Lecture Notes in Math. 278, Springer, 1972. Zbl0243.12005MR562503
  11. [11] H. Jacquet & R. P. Langlands, Automorphic forms on GL ( 2 ) , Lecture Notes in Math. 114, Springer, 1970. Zbl0236.12010MR401654
  12. [12] B. Mazur & K. Rubin, Finding large Selmer rank via an arithmetic theory of local constants, Ann. of Math.166 (2007), 579–612. Zbl1219.11084MR2373150
  13. [13] J. Nekovář, Selmer complexes, Astérisque 310 (2006), 559. Zbl1211.11120MR2333680
  14. [14] J. Nekovář, On the parity of ranks of Selmer groups. III, Doc. Math. 12 (2007), 243–274. Zbl1201.11067MR2350290
  15. [15] J. Nekovář, The Euler system method for CM points on Shimura curves, in L -functions and Galois representations (Durham, July 2004), LMS Lecture Note Series 320, Cambridge Univ. Press, 2007, 471–547. Zbl1152.11023MR2392363
  16. [16] J. Nekovář, On the parity of ranks of Selmer groups IV, to appear in Compositio Math.. Zbl1221.11150MR2575086
  17. [17] H. Saito, On Tunnell’s formula for characters of GL ( 2 ) , Compositio Math.85 (1993), 99–108. Zbl0795.22009MR1199206
  18. [18] J. B. Tunnell, Local ϵ -factors and characters of GL ( 2 ) , Amer. J. Math.105 (1983), 1277–1307. Zbl0532.12015MR721997
  19. [19] J.-L. Waldspurger, Correspondances de Shimura et quaternions, Forum Math.3 (1991), 219–307. Zbl0724.11026MR1103429

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