Root numbers, Selmer groups, and non-commutative Iwasawa theory

Coates, John; Fukaya, Takako; Kato, Kazuya; Sujatha, Ramdorai

doi:10.1090/S1056-3911-09-00504-9

Authors: John Coates, Takako Fukaya, Kazuya Kato and Ramdorai Sujatha
Journal: J. Algebraic Geom. 19 (2010), 19-97
DOI: https://doi.org/10.1090/S1056-3911-09-00504-9
Published electronically: April 15, 2009
MathSciNet review: 2551757
Full-text PDF

Abstract | References | Additional Information

Abstract: Let $E$ be an elliptic curve over a number field $F$, and let $F_\infty$ be a Galois extension of $F$ whose Galois group $G$ is a $p$-adic Lie group. The aim of the present paper is to provide some evidence that, in accordance with the main conjectures of Iwasawa theory, there is a close connection between the action of the Selmer group of $E$ over $F_\infty$, and the global root numbers attached to the twists of the complex $L$-function of $E$ by Artin representations of $G$.

References

Hyman Bass, Algebraic $K$-theory, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR 0249491
Nicolas Bourbaki, Commutative algebra. Chapters 1–7, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1989. Translated from the French; Reprint of the 1972 edition. MR 979760
Christophe Breuil, Groupes $p$-divisibles, groupes finis et modules filtrés, Ann. of Math. (2) 152 (2000), no. 2, 489–549 (French, with French summary). MR 1804530, DOI https://doi.org/10.2307/2661391
B. J. Birch and N. M. Stephens, The parity of the rank of the Mordell-Weil group, Topology 5 (1966), 295–299. MR 201379, DOI https://doi.org/10.1016/0040-9383%2866%2990021-8
Henri Cartan and Samuel Eilenberg, Homological algebra, Princeton University Press, Princeton, N. J., 1956. MR 0077480
Charles W. Curtis and Irving Reiner, Methods of representation theory. Vol. I, John Wiley & Sons, Inc., New York, 1981. With applications to finite groups and orders; Pure and Applied Mathematics; A Wiley-Interscience Publication. MR 632548
J. H. Coates and S. Howson, Euler characteristics and elliptic curves. II, J. Math. Soc. Japan 53 (2001), no. 1, 175–235. MR 1800527, DOI https://doi.org/10.2969/jmsj/05310175
John Coates, Takako Fukaya, Kazuya Kato, Ramdorai Sujatha, and Otmar Venjakob, The $\rm GL_2$ main conjecture for elliptic curves without complex multiplication, Publ. Math. Inst. Hautes Études Sci. 101 (2005), 163–208. MR 2217048, DOI https://doi.org/10.1007/s10240-004-0029-3
J. Coates and R. Greenberg, Kummer theory for abelian varieties over local fields, Invent. Math. 124 (1996), no. 1-3, 129–174. MR 1369413, DOI https://doi.org/10.1007/s002220050048
John Coates, Peter Schneider, and Ramdorai Sujatha, Links between cyclotomic and ${\rm GL}_2$ Iwasawa theory, Doc. Math. Extra Vol. (2003), 187–215. Kazuya Kato’s fiftieth birthday. MR 2046599
J. Coates and R. Sujatha, Galois cohomology of elliptic curves, Tata Institute of Fundamental Research Lectures on Mathematics, vol. 88, Published by Narosa Publishing House, New Delhi; for the Tata Institute of Fundamental Research, Mumbai, 2000. MR 1759312
John Coates, Ramdorai Sujatha, and Jean-Pierre Wintenberger, On the Euler-Poincaré characteristics of finite dimensional $p$-adic Galois representations, Publ. Math. Inst. Hautes Études Sci. 93 (2001), 107–143. MR 1863736, DOI https://doi.org/10.1007/s10240-001-8189-x
P. Deligne, Les constantes des équations fonctionnelles des fonctions $L$, Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Springer, Berlin, 1973, pp. 501–597. Lecture Notes in Math., Vol. 349 (French). MR 0349635

Darmon, H., Tian, Y., Heegner points over false Tate curve extensions, Preprint.

Vladimir Dokchitser, Root numbers of non-abelian twists of elliptic curves, Proc. London Math. Soc. (3) 91 (2005), no. 2, 300–324. With an appendix by Tom Fisher. MR 2167089, DOI https://doi.org/10.1112/S0024611505015261

T. Dokchitser and V. Dokchitser, Computations in non-commutative Iwasawa theory, Proc. Lond. Math. Soc. (3) 94 (2007), no. 1, 211–272. With an appendix by J. Coates and R. Sujatha. MR 2294995, DOI https://doi.org/10.1112/plms/pdl014

Dokchitser, T., Dokchitser, V., Ranks of elliptic curves with a cyclic isogeny, Journal of Number Theory 128 (2008), 662–679.

Dokchitser, T., Dokchitser, V., On the Birch Swinnerton-Dyer quotients modulo squares, arXiv:math/0610290v2, to appear in Annals of Mathematics.

Dokchitser, T., Dokchitser, V., Regulator constants and the parity conjecture, arXiv:math 0709.2852, to appear in Inventiones Mathematicae.

Matthias Flach, A generalisation of the Cassels-Tate pairing, J. Reine Angew. Math. 412 (1990), 113–127. MR 1079004, DOI https://doi.org/10.1515/crll.1990.412.113

Ralph Greenberg, Iwasawa theory for elliptic curves, Arithmetic theory of elliptic curves (Cetraro, 1997) Lecture Notes in Math., vol. 1716, Springer, Berlin, 1999, pp. 51–144. MR 1754686, DOI https://doi.org/10.1007/BFb0093453

Ralph Greenberg, Iwasawa theory for $p$-adic representations, Algebraic number theory, Adv. Stud. Pure Math., vol. 17, Academic Press, Boston, MA, 1989, pp. 97–137. MR 1097613, DOI https://doi.org/10.2969/aspm/01710097

Greenberg, R., Iwasawa theory, projective modules, and modular representations, to appear in Memoirs of Amer. Math. Soc.

Li Guo, General Selmer groups and critical values of Hecke $L$-functions, Math. Ann. 297 (1993), no. 2, 221–233. MR 1241803, DOI https://doi.org/10.1007/BF01459498

Yoshitaka Hachimori and Kazuo Matsuno, An analogue of Kida’s formula for the Selmer groups of elliptic curves, J. Algebraic Geom. 8 (1999), no. 3, 581–601. MR 1689359

Yoshitaka Hachimori and Otmar Venjakob, Completely faithful Selmer groups over Kummer extensions, Doc. Math. Extra Vol. (2003), 443–478. Kazuya Kato’s fiftieth birthday. MR 2046605

Kazuya Kato, $p$-adic Hodge theory and values of zeta functions of modular forms, Astérisque 295 (2004), ix, 117–290 (English, with English and French summaries). Cohomologies $p$-adiques et applications arithmétiques. III. MR 2104361

Kim, B.-D., The parity conjecture and algebraic functional equations for elliptic curves at supersingular reduction primes, Ph.D Thesis, Stanford University (2005).

Kazuo Matsuno, Finite $\Lambda $-submodules of Selmer groups of abelian varieties over cyclotomic $\Bbb Z_p$-extensions, J. Number Theory 99 (2003), no. 2, 415–443. MR 1969183, DOI https://doi.org/10.1016/S0022-314X%2802%2900078-1

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

Mazur, M., Rubin, K., Growth of Selmer ranks in nonabelian extensions of number fields, Duke Math. J. 143 (2008), 437–461.

J. S. Milne, Arithmetic duality theorems, Perspectives in Mathematics, vol. 1, Academic Press, Inc., Boston, MA, 1986. MR 881804

P. Monsky, Generalizing the Birch-Stephens theorem. I. Modular curves, Math. Z. 221 (1996), no. 3, 415–420. MR 1381589, DOI https://doi.org/10.1007/PL00004518

Jan Nekovář, Selmer complexes, Astérisque 310 (2006), viii+559 (English, with English and French summaries). MR 2333680

Jan Nekovář, On the parity of ranks of Selmer groups. II, C. R. Acad. Sci. Paris Sér. I Math. 332 (2001), no. 2, 99–104 (English, with English and French summaries). MR 1813764, DOI https://doi.org/10.1016/S0764-4442%2800%2901808-5

Michel Raynaud, Variétés abéliennes et géométrie rigide, Actes du Congrès International des Mathématiciens (Nice, 1970) Gauthier-Villars, Paris, 1971, pp. 473–477. MR 0427326

David E. Rohrlich, Galois theory, elliptic curves, and root numbers, Compositio Math. 100 (1996), no. 3, 311–349. MR 1387669

David E. Rohrlich, Scarcity and abundance of trivial zeros in division towers, J. Algebraic Geom. 17 (2008), no. 4, 643–675. MR 2424923, DOI https://doi.org/10.1090/S1056-3911-08-00462-1

Jean-Pierre Serre, Propriétés galoisiennes des points d’ordre fini des courbes elliptiques, Invent. Math. 15 (1972), no. 4, 259–331 (French). MR 387283, DOI https://doi.org/10.1007/BF01405086

Jean-Pierre Serre, Linear representations of finite groups, Springer-Verlag, New York-Heidelberg, 1977. Translated from the second French edition by Leonard L. Scott; Graduate Texts in Mathematics, Vol. 42. MR 0450380

Jean-Pierre Serre, Sur la dimension cohomologique des groupes profinis, Topology 3 (1965), 413–420 (French). MR 180619, DOI https://doi.org/10.1016/0040-9383%2865%2990006-6

Grothendieck, A., et al., Groupes de monodromie en géométrie algébrique (SGA 7, Vol 1), Lecture Notes in Math. 288 (1972).

Shuter, M., Descent on division fields of elliptic curves, Cambridge Ph.D Thesis (2006).

Richard G. Swan, The Grothendieck ring of a finite group, Topology 2 (1963), 85–110. MR 153722, DOI https://doi.org/10.1016/0040-9383%2863%2990025-9

Richard G. Swan, $K$-theory of finite groups and orders, Lecture Notes in Mathematics, Vol. 149, Springer-Verlag, Berlin-New York, 1970. MR 0308195

Additional Information

John Coates
Affiliation: DPMMS, University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, England
MR Author ID: 50035
Email: J.H.Coates@dpmms.cam.ac.uk

Takako Fukaya
Affiliation: Keio University, Hiyoshi, Kohoku-ku, Yokohama, 223-8521, Japan
Email: takakof@hc.cc.keio.ac.jp

Kazuya Kato
Affiliation: Department of Mathematics, Kyoto University, Kyoto, 606-8502, Japan
Email: kzkt@math.kyoto-u.ac.jp

Ramdorai Sujatha
Affiliation: School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400 005, India
MR Author ID: 293023
ORCID: 0000-0003-1221-0710
Email: sujatha@math.tifr.res.in

Received by editor(s): July 1, 2007
Received by editor(s) in revised form: October 24, 2007
Published electronically: April 15, 2009
Additional Notes: The second author gratefully acknowledges support from the JSPS Postdoctoral Fellowship for research abroad