On the structure of Selmer groups over $p$-adic Lie extensions
Authors:
Yoshihiro Ochi and Otmar Venjakob
Journal:
J. Algebraic Geom. 11 (2002), 547-580
DOI:
https://doi.org/10.1090/S1056-3911-02-00297-7
Published electronically:
March 18, 2002
MathSciNet review:
1894938
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Abstract: The goal of this paper is to prove that the Pontryagin dual of the Selmer group over the trivializing extension of an elliptic curve without complex multiplication does not have any nonzero pseudo-null submodule. The main point is to extend the definition of pseudo-null to modules over the completed group ring $\mathbb {Z}_p[[G]]$ of an arbitrary $p$-adic Lie group $G$ without $p$-torsion. For this purpose we prove that $\mathbb {Z}_p[[G]]$ is an Auslander regular ring. For the proof we also extend some results of Jannsen’s homotopy theory of modules and study intensively higher Iwasawa adjoints.
[Au]aus M. Auslander and M. Bridger. Stable Module Theory, volume 94 of Memoirs of the AMS. AMS, 1969.
[Bj1]erik J.-E. Björk, Filtered Noetherian rings, In Noetherian rings and their applications, Conf. Oberwolfach/FRG 1983, Math. Surv. Monogr., 24, 59-97, 1987.
[Bj2]erik1 ---, Rings of Differential Operators, North-Holland Math. Library, 21, 1979.
[Br]Br A. Brumer, Pseudocompact algebras, profinite groups and class formations, J. Algebra, 4, 442-470, 1966.
[Brun]bruns W. Bruns and J. Herzog. Cohen-Macaulay Rings, Cambridge studies in advance mathematics 39, Cambridge University Press, 1993.
[Co]Co J. Coates, Fragments of the $GL_2$ Iwasawa theory of elliptic curve without complex multiplication, in Arithmetic of Elliptic Curves, LNM 1716, Springer 1999.
[CG]CG J. Coates and R. Greenberg. Kummer theory of abelian varieties over local fields, Invent. Math., 124 (1996), 129-174.
[CH]CH J. Coates and S. Howson, Euler characteristics and elliptic curves II, J. Math. Soc. Japan, 53 (1), 175–235, 2001.
[DSMS]dsms J.D. Dixon, M.P.F. du Sautoy, A. Mann, D. Segal. Analytic Pro-p Groups, Cambridge Studies in Advanced Mathematics 61, Cambridge University Press, 2nd edition, 1999.
[Gr1]green R. Greenberg, On the structure of certain Galois groups, Invent. Math. 47, 85-99, 1978.
[Gr2]Gr ---, Iwasawa theory for p-adic representations, Advanced Studies in Pure Mathematics 17, Algebraic Number Theory-in honour of K. Iwasawa 97-137, 1989 .
[Gr3]Gr3 ---, Iwasawa theory for elliptic curves, in Arithmetic of Elliptic Curves, LNM 1716, Springer, 1999.
[Ha1]Ha M. Harris, p-adic Representations arising from descent on abelian varieties, Compositio Math., 39 (1979), 177-245.
[Ha2]Ha2 ---, Systematic growth of Mordell-Weil groups of abelian varieties in towers of number fields, Invent Math., 51 (1979), 123-141.
[HM]HM Y. Hachimori and K. Matsuno, On finite $\Lambda$-submodules of Selmer groups of elliptic curves, Proc. Amer. Math. Soc. 128 (2000), 2539-2541.
[Iw]Iw K. Iwasawa, On $\operatorname {\mathbb {Z}} _{\ell }$-extensions of algebraic number fields, Ann of Math., 98, 246-326, 1973.
[Ja1]Ja1 U. Jannsen, Iwasawa modules up to isomorphism, Advanced Studies in Pure Mathematics 17, Algebraic Number Theory-in honour of K. Iwasawa 171-207, 1989.
[Ja2]Ja2 ---, On the $\ell$-adic cohomology of varieties over number fields and its Galois cohomology, Galois Groups over $\operatorname {\mathbb {Q}},$ 315-360, Springer, 1989.
[Ja3]Ja3 ---, Continuous etale cohomology, Mathematische Annalen 280 (1988), 207-245.
[Ja4]Ja4 ---, A spectral sequence for Iwasawa adjoints, unpublished notes 1994.
[La]Lazard M. Lazard, Groupes analytiques p-adiques, Publ. Math. I. H. E. S., 26, 1965.
[Le]leva Thierry Levasseur, Grade des modules sur certains anneaux filtres. Commun. Algebra, 9(15), 1519–1532, 1981.
[Ma]Ma B. Mazur, Rational points of abelian varieties with values in towers of number fields, Invent. Math., 18 (1972), 183-226.
[Ng]Ng T. Nguyen-Quang-Do, Formations de classes et modules d’Iwasawa, in: Number Theory Noordwigerhout 1983, LNM 1068, Springer 1984.
[NSW]NSW J. Neukirch, A. Schmidt, K. Wingberg, Cohomology of Number Fields, Springer 2000.
[Oc]Oc Y. Ochi, Iwasawa modules via homotopy theory, PhD thesis, University of Cambridge, 1999.
[OV]OV Y. Ochi and O. Venjakob, On the ranks of Iwasawa modules over $p$-adic Lie extensions, to appear in Math. Proc. Camb. Philos. Soc.
[Pe]PR B. Perrin-Riou, Groupe de Selmer d’une courbe elliptique a multiplication complexe, Compo. Math., vol. 43 (1981), 387-417.
[Sc1]Sc1 P. Schneider, Über gewisse Galoiscohomologiegruppen, Math. Z. 168 (1979), 181-205.
[Sc2]Sc2 ---, Iwasawa $L$-functions of varieties over algebraic number fields, a first approach, Invent. Math. 71 (1983), 251-293.
[Sc3]Sc3 ---, Motivic Iwasawa theory, Advanced Studies in Pure Mathematics 17, Algebraic Number Theory-in honour of K. Iwasawa, pp. 421-456, 1989.
[Se1]Se1 J-P. Serre, Cohomologie Galoisienne, Lecture Notes in Math., 5, Springer 1965.
[Se2]Se2 ---, Abelian $\ell$-adic representations and Elliptic Curves, W. A. Benjamin. INC, 1968.
[ST]ST J-P. Serre and J. Tate, Good reduction of abelian varieties, Ann. of Math., 88 pp. 492-517, (1968).
[Si]Si J. Silverman, The Arithmetic of Elliptic Curves, Springer 1986.
[Su]Su R. Sujatha, Euler-Poincare characteristics of $p$-adic Lie groups and arithmetic, preprint, 2000.
[Ve]Ve O. Venjakob, Iwasawa Theory of $p$-adic Lie extensions, Dissertation, University of Heidelberg, 2000.
[We]We C. Weibel, Introduction to Homological Algebra, Cambridge U.P., 1994.
[Wil]wi J.S. Wilson, Profinite Groups, volume 19 of London Mathematical Society Monographs New Series. Oxford University Press, 1st edition, 1998.
[Wi1]Wi1 K. Wingberg, Duality theorems for $\Gamma$-extensions of algebraic number fields, Composito Math., 55 (1985), 333-381 .
[Wi2]Wi2 ---, On the rational points of abelian varieties over $\operatorname {\mathbb {Z}}_p$-extensions of number fields, Math. Ann. 279, (1987) 279-324.
[Wi3]Wi3 ---, Duality theorems for abelian varieties over $\operatorname {\mathbb {Z}}_p$-extensions, Advanced Studies in Pure Mathematics 17, Algebraic Number Theory-in honour of K. Iwasawa pp. 471-492, 1989.
[Au]aus M. Auslander and M. Bridger. Stable Module Theory, volume 94 of Memoirs of the AMS. AMS, 1969.
[Bj1]erik J.-E. Björk, Filtered Noetherian rings, In Noetherian rings and their applications, Conf. Oberwolfach/FRG 1983, Math. Surv. Monogr., 24, 59-97, 1987.
[Bj2]erik1 ---, Rings of Differential Operators, North-Holland Math. Library, 21, 1979.
[Br]Br A. Brumer, Pseudocompact algebras, profinite groups and class formations, J. Algebra, 4, 442-470, 1966.
[Brun]bruns W. Bruns and J. Herzog. Cohen-Macaulay Rings, Cambridge studies in advance mathematics 39, Cambridge University Press, 1993.
[Co]Co J. Coates, Fragments of the $GL_2$ Iwasawa theory of elliptic curve without complex multiplication, in Arithmetic of Elliptic Curves, LNM 1716, Springer 1999.
[CG]CG J. Coates and R. Greenberg. Kummer theory of abelian varieties over local fields, Invent. Math., 124 (1996), 129-174.
[CH]CH J. Coates and S. Howson, Euler characteristics and elliptic curves II, J. Math. Soc. Japan, 53 (1), 175–235, 2001.
[DSMS]dsms J.D. Dixon, M.P.F. du Sautoy, A. Mann, D. Segal. Analytic Pro-p Groups, Cambridge Studies in Advanced Mathematics 61, Cambridge University Press, 2nd edition, 1999.
[Gr1]green R. Greenberg, On the structure of certain Galois groups, Invent. Math. 47, 85-99, 1978.
[Gr2]Gr ---, Iwasawa theory for p-adic representations, Advanced Studies in Pure Mathematics 17, Algebraic Number Theory-in honour of K. Iwasawa 97-137, 1989 .
[Gr3]Gr3 ---, Iwasawa theory for elliptic curves, in Arithmetic of Elliptic Curves, LNM 1716, Springer, 1999.
[Ha1]Ha M. Harris, p-adic Representations arising from descent on abelian varieties, Compositio Math., 39 (1979), 177-245.
[Ha2]Ha2 ---, Systematic growth of Mordell-Weil groups of abelian varieties in towers of number fields, Invent Math., 51 (1979), 123-141.
[HM]HM Y. Hachimori and K. Matsuno, On finite $\Lambda$-submodules of Selmer groups of elliptic curves, Proc. Amer. Math. Soc. 128 (2000), 2539-2541.
[Iw]Iw K. Iwasawa, On $\operatorname {\mathbb {Z}} _{\ell }$-extensions of algebraic number fields, Ann of Math., 98, 246-326, 1973.
[Ja1]Ja1 U. Jannsen, Iwasawa modules up to isomorphism, Advanced Studies in Pure Mathematics 17, Algebraic Number Theory-in honour of K. Iwasawa 171-207, 1989.
[Ja2]Ja2 ---, On the $\ell$-adic cohomology of varieties over number fields and its Galois cohomology, Galois Groups over $\operatorname {\mathbb {Q}},$ 315-360, Springer, 1989.
[Ja3]Ja3 ---, Continuous etale cohomology, Mathematische Annalen 280 (1988), 207-245.
[Ja4]Ja4 ---, A spectral sequence for Iwasawa adjoints, unpublished notes 1994.
[La]Lazard M. Lazard, Groupes analytiques p-adiques, Publ. Math. I. H. E. S., 26, 1965.
[Le]leva Thierry Levasseur, Grade des modules sur certains anneaux filtres. Commun. Algebra, 9(15), 1519–1532, 1981.
[Ma]Ma B. Mazur, Rational points of abelian varieties with values in towers of number fields, Invent. Math., 18 (1972), 183-226.
[Ng]Ng T. Nguyen-Quang-Do, Formations de classes et modules d’Iwasawa, in: Number Theory Noordwigerhout 1983, LNM 1068, Springer 1984.
[NSW]NSW J. Neukirch, A. Schmidt, K. Wingberg, Cohomology of Number Fields, Springer 2000.
[Oc]Oc Y. Ochi, Iwasawa modules via homotopy theory, PhD thesis, University of Cambridge, 1999.
[OV]OV Y. Ochi and O. Venjakob, On the ranks of Iwasawa modules over $p$-adic Lie extensions, to appear in Math. Proc. Camb. Philos. Soc.
[Pe]PR B. Perrin-Riou, Groupe de Selmer d’une courbe elliptique a multiplication complexe, Compo. Math., vol. 43 (1981), 387-417.
[Sc1]Sc1 P. Schneider, Über gewisse Galoiscohomologiegruppen, Math. Z. 168 (1979), 181-205.
[Sc2]Sc2 ---, Iwasawa $L$-functions of varieties over algebraic number fields, a first approach, Invent. Math. 71 (1983), 251-293.
[Sc3]Sc3 ---, Motivic Iwasawa theory, Advanced Studies in Pure Mathematics 17, Algebraic Number Theory-in honour of K. Iwasawa, pp. 421-456, 1989.
[Se1]Se1 J-P. Serre, Cohomologie Galoisienne, Lecture Notes in Math., 5, Springer 1965.
[Se2]Se2 ---, Abelian $\ell$-adic representations and Elliptic Curves, W. A. Benjamin. INC, 1968.
[ST]ST J-P. Serre and J. Tate, Good reduction of abelian varieties, Ann. of Math., 88 pp. 492-517, (1968).
[Si]Si J. Silverman, The Arithmetic of Elliptic Curves, Springer 1986.
[Su]Su R. Sujatha, Euler-Poincare characteristics of $p$-adic Lie groups and arithmetic, preprint, 2000.
[Ve]Ve O. Venjakob, Iwasawa Theory of $p$-adic Lie extensions, Dissertation, University of Heidelberg, 2000.
[We]We C. Weibel, Introduction to Homological Algebra, Cambridge U.P., 1994.
[Wil]wi J.S. Wilson, Profinite Groups, volume 19 of London Mathematical Society Monographs New Series. Oxford University Press, 1st edition, 1998.
[Wi1]Wi1 K. Wingberg, Duality theorems for $\Gamma$-extensions of algebraic number fields, Composito Math., 55 (1985), 333-381 .
[Wi2]Wi2 ---, On the rational points of abelian varieties over $\operatorname {\mathbb {Z}}_p$-extensions of number fields, Math. Ann. 279, (1987) 279-324.
[Wi3]Wi3 ---, Duality theorems for abelian varieties over $\operatorname {\mathbb {Z}}_p$-extensions, Advanced Studies in Pure Mathematics 17, Algebraic Number Theory-in honour of K. Iwasawa pp. 471-492, 1989.
Additional Information
Yoshihiro Ochi
Affiliation:
Universität Heidelberg, Mathematisches Institut, Im Neuenheimer Feld 288, 69120 Heidelberg, Germany
Address at time of publication:
Korea Institute for Advanced Study (KIAS), 207-43 Cheongryangri-dong, Dongdaemun-gu, Seoul 130-012, South Korea
Email:
ochi@kias.re.kr
Otmar Venjakob
Affiliation:
Universität Heidelberg, Mathematisches Institut, Im Neuenheimer Feld 288, 69120 Heidelberg, Germany
Address at time of publication:
Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 OWB, United Kingdom
Email:
otmar@mathi.uni-heidelberg.de
Received by editor(s):
May 14, 2000
Received by editor(s) in revised form:
September 6, 2000
Published electronically:
March 18, 2002
Additional Notes:
During this research, Y. Ochi has been supported by the Deutsche Forschungsgemeinschaft (DFG) “Forschergruppe Arithmetik" at the Mathematical Institute, Heidelberg.
