Conjecture A and $\mu $-invariant for Selmer groups of supersingular elliptic curves

Parham Hamidi; Jishnu Ray

doi:10.5802/jtnb.1181

Conjecture A and

μ

-invariant for Selmer groups of supersingular elliptic curves

Parham Hamidi ¹ ; Jishnu Ray ²

¹ Department of Mathematics, The University of British Columbia Room 121, 1984 Mathematics Road, Vancouver, BC Canada V6T 1Z2
² School of Mathematics, Tata Institute of Fundamental Research, Dr Homi Bhabha Road, Navy Nagar, Colaba, Mumbai, Maharashtra 400005, India

Journal de théorie des nombres de Bordeaux, Tome 33 (2021) no. 3.1, pp. 853-886.

Résumé
Abstract

Soit $p$ un nombre premier impair et soit $E$ une courbe elliptique sur un corps de nombres $F$ ayant bonne réduction en toute place au-dessus de $p .$ Dans cet article de synthèse, nous donnons un aperçu de certains des résultats importants sur le groupe de Selmer fin et les groupes de Selmer signés dans les tours cyclotomiques aussi que sur les groupes de Selmer signés dans les $ℤ_{p}^{2}$ -extensions d’un corps quadratique imaginaire, où $p$ est complètement décomposé. Nous discutons uniquement des aspects algébriques en utilisant des outils de la théorie d’Iwasawa. Nous donnons un survol de certains des résultats récents impliquant l’annulation de l’invariant $μ$ sous l’hypothèse de la conjecture A. En outre, nous esquissons une analogie entre le groupe de Selmer classique dans le cas de bonne réduction ordinaire et le groupe de Selmer signé de Kobayashi dans le cas supersingulier. Nous mettons l’accent sur les propriétés des groupes de Selmer signés dans le cas où $E$ a bonne réduction supersingulière, qui sont complètement analogues à celles des groupes de Selmer classiques quand $E$ a bonne réduction ordinaire. Cet article ne contient pas de démonstrations, cependant pour le lecteur intéressé, nous donnons des références pour les résultats exposés dans le texte.

Let $p$ be an odd prime and let $E$ be an elliptic curve defined over a number field $F$ with good reduction at the primes above $p$ . In this survey article, we give an overview of some of the important results proven for the fine Selmer group and the signed Selmer groups over cyclotomic towers as well as the signed Selmer groups over $ℤ_{p}^{2}$ -extensions of an imaginary quadratic field where $p$ splits completely. We only discuss the algebraic aspects of these objects through Iwasawa theory. We also attempt to give some of the recent results implying the vanishing of the $μ$ -invariant under the hypothesis of Conjecture A. Moreover, we draw an analogy between the classical Selmer group in the ordinary reduction case and that of the signed Selmer groups of Kobayashi in the supersingular reduction case. We highlight properties of signed Selmer groups, when $E$ has good supersingular reduction, which are completely analogous to the classical Selmer group, when $E$ has good ordinary reduction. In this survey paper we do not present any proofs, however, we have tried to give references of the discussed results for the interested reader.

Reçu le : 2020-02-26
Révisé le : 2020-10-09
Accepté le : 2021-02-01
Publié le : 2022-01-26

DOI : 10.5802/jtnb.1181

Classification : 11G40, 11R23, 14H52, 14G42
Mots clés : Elliptic curves, Iwasawa theory, Selmer group

Affiliations des auteurs :

Parham Hamidi ¹ ; Jishnu Ray ²

¹ Department of Mathematics, The University of British Columbia Room 121, 1984 Mathematics Road, Vancouver, BC Canada V6T 1Z2
² School of Mathematics, Tata Institute of Fundamental Research, Dr Homi Bhabha Road, Navy Nagar, Colaba, Mumbai, Maharashtra 400005, India

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Parham Hamidi and Jishnu Ray},
     title = {Conjecture~A and $\mu $-invariant for {Selmer} groups of supersingular elliptic curves},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {853--886},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
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Parham Hamidi; Jishnu Ray. Conjecture A and $\mu $-invariant for Selmer groups of supersingular elliptic curves. Journal de théorie des nombres de Bordeaux, Tome 33 (2021) no. 3.1, pp. 853-886. doi : 10.5802/jtnb.1181. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1181/

Bibliographie
Cité par

[1] Victor Abrashkin An analogue of the field-of-norms functor and of the Grothendieck conjecture, J. Algebr. Geom., Volume 16 (2007) no. 4, pp. 671-730 | DOI | MR | Zbl

[2] Suman Ahmed; Meng Fai Lim On the Euler characteristics of signed Selmer groups, Bull. Aust. Math. Soc., Volume 101 (2020) no. 2, pp. 238-246 | DOI | MR | Zbl

[3] Fabrizio Andreatta Generalized ring of norms and generalized $(φ, Γ)$ -modules, Ann. Sci. Éc. Norm. Supér., Volume 39 (2006) no. 4, pp. 599-647 | DOI | Numdam | MR | Zbl

[4] Konstantin Ardakov; Kenneth A. Brown Ring-theoretic properties of Iwasawa algebras: a survey, Doc. Math., Volume Extra Vol. (2006), pp. 7-33 | MR

[5] Denis Benois On Iwasawa theory of crystalline representations, Duke Math. J., Volume 104 (2000) no. 2, pp. 211-267 | MR | Zbl

[6] Laurent Berger Bloch and Kato’s exponential map: three explicit formulas, Doc. Math., Volume Extra Vol. (2003), pp. 99-129 (Kazuya Kato’s fiftieth birthday) | MR | Zbl

[7] Laurent Berger Limites de représentations cristallines, Compos. Math., Volume 140 (2004) no. 6, pp. 1473-1498 | DOI | MR | Zbl

[8] Laurent Berger Lifting the field of norms, J. Éc. Polytech., Math., Volume 1 (2014), pp. 29-38 | DOI | Numdam | MR | Zbl

[9] Laurent Berger; Lionel Fourquaux Iwasawa theory and $F$ -analytic Lubin-Tate $(φ, Γ)$ -modules, Doc. Math., Volume 22 (2017), pp. 999-1030 | MR | Zbl

[10] Patrick Billot Quelques aspects de la descente sur une courbe elliptique dans le cas de réduction supersingulière, Compos. Math., Volume 58 (1986) no. 3, pp. 341-369 | Numdam | MR | Zbl

[11] Nicolas Bourbaki Elements of Mathematics. Algebra. II. Chapters 4–7, Springer, 1990, vii+461 pages (Translated from the French by P. M. Cohn and J. Howie) | MR

[12] Kâzım Büyükboduk; Antonio Lei Integral Iwasawa theory of Galois representations for non-ordinary primes, Math. Z., Volume 286 (2017) no. 1-2, pp. 361-398 | DOI | MR | Zbl

[13] Frédéric Cherbonnier; Pierre Colmez Représentations $p$ -adiques surconvergentes, Invent. Math., Volume 133 (1998) no. 3, pp. 581-611 | DOI | MR | Zbl

[14] Frédéric Cherbonnier; Pierre Colmez Théorie d’Iwasawa des représentations $p$ -adiques d’un corps local, J. Am. Math. Soc., Volume 12 (1999) no. 1, pp. 241-268 | DOI | MR | Zbl

[15] John Coates; Ralph Greenberg Kummer theory for abelian varieties over local fields, Invent. Math., Volume 124 (1996) no. 1-3, pp. 129-174 | DOI | MR | Zbl

[16] John Coates; Susan Howson Euler characteristics and elliptic curves. II, J. Math. Soc. Japan, Volume 53 (2001) no. 1, pp. 175-235 | DOI | MR | Zbl

[17] John Coates; Peter Schneider; Ramdorai Sujatha Links between cyclotomic and ${GL}_{2}$ Iwasawa theory, Doc. Math., Volume Extra Vol. (2003), pp. 187-215 (Kazuya Kato’s fiftieth birthday) | MR | Zbl

[18] John Coates; Peter Schneider; Ramdorai Sujatha Modules over Iwasawa algebras, J. Inst. Math. Jussieu, Volume 2 (2003) no. 1, pp. 73-108 | DOI | MR

[19] John Coates; Ramdorai Sujatha Fine Selmer groups of elliptic curves over $p$ -adic Lie extensions, Math. Ann., Volume 331 (2005) no. 4, pp. 809-839 | DOI | MR | Zbl

[20] John Coates; Ramdorai Sujatha Galois cohomology of elliptic curves, Narosa Publishing House, 2010, xii+98 pages | MR

[21] Bruce Ferrero; Lawrence C. Washington The Iwasawa Invariant $μ_{p}$ Vanishes for Abelian Number Fields, Ann. Math., Volume 109 (1979) no. 2, pp. 377-395 | DOI | MR | Zbl

[22] Jean-Marc Fontaine Sur certains types de représentations $p$ -adiques du groupe de Galois d’un corps local; construction d’un anneau de Barsotti-Tate, Ann. Math., Volume 115 (1982) no. 3, pp. 529-577 | DOI | MR | Zbl

[23] Jean-Marc Fontaine Représentations $p$ -adiques des corps locaux. I, The Grothendieck Festschrift, Vol. II (Progress in Mathematics), Volume 87, Birkhäuser, 1990, pp. 249-309 | MR | Zbl

[24] Jean-Marc Fontaine; Jean-Pierre Wintenberger Extensions algébrique et corps des normes des extensions APF des corps locaux, C. R. Math. Acad. Sci. Paris, Volume 288 (1979) no. 8, p. A441-A444 | MR | Zbl

[25] Ofer Gabber; Lorenzo Ramero Almost ring theory, Lecture Notes in Mathematics, 1800, Springer, 2003, vi+307 pages | DOI | MR

[26] Ralph Greenberg The structure of Selmer groups, Proc. Natl. Acad. Sci. USA, Volume 94 (1997) no. 21, pp. 11125-11128 | DOI | MR | Zbl

[27] Ralph Greenberg Iwasawa theory for elliptic curves, Arithmetic theory of elliptic curves (Cetraro, 1997) (Lecture Notes in Mathematics), Volume 1716, Springer, 1999, pp. 51-144 | DOI | MR | Zbl

[28] Laurent Herr Sur la cohomologie galoisienne des corps $p$ -adiques, Bull. Soc. Math. Fr., Volume 126 (1998) no. 4, pp. 563-600 | DOI | Numdam | MR | Zbl

[29] Florian Ito Sprung Iwasawa theory for elliptic curves at supersingular primes: A pair of main conjectures, J. Number Theory, Volume 132 (2012) no. 7, pp. 1483-1506 | DOI | MR | Zbl

[30] Florian Ito Sprung Chromatic Selmer groups and arithmetic invariants of elliptic curves, J. Théor. Nombres Bordeaux, Volume 33 (2021) no. 3.2, pp. 1103-1114

[31] Kenkichi Iwasawa On $Γ$ -extensions of algebraic number fields, Bull. Am. Math. Soc., Volume 65 (1959), pp. 183-226 | DOI | MR | Zbl

[32] Kenkichi Iwasawa On $ℤ_{l}$ -Extensions of Algebraic Number Fields, Ann. Math., Volume 98 (1973) no. 2, pp. 246-326 | DOI | Zbl

[33] Kazuya Kato $p$ -adic Hodge theory and values of zeta functions of modular forms, Cohomologies $p$ -adiques et applications arithmétiques. III (Astérisque), Volume 295, Société Mathématique de France, 2004, pp. 117-290 | Numdam | MR | Zbl

[34] Byoung Du Kim The plus/minus Selmer groups for supersingular primes, J. Aust. Math. Soc., Volume 95 (2013) no. 2, pp. 189-200 | MR | Zbl

[35] Byoung Du Kim Signed-Selmer groups over the $ℤ_{p}^{2}$ -extension of an imaginary quadratic field, Can. J. Math., Volume 66 (2014) no. 4, pp. 826-843 | DOI | MR | Zbl

[36] Takahiro Kitajima; Rei Otsuki On the plus and the minus Selmer groups for elliptic curves at supersingular primes, Tokyo J. Math., Volume 41 (2018) no. 1, pp. 273-303 | MR | Zbl

[37] Shin-ichi Kobayashi Iwasawa theory for elliptic curves at supersingular primes, Invent. Math., Volume 152 (2003) no. 1, pp. 1-36 | DOI | MR | Zbl

[38] Benjamin Kupferer; Otmar Venjakob Herr-complexes in the Lubin-Tate setting, 2020 | arXiv

[39] Antonio Lei Iwasawa theory for modular forms at supersingular primes, Compos. Math., Volume 147 (2011) no. 3, pp. 803-838 | DOI | MR | Zbl

[40] Antonio Lei; David Loeffler; Sarah Livia Zerbes Coleman maps and the $p$ -adic regulator, Algebra Number Theory, Volume 5 (2011) no. 8, pp. 1095-1131 | DOI | MR | Zbl

[41] Antonio Lei; Ramdorai Sujatha On Selmer groups in the supersingular reduction case, Tokyo J. Math., Volume 43 (2020) no. 2, pp. 455-479 | MR | Zbl

[42] Antonio Lei; Sarah Livia Zerbes Signed Selmer groups over $p$ -adic Lie extensions, J. Théor. Nombres Bordeaux, Volume 24 (2012) no. 2, pp. 377-403 | Numdam | MR | Zbl

[43] Ahmed Matar On the $Λ$ -cotorsion subgroup of the Selmer group, Asian J. Math., Volume 24 (2020) no. 3, pp. 437-456 | DOI | MR | Zbl

[44] Barry Mazur Rational points of abelian varieties with values in towers of number fields, Invent. Math., Volume 18 (1972), pp. 183-266 | DOI | MR | Zbl

[45] Barry Mazur; Andrew Wiles Class fields of abelian extensions of $ℚ$ , Invent. Math., Volume 76 (1984) no. 2, pp. 179-330 | DOI | MR

[46] Jürgen Neukirch; Alexander Schmidt; Kay Wingberg Cohomology of number fields, Grundlehren der Mathematischen Wissenschaften, 323, Springer, 2008, xvi+825 pages | DOI | MR

[47] Yoshihiro Ochi; Otmar Venjakob On the structure of Selmer groups over $p$ -adic Lie extensions, J. Algebr. Geom., Volume 11 (2002) no. 3, pp. 547-580 | DOI | MR | Zbl

[48] Bernadette Perrin-Riou Groupe de Selmer d’une courbe elliptique à multiplication complexe, Compos. Math., Volume 43 (1981) no. 3, pp. 387-417 | MR | Zbl

[49] Bernadette Perrin-Riou Arithmétique des courbes elliptiques et théorie d’Iwasawa, Mém. Soc. Math. Fr., Nouv. Sér. (1984) no. 17, p. 130 | MR | Zbl

[50] Bernadette Perrin-Riou Théorie d’Iwasawa et hauteurs $p$ -adiques (cas des variétés abéliennes), Séminaire de Théorie des Nombres, Paris, 1990–91 (Progress in Mathematics), Volume 108, Birkhäuser, 1993, pp. 203-220 | DOI | MR | Zbl

[51] Bernadette Perrin-Riou Théorie d’Iwasawa des représentations $p$ -adiques sur un corps local, Invent. Math., Volume 115 (1994) no. 1, pp. 81-161 (with an appendix by Jean-Marc Fontaine) | DOI | MR | Zbl

[52] Bernadette Perrin-Riou Fonctions $L$ $p$ -adiques des représentations $p$ -adiques, Astérisque, 229, Société Mathématique de France, 1995, 198 pages | Numdam

[53] Bernadette Perrin-Riou Arithmétique des courbes elliptiques à réduction supersingulière en $p$ , Exp. Math., Volume 12 (2003) no. 2, pp. 155-186 | DOI | MR | Zbl

[54] Robert Pollack On the $p$ -adic $L$ -function of a modular form at a supersingular prime, Duke Math. J., Volume 118 (2003) no. 3, pp. 523-558 | DOI | MR | Zbl

[55] Gautier Ponsinet Universal norms and the Fargues–Fontaine curve (2020) (https://arxiv.org/abs/2010.02292)

[56] Sujatha Ramdorai On the $μ$ -invariant in Iwasawa theory, WIN—women in numbers (Fields Institute Communications), Volume 60, American Mathematical Society, 2011, pp. 265-276 | MR | Zbl

[57] Peter Schneider Galois representations and $(φ, Γ)$ -modules, Cambridge Studies in Advanced Mathematics, 164, Cambridge University Press, 2017, vii+148 pages | DOI | MR

[58] Peter Schneider; Otmar Venjakob Coates-Wiles homomorphisms and Iwasawa cohomology for Lubin-Tate extensions, Elliptic curves, modular forms and Iwasawa theory (Springer Proceedings in Mathematics & Statistics), Volume 188, Springer, 2016, pp. 401-468 | DOI | MR | Zbl

[59] Anthony J. Scholl Higher fields of norms and $(φ, Γ)$ -modules, Doc. Math., Volume Extra Vol. (2006), pp. 685-709 | MR

[60] Peter Scholze Perfectoid spaces, Publ. Math., Inst. Hautes Étud. Sci., Volume 116 (2012), pp. 245-313 | DOI | Numdam | MR | Zbl

[61] Shankar Sen Ramification in $p$ -adic Lie extensions, Invent. Math., Volume 17 (1972), pp. 44-50 | DOI | MR | Zbl

[62] Jean-Pierre Serre Sur la dimension cohomologique des groupes profinis, Topology, Volume 3 (1965), pp. 413-420 | DOI | MR | Zbl

[63] Ramdorai Sujatha; Malte Witte Fine Selmer groups and isogeny invariance, Geometry, algebra, number theory, and their information technology applications (Springer Proceedings in Mathematics & Statistics), Volume 251, Springer, 2018, pp. 419-444 | DOI | MR | Zbl

[64] Otmar Venjakob On the structure theory of the Iwasawa algebra of a $p$ -adic Lie group, J. Eur. Math. Soc., Volume 4 (2002) no. 3, pp. 271-311 | DOI | MR | Zbl

[65] Lawrence C. Washington Introduction to cyclotomic fields, Graduate Texts in Mathematics, 83, Springer, 1997, xiv+487 pages | DOI | MR

[66] Kay Wingberg Duality theorems for abelian varieties over $Z_{p}$ -extensions, Algebraic number theory (Advanced Studies in Pure Mathematics), Volume 17, Academic Press Inc., 1989, pp. 471-492 | DOI | MR | Zbl

[67] Christian Wuthrich Iwasawa theory of the fine Selmer group, J. Algebr. Geom., Volume 16 (2007) no. 1, pp. 83-108 | DOI | MR | Zbl

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