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

Hamidi, Parham; Ray, Jishnu

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

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 ${\mathbb{Z}}_p^2$-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 $\mu$ 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 ${\mathbb{Z}}_p^2$-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 $\mu$-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.

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

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

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

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

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

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

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

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

[10] Billot, Patrick 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] Bourbaki, Nicolas 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] Büyükboduk, Kâzım; Lei, Antonio 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] Cherbonnier, Frédéric; Colmez, Pierre Représentations $p$-adiques surconvergentes, Invent. Math., Volume 133 (1998) no. 3, pp. 581-611 | DOI | MR | Zbl

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

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

[17] Coates, John; Schneider, Peter; Sujatha, Ramdorai Links between cyclotomic and ${\mathrm{GL}}_2$ Iwasawa theory, Doc. Math., Volume Extra Vol. (2003), pp. 187-215 (Kazuya Kato’s fiftieth birthday) | MR | Zbl

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

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

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

[22] Fontaine, Jean-Marc 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] Fontaine, Jean-Marc 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] Fontaine, Jean-Marc; Wintenberger, Jean-Pierre 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] Gabber, Ofer; Ramero, Lorenzo Almost ring theory, Lecture Notes in Mathematics, 1800, Springer, 2003, vi+307 pages | DOI | MR

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[58] Schneider, Peter; Venjakob, Otmar 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] Scholl, Anthony J. Higher fields of norms and $(\phi ,\Gamma )$-modules, Doc. Math., Volume Extra Vol. (2006), pp. 685-709 | MR

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

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

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

[63] Sujatha, Ramdorai; Witte, Malte 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] Venjakob, Otmar 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] Washington, Lawrence C. Introduction to cyclotomic fields, Graduate Texts in Mathematics, 83, Springer, 1997, xiv+487 pages | DOI | MR

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

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