Control Theorems for Fine Selmer Groups

Debanjana Kundu; Meng Fai Lim

doi:10.5802/jtnb.1231

Debanjana Kundu ¹; Meng Fai Lim ²

¹ Mathematics Department, 1984, Mathematics Road, University of British Columbia, Vancouver, Canada, V6T1Z2
² School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical Sciences Central China Normal University, Wuhan, 430079, P.R.China.

Journal de théorie des nombres de Bordeaux, Volume 34 (2022) no. 3, pp. 851-880.

Abstract
Résumé

We study the growth of the $p$ -primary fine Selmer group, $R (E / F^{'})$ , of an elliptic curve over an intermediate sub-extension $F^{'}$ of a $p$ -adic Lie extension, $ℒ / F$ . We estimate the $ℤ_{p}$ -corank of the kernel and cokernel of the restriction map $r_{ℒ / F^{'}} : R (E / F^{'}) \to R {(E / ℒ)}^{Gal (ℒ / F^{'})}$ with $F^{'}$ a finite extension of $F$ contained in $ℒ$ . We show that the growth of the fine Selmer groups in these intermediate sub-extension is related to the structure of the fine Selmer group over the infinite level. On specializing to certain (possibly non-commutative) $p$ -adic Lie extensions, we prove finiteness of the kernel and cokernel and provide growth estimates on their orders.

Nous étudions la croissance du groupe de Selmer fin $p$ -primaire $R (E / F^{'})$ d’une courbe elliptique sur une sous-extension intermédiaire $F^{'}$ d’une extension de Lie $p$ -adique, $ℒ / F$ . Nous estimons le $ℤ_{p}$ -corang du noyau et du conoyau de l’application de restriction $r_{ℒ / F^{'}} : R (E / F^{'}) \to R {(E / ℒ)}^{Gal (ℒ / F^{'})},$ où $F^{'}$ est une extension finie de $F$ contenue dans $ℒ$ . Nous montrons également que la croissance des groupes de Selmer fins dans ces sous-extensions intermédiaires est liée à la structure du groupe de Selmer fin au niveau infini. En spécialisant au cas des extensions de Lie $p$ -adiques classiques (éventuellement non commutatives), nous prouvons la finitude du noyau et du conoyau et fournissons des estimations de croissance de leurs ordres.

Received: 2021-06-14
Revised: 2021-10-16
Accepted: 2021-10-29
Published online: 2023-01-27

DOI: 10.5802/jtnb.1231

Classification: 11R23
Keywords: control theorem, fine Selmer groups

Author's affiliations:

Debanjana Kundu ¹; Meng Fai Lim ²

¹ Mathematics Department, 1984, Mathematics Road, University of British Columbia, Vancouver, Canada, V6T1Z2
² School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical Sciences Central China Normal University, Wuhan, 430079, P.R.China.

License:

CC-BY-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

@article{JTNB_2022__34_3_851_0,
     author = {Debanjana Kundu and Meng Fai Lim},
     title = {Control {Theorems} for {Fine} {Selmer} {Groups}},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {851--880},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {34},
     number = {3},
     year = {2022},
     doi = {10.5802/jtnb.1231},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1231/}
}

TY  - JOUR
AU  - Debanjana Kundu
AU  - Meng Fai Lim
TI  - Control Theorems for Fine Selmer Groups
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2022
SP  - 851
EP  - 880
VL  - 34
IS  - 3
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1231/
DO  - 10.5802/jtnb.1231
LA  - en
ID  - JTNB_2022__34_3_851_0
ER  -

%0 Journal Article
%A Debanjana Kundu
%A Meng Fai Lim
%T Control Theorems for Fine Selmer Groups
%J Journal de théorie des nombres de Bordeaux
%D 2022
%P 851-880
%V 34
%N 3
%I Société Arithmétique de Bordeaux
%U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1231/
%R 10.5802/jtnb.1231
%G en
%F JTNB_2022__34_3_851_0

Debanjana Kundu; Meng Fai Lim. Control Theorems for Fine Selmer Groups. Journal de théorie des nombres de Bordeaux, Volume 34 (2022) no. 3, pp. 851-880. doi : 10.5802/jtnb.1231. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1231/

References
Cited by

[1] Chandrakant S. Aribam On the $μ$ -invariant of fine Selmer groups, J. Number Theory, Volume 135 (2014), pp. 284-300 | DOI | Zbl

[2] Paul N. Balister; Susan Howson Note on Nakayama’s lemma for compact $Λ$ -modules, Asian J. Math., Volume 1 (1997) no. 2, pp. 224-229 | DOI | Zbl

[3] Amala Bhave Analogue of Kida’s formula for certain strongly admissible extensions, J. Number Theory, Volume 122 (2007) no. 1, pp. 100-120 | DOI | Zbl

[4] John Coates Fragments of the GL $_{2}$ Iwasawa theory of elliptic curves without complex multiplication., Arithmetic theory of elliptic curves (Cetraro, 1997) (Lecture Notes in Mathematics), Volume 1716, Springer, 1999, pp. 1-50 | DOI | Zbl

[5] John Coates; Ramdorai Sujatha Euler–Poincaré characteristics of abelian varieties, C. R. Math. Acad. Sci. Paris, Volume 329 (1999) no. 4, pp. 309-313 | DOI | Zbl

[6] 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 | Zbl

[7] Albert A. Cuoco; Paul Monsky Class numbers in $ℤ_{p}^{d}$ -extensions, Math. Ann., Volume 255 (1981) no. 2, pp. 235-258 | DOI | Zbl

[8] John Dixon; Marcus du Sautoy; Avinoam Mann; Dan Segal Analytic pro- $p$ groups, Cambridge Studies in Advanced Mathematics, 61, Cambridge University Press, 1999, xviii+368 pages | DOI

[9] Kenneth R. Goodearl; Robert B. Warfield An introduction to non-commutative Noetherian rings, London Mathematical Society Student Texts, 61, Cambridge University Press, 2004, xxiv+344 pages | DOI

[10] Ralph Greenberg Introduction to Iwasawa theory for elliptic curves, Arithmetic algebraic geometry (Park City, UT, 1999) (IAS/Park City Mathematics Series), Volume 9, American Mathematical Society, 1999, pp. 407-464 | DOI

[11] Ralph Greenberg Galois theory for the Selmer group of an abelian variety, Compos. Math., Volume 136 (2003) no. 3, pp. 255-297 | DOI | Zbl

[12] Yoshitaka Hachimori; Otmar Venjakob Completely faithful Selmer groups over Kummer extensions, Doc. Math. (2003), pp. 443-478 (Extra Volume: Kazuya Kato’s Fiftieth Birthday) | Zbl

[13] Michael Harris Correction to $p$ -adic representations arising from descent on Abelian varieties, Compos. Math., Volume 121 (2000) no. 1, pp. 105-108 | DOI | Zbl

[14] Susan Howson Euler characteristics as invariants of Iwasawa modules, Proc. Lond. Math. Soc., Volume 85 (2002) no. 3, pp. 634-658 | DOI | Zbl

[15] Susan Howson Structure of central torsion Iwasawa modules, Bull. Soc. Math. Fr., Volume 130 (2002) no. 4, pp. 507-535 | DOI | Numdam | Zbl

[16] Kenkichi Iwasawa On $Γ$ -extensions of algebraic number fields, Bull. Am. Math. Soc., Volume 65 (1959) no. 4, pp. 183-226 | DOI | Zbl

[17] Somnath Jha Fine Selmer group of Hida deformations over non-commutative $p$ -adic Lie extensions, Asian J. Math., Volume 16 (2012) no. 2, pp. 353-365 | Zbl

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

[19] Yusuke Kubo; Yuichiro Taguchi A generalization of a theorem of Imai and its applications to Iwasawa theory, Math. Z., Volume 275 (2013) no. 3-4, pp. 1181-1195 | DOI | Zbl

[20] Tsit Y. Lam Lectures on modules and rings, Graduate Texts in Mathematics, 189, Springer, 1999, xxiv+557 pages

[21] Arthur Lannuzel; Thong Nguyen Quang Do Conjectures de Greenberg et extensions pro- $p$ -libres d’un corps de nombres, Manuscr. Math., Volume 102 (2000) no. 2, pp. 187-209 | DOI | Zbl

[22] Dingli Liang; Meng Fai Lim On the Iwasawa asymptotic class number formula for $ℤ_{p}^{r} ⋊ ℤ_{p}$ -extensions, Acta Arith., Volume 189 (2019) no. 2, pp. 191-208 | DOI | Zbl

[23] Meng Fai Lim On the pseudo-nullity of the dual fine Selmer groups, Int. J. Number Theory, Volume 11 (2015) no. 7, pp. 2055-2063 | Zbl

[24] Meng Fai Lim Comparing the $π$ -primary submodules of the dual Selmer groups, Asian J. Math., Volume 21 (2017) no. 6, pp. 1153-1181 | Zbl

[25] Meng Fai Lim Notes on the fine Selmer groups, Asian J. Math., Volume 21 (2017) no. 2, pp. 337-361 | Zbl

[26] Meng Fai Lim A note on asymptotic class number upper bounds in $p$ -adic Lie extensions, Acta Math. Sin., Engl. Ser., Volume 35 (2019) no. 9, pp. 1481-1490 | Zbl

[27] Meng Fai Lim On the control theorem for fine Selmer groups and the growth of fine Tate-Shafarevich groups in $ℤ_{p}$ -extensions, Doc. Math., Volume 25 (2020), pp. 2445-2471 | Zbl

[28] Meng Fai Lim; V. Kumar Murty The growth of the Selmer group of an elliptic curve with split multiplicative reduction, Int. J. Number Theory, Volume 10 (2014) no. 3, pp. 675-687 | Zbl

[29] Meng Fai Lim; V. Kumar Murty Growth of Selmer Groups of CM Abelian Varieties, Can. J. Math., Volume 67 (2015) no. 3, pp. 654-666 | DOI | Zbl

[30] Meng Fai Lim; V. Kumar Murty The growth of fine Selmer groups, J. Ramanujan Math. Soc., Volume 31 (2016) no. 1, pp. 79-94 | Zbl

[31] Meng Fai Lim; Ramdorai Sujatha On the structure of fine Selmer groups and Selmer groups of CM elliptic curves (to appear in Proceeding of Ropar Conference, RMS-Lecture Notes Series)

[32] Barry Mazur Rational points of abelian varieties with values in towers of number fields, Invent. Math., Volume 18 (1972) no. 3-4, pp. 183-266 | DOI | Zbl

[33] John S. Milne Arithmetic duality theorems, BookSurge, 2006

[34] Jurgen Neukirch; Alexander Schmidt; Kay Wingberg Cohomology of number fields, Grundlehren der Mathematischen Wissenschaften, 323, Springer, 2008 | DOI

[35] Andreas Neumann Completed group algebras without zero divisors, Arch. Math., Volume 51 (1988) no. 6, pp. 496-499 | DOI | Zbl

[36] Yoshihiro Ochi A remark on the pseudo-nullity conjecture for fine Selmer groups of elliptic curves, Comment. Math. Univ. St. Pauli, Volume 58 (2009) no. 1, pp. 1-7 | Zbl

[37] Guillaume Perbet Sur les invariants d’Iwasawa dans les extensions de Lie $p$ -adiques, Algebra Number Theory, Volume 5 (2012) no. 6, pp. 819-848 | DOI | Zbl

[38] Karl Rubin The “main conjectures” of Iwasawa theory for imaginary quadratic fields, Invent. Math., Volume 103 (1991) no. 1, pp. 25-68 | DOI | Zbl

[39] Karl Rubin Euler systems, Princeton University Press, 2000 no. 147 | DOI

[40] Jean-Pierre Serre Sur les groupes de congruence des variétés abéliennes. II, Izv. Akad. Nauk SSSR, Ser. Mat., Volume 35 (1971) no. 4, pp. 731-737 | Zbl

[41] Jean-Pierre Serre Abelian $l$ -adic representations and elliptic curves, Research Notes in Mathematics, 7, A K Peters, 1998, 199 pages (with the collaboration of Willem Kuyk and John Labute, revised reprint of the 1968 original)

[42] Jean-Pierre Serre Local Algebra, Springer Monographs in Mathematics, Springer, 2000

[43] Sudhanshu Shekhar Comparing the corank of fine Selmer group and Selmer group of elliptic curves, J. Ramanujan Math. Soc., Volume 33 (2018) no. 2, pp. 205-217 | Zbl

[44] Joseph H. Silverman The arithmetic of elliptic curves, Graduate Texts in Mathematics, 106, Springer, 2009 | DOI

[45] Christopher Skinner; Eric. Urban The Iwasawa main conjectures for ${GL}_{2}$ , Invent. Math., Volume 195 (2014) no. 1, pp. 1-277 | DOI | Zbl

[46] John Tate Letter from Tate to Iwasawa on a relation between $K_{2}$ and Galois cohomology, Algebraic $K$ -theory, II: “Classical” algebraic $K$ -theory and connections with arithmetic (Proc. Conf., Seattle Res. Center, Battelle Memorial Inst., 1972) (Lecture Notes in Mathematics), Volume 342, Springer, 1973, pp. 524-527 | Zbl

[47] 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 | Zbl

[48] Otmar Venjakob A non-commutative Weierstrass preparation theorem and applications to Iwasawa theory, J. Reine Angew. Math., Volume 559 (2003), pp. 153-191 (with an appendix by Denis Vogel) | Zbl

[49] Christian Wuthrich The fine Selmer group and height pairings, Ph. D. Thesis, University of Cambridge (2004)

[50] Christian Wuthrich The fine Tate-Shafarevich group, Math. Proc. Camb. Philos. Soc., Volume 142 (2007) no. 1, pp. 1-12 | DOI | Zbl

[51] Christian Wuthrich Iwasawa theory of the fine Selmer group, J. Algebr. Geom., Volume 16 (2007) no. 1, pp. 83-109 | DOI | Zbl

[52] Sarah Livia Zerbes Selmer groups over $p$ -adic Lie extensions. I., J. Lond. Math. Soc., Volume 70 (2004) no. 3, pp. 586-608 | DOI | Zbl

Cited by Sources: