Chromatic Selmer groups and arithmetic invariants of elliptic curves

Florian Ito Sprung

doi:10.5802/jtnb.1190

Florian Ito Sprung ¹

¹School of Mathematical and Statistical Sciences Arizona State University Tempe, AZ 85287-1804, USA

Journal de théorie des nombres de Bordeaux, Tome 33 (2021) no. 3.2, pp. 1103-1114

Abstract (VO)
Résumé

Chromatic Selmer groups are modified Selmer groups with local information for supersingular primes $p$ . We sketch their role in establishing the $p$ -primary part of the Birch–Swinnerton-Dyer formula in Sections 2–5, and then study the growth of the Mordell–Weil rank along the $ℤ_{p}^{2}$ -extension of a quadratic imaginary number field in which $p$ splits in Section 6.

Les groupes de Selmer chromatiques sont des modifications des groupes de Selmer, qui contiennent des informations locales pour les nombres premiers $p$ supersinguliers. Dans les sections 2–5, on esquisse leur rôle dans la démonstration de la partie $p$ -primaire de la formule de Birch et Swinnerton-Dyer, et ensuite, dans la section 6, on étudie la croissance du rang de Mordell–Weil le long de la $ℤ_{p}^{2}$ -extension d’un corps quadratique imaginaire dans lequel $p$ est décomposé.

Détail
Citer cet article

Reçu le : 2019-12-12
Révisé le : 2020-06-09
Accepté le : 2020-07-28
Publié le : 2022-01-26

DOI : 10.5802/jtnb.1190

Classification : 11G40, 11R23, 14H52
Keywords: Elliptic curves, Selmer group, Mordell–Weil rank

Affiliations des auteurs :

Florian Ito Sprung ¹

¹ School of Mathematical and Statistical Sciences Arizona State University Tempe, AZ 85287-1804, USA

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

Florian Ito Sprung. Chromatic Selmer groups and arithmetic invariants of elliptic curves. Journal de théorie des nombres de Bordeaux, Tome 33 (2021) no. 3.2, pp. 1103-1114. doi: 10.5802/jtnb.1190

@article{JTNB_2021__33_3.2_1103_0,
     author = {Florian Ito Sprung},
     title = {Chromatic {Selmer} groups and arithmetic invariants of elliptic curves},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {1103--1114},
     year = {2021},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {33},
     number = {3.2},
     doi = {10.5802/jtnb.1190},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1190/}
}

TY  - JOUR
AU  - Florian Ito Sprung
TI  - Chromatic Selmer groups and arithmetic invariants of elliptic curves
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2021
SP  - 1103
EP  - 1114
VL  - 33
IS  - 3.2
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1190/
DO  - 10.5802/jtnb.1190
LA  - en
ID  - JTNB_2021__33_3.2_1103_0
ER  -

%0 Journal Article
%A Florian Ito Sprung
%T Chromatic Selmer groups and arithmetic invariants of elliptic curves
%J Journal de théorie des nombres de Bordeaux
%D 2021
%P 1103-1114
%V 33
%N 3.2
%I Société Arithmétique de Bordeaux
%U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1190/
%R 10.5802/jtnb.1190
%G en
%F JTNB_2021__33_3.2_1103_0

Bibliographie
Cité par

[1] Adebisi Agboola; Benjamin Howard Anticyclotomic Iwasawa theory of CM elliptic curves. II, Math. Res. Lett., Volume 12 (2005) no. 5-6, pp. 611-621 | Zbl | MR | DOI

[2] Massimo Bertolini Selmer groups and Heegner points in anticyclotomic $Z_{p}$ -extensions, Compos. Math., Volume 99 (1995) no. 2, pp. 153-182 | Zbl | MR

[3] Li Cai; Chao Li; Shuai Zhai On the 2-part of the Birch and Swinnerton-Dyer conjecture for quadratic twists of elliptic curves, J. Lond. Math. Soc., Volume 101 (2020) no. 2, pp. 714-734 | arXiv | Zbl | MR | DOI

[4] John Coates; Peter Schneider; Ramdorai Sujatha Links between cyclotomic and ${GL}_{2}$ Iwasawa theory, Doc. Math. (2003) no. Extra Vol., pp. 187-215 (Kazuya Kato’s fiftieth birthday) | Zbl | MR

[5] John H. Coates; Kenneth A. Ribet; Ralph Greenberg; Karl Rubin Arithmetic theory of elliptic curves, Lecture Notes in Mathematics, 1716, Springer, 1999, viii+234 pages Lectures given at the 3rd Session of the Centro Internazionale Matematico Estivo (C.I.M.E.) held in Cetraro, Italy, July 12–19, 1997 | MR | DOI

[6] Christophe Cornut Mazur’s conjecture on higher Heegner points, Invent. Math., Volume 148 (2002) no. 3, pp. 495-523 | Zbl | MR | DOI

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

[8] Ralph Greenberg Iwasawa theory and $p$ -adic deformations of motives, Motives (Seattle, WA, 1991) (Proceedings of Symposia in Pure Mathematics), Volume 55, American Mathematical Society, 1994, pp. 193-223 | Zbl | MR

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

[10] Yoshitaka Hachimori; Otmar Venjakob Completely faithful Selmer groups over Kummer extensions, Doc. Math. (2003) no. Extra Vol., pp. 443-478 (Kazuya Kato’s fiftieth birthday) | Zbl | MR

[11] Parham Hamidi; Jishnu Ray Conjecture $A$ and $μ$ -invariant for Selmer groups of supersingular elliptic curves, J. Théor. Nombres Bordeaux, Volume 33 (2021) no. 3.1, pp. 853-886

[12] Ming-Lun Hsieh Eisenstein congruence on unitary groups and Iwasawa main conjectures for CM fields, J. Am. Math. Soc., Volume 27 (2014) no. 3, pp. 753-862 | Zbl | MR | DOI

[13] Pin-Chi Hung; Meng Fai Lim On the growth of Mordell-Weil ranks in $p$ -adic Lie extensions (2019) (https://arxiv.org/abs/1902.01068)

[14] 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 | Zbl | MR

[15] 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 | Zbl | MR | DOI

[16] Guido Kings; David Loeffler; Sarah Livia Zerbes Rankin-Eisenstein classes and explicit reciprocity laws, Camb. J. Math., Volume 5 (2017) no. 1, pp. 1-122 | Zbl | MR | DOI

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

[18] Antonio Lei Factorisation of two-variable $p$ -adic $L$ -functions, Can. Math. Bull., Volume 57 (2014) no. 4, pp. 845-852 | Zbl | MR | DOI

[19] Antonio Lei; Gautier Ponsinet On the Mordell–Weil ranks of supersingular abelian varieties in cyclotomic extensions, Proc. Am. Math. Soc., Ser. B, Volume 7 (2020), pp. 1-16 | Zbl | MR | DOI

[20] Antonio Lei; Florian Sprung Ranks of elliptic curves over $ℤ_{p}^{2}$ -extensions, Isr. J. Math., Volume 236 (2020) no. 1, pp. 183-206 | Zbl | MR | DOI

[21] David Loeffler; Sarah Livia Zerbes Iwasawa theory and $p$ -adic $L$ -functions over $ℤ_{p}^{2}$ -extensions, Int. J. Number Theory, Volume 10 (2014) no. 8, pp. 2045-2095 | Zbl | MR | DOI

[22] Matteo Longo; Stefano Vigni Plus/minus Heegner points and Iwasawa theory of elliptic curves at supersingular primes, Boll. Unione Mat. Ital., Volume 12 (2019) no. 3, pp. 315-347 | Zbl | MR | DOI

[23] Bernadette Perrin-Riou Fonctions $L$ $p$ -adiques d’une courbe elliptique et points rationnels, Ann. Inst. Fourier, Volume 43 (1993) no. 4, pp. 945-995 | Zbl | Numdam | DOI | MR

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

[25] 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 | Zbl | MR | DOI

[26] David E. Rohrlich On $L$ -functions of elliptic curves and cyclotomic towers, Invent. Math., Volume 75 (1984) no. 3, pp. 409-423 | Zbl | MR | DOI

[27] Christopher Skinner Multiplicative reduction and the cyclotomic main conjecture for ${GL}_{2}$ , Pac. J. Math., Volume 283 (2016) no. 1, pp. 171-200 | Zbl | MR | DOI

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

[29] Florian 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 | Zbl | MR | DOI

[30] Florian Sprung The Šafarevič–Tate group in cyclotomic $ℤ_{p}$ -extensions at supersingular primes, J. Reine Angew. Math., Volume 681 (2013), pp. 199-218 | Zbl | MR

[31] Florian Sprung The Iwasawa Main Conjecture for Elliptic Curves at odd supersingular primes (2016) (https://arxiv.org/abs/1610.10017, submitted)

[32] Florian Sprung On pairs of $p$ -adic $L$ -functions for weight-two modular forms, Algebra Number Theory, Volume 11 (2017) no. 4, pp. 885-928 | Zbl | MR | DOI

[33] Jeanine Van Order Some remarks on the two-variable main conjecture of Iwasawa theory for elliptic curves without complex multiplication, J. Algebra, Volume 350 (2012), pp. 273-299 | Zbl | MR | DOI

[34] Vinayak Vatsal Special values of anticyclotomic $L$ -functions, Duke Math. J., Volume 116 (2003) no. 2, pp. 219-261 | Zbl | MR | DOI

[35] Xin Wan Iwasawa Main Conjecture and BSD Conjecture (2014) (https://arxiv.org/abs/1411.6352, submitted)

Cité par Sources :