Chromatic Selmer groups and arithmetic invariants of elliptic curves
Journal de théorie des nombres de Bordeaux, Volume 33 (2021) no. 3.2, pp. 1103-1114.

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 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 p 2 -extension d’un corps quadratique imaginaire dans lequel p est décomposé.

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DOI: 10.5802/jtnb.1190
Classification: 11G40,  11R23,  14H52
Keywords: Elliptic curves, Selmer group, Mordell–Weil rank
Florian Ito Sprung 1

1 School of Mathematical and Statistical Sciences Arizona State University Tempe, AZ 85287-1804, USA
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Florian Ito Sprung. Chromatic Selmer groups and arithmetic invariants of elliptic curves. Journal de théorie des nombres de Bordeaux, Volume 33 (2021) no. 3.2, pp. 1103-1114. doi : 10.5802/jtnb.1190. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1190/

[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 | DOI | MR | Zbl

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

[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 | DOI | MR | Zbl

[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) | MR | Zbl

[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 | DOI | MR

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

[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 | MR | Zbl

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

[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) | MR | Zbl

[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 | DOI | MR | Zbl

[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 | MR | Zbl

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

[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 | DOI | MR | Zbl

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

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

[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 | DOI | MR | Zbl

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

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

[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 | DOI | MR | Zbl

[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 | DOI | Numdam | MR | Zbl

[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 | DOI | MR | Zbl

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

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

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

[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 | DOI | MR | Zbl

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

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

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

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

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