p r -Selmer companion modular forms
[p r -Selmer compagnon formes modulaires]
Annales de l'Institut Fourier, Online first, 35 p.

L’étude de groupes n-Selmer de courbes elliptiques sur des corps de nombres algébriques dans un passé récent a conduit à la découverte de certains résultats profonds en arithmétique des courbes elliptiques. Étant données deux courbes elliptiques E 1 et E 2 sur un corps de nombres algébriques K, Mazur–Rubin les a définies comme n-Selmer compagnon si pour chaque caractère quadratique χ de K, les groupes n-Selmer de E 1 χ et E 2 χ sur K sont isomorphes. Étant donné un nombre premier p, ils ont donné des conditions suffisantes pour que deux courbes elliptiques soient des compagnons p r -Selmer en termes de congruences mod-p r entre les courbes. Nous discutons d’un analogue de ce résultat pour les groupes p r -Bloch–Kato Selmer de formes modulaires. Nous comparons le groupe Bloch–Kato Selmer d’une forme modulaire respectivement avec le groupe Greenberg Selmer lorsque la forme modulaire est p-ordinaire et avec les groupes de Selmer signés de Lei–Loeffler–Zerbes lorsque la forme modulaire est non ordinaire en p. Nous relions aussi nos résultats de congruence bien connus pour les valeurs spéciales des fonctions L correspondantes dues à Vatsal.

The study of n-Selmer groups of elliptic curves over number fields in recent past has led to the discovery of some deep results in the arithmetic of elliptic curves. Given two elliptic curves E 1 and E 2 over a number field K, Mazur–Rubin have defined them to be n-Selmer companion if for every quadratic character χ of K, the n-Selmer groups of E 1 χ and E 2 χ over K are isomorphic. Given a prime p, they have given sufficient conditions for two elliptic curves to be p r -Selmer companion in terms of mod-p r congruences between the curves. We discuss an analogue of this for Bloch–Kato p r -Selmer groups of modular forms. We compare the Bloch–Kato Selmer group of a modular form respectively with the Greenberg Selmer group when the modular form is p-ordinary and with the signed Selmer groups of Lei–Loeffler–Zerbes when the modular form is non-ordinary at p. We also indicate the relation between our results and the well-known congruence results for the special values of the corresponding L-functions due to Vatsal.

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DOI : https://doi.org/10.5802/aif.3392
Classification : 11F33,  11R23,  11R34,  11S25,  11G40
Mots clés : Groupe résiduel Bloch–Kato Selmer, congruence des formes modulaires
@unpublished{AIF_0__0_0_A29_0,
     author = {Jha, Somnath and Majumdar, Dipramit and Shekhar, Sudhanshu},
     title = {$p^r$-Selmer companion modular forms},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     year = {2021},
     doi = {10.5802/aif.3392},
     language = {en},
     note = {Online first},
}
Jha, Somnath; Majumdar, Dipramit; Shekhar, Sudhanshu. $p^r$-Selmer companion modular forms. Annales de l'Institut Fourier, Online first, 35 p.

[1] Bellaïche, Joël An introduction to the conjecture of Bloch and Kato, Lectures at the Clay Mathematical Institute summer School, Honolulu, Hawaii, available at https://www.claymath.org/sites/default/files/bellaiche.pdf, 2009 https://www.claymath.org/sites/default/files/bellaiche.pdf

[2] Bloch, Spencer; Kato, Kazuya L-functions and Tamagawa numbers of motives, The Grothendieck Festschrift, Vol. I (Progress in Mathematics), Volume 86, Birkhäuser, 1990, pp. 333-400 | MR 1086888

[3] Cremona, John E.; Mazur, Barry Visualizing elements in the Shafarevich–Tate group, Exp. Math., Volume 9 (2000) no. 1, pp. 13-28 | MR 1758797

[4] Darmon, Henri; Diamond, Fred; Taylor, Richard Fermat’s last theorem, Elliptic curves, modular forms & Fermat’s last theorem (Hong Kong, 1993), International Press, 1997, pp. 2-140 | MR 1605752

[5] Dummigan, Neil; Stein, William; Watkins, Mark Constructing elements in Shafarevich–Tate groups of modular motives, Number theory and algebraic geometry (London Mathematical Society Lecture Note Series), Volume 303, Cambridge University Press, 2003, pp. 91-118 | MR 2053457

[6] Emerton, Matthew; Pollack, Robert; Weston, Tom Variation of Iwasawa invariants in Hida families, Invent. Math., Volume 163 (2006) no. 3, pp. 523-580 | Article | MR 2207234

[7] Fontaine, Jean-Marc Le corps des périodes p-adiques, Périodes p-adiques (Bures-sur-Yvette, 1988) (Astérisque), Volume 223, Société Mathématique de France, 1994, pp. 59-111 (With an appendix by Pierre Colmez) | MR 1293971

[8] Fontaine, Jean-Marc; Ouyang, Yi Theory of p-adic Galois Representations, available at https://www.math.u-psud.fr/~fontaine/galoisrep.pdf https://www.math.u-psud.fr/~fontaine/galoisrep.pdf

[9] Fukaya, Takako; Kato, Kazuya A formulation of conjectures on p-adic zeta functions in noncommutative Iwasawa theory, Proceedings of the St. Petersburg Mathematical Society. Vol. XII (Translations. Series 2), Volume 219 (2006), pp. 1-85 | Article | MR 2276851

[10] 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 | Article | MR 1754686

[11] Hachimori, Yoshitaka; Matsuno, Kazuo An analogue of Kida’s formula for the Selmer groups of elliptic curves, J. Algebr. Geom., Volume 8 (1999) no. 3, pp. 581-601 | MR 1689359

[12] Hatley, Jeffrey; Lei, Antonio Arithmetic properties of signed Selmer groups at non-ordinary primes, Ann. Inst. Fourier, Volume 69 (2019) no. 3, pp. 1259-1294 | MR 3986915

[13] Hida, Haruzo Modular forms and Galois cohomology, Cambridge Studies in Advanced Mathematics, 69, Cambridge University Press, 2000, x+343 pages | Article | MR 1779182

[14] Kobayashi, Shin-ichi Iwasawa theory for elliptic curves at supersingular primes, Invent. Math., Volume 152 (2003) no. 1, pp. 1-36 | Article | MR 1965358

[15] Lei, Antonio Iwasawa theory for modular forms at supersingular primes, Compos. Math., Volume 147 (2011) no. 3, pp. 803-838 | Article | MR 2801401

[16] 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 | Article | MR 2948474

[17] Lei, Antonio; Loeffler, David; Zerbes, Sarah Livia On the asymptotic growth of Bloch–Kato–Shafarevich–Tate groups of modular forms over cyclotomic extensions, Can. J. Math., Volume 69 (2017) no. 4, pp. 826-850 | Article | MR 3679697

[18] Livné, Ron On the conductors of mod l Galois representations coming from modular forms, J. Number Theory, Volume 31 (1989) no. 2, pp. 133-141 | Article | MR 987567

[19] The LMFDB Collaboration The L-functions and Modular Forms Database, 2013 (http://www.lmfdb.org)

[20] Mazur, Barry; Rubin, Karl Selmer companion curves, Trans. Am. Math. Soc., Volume 367 (2015) no. 1, pp. 401-421 | Article | MR 3271266

[21] Mazur, Barry; Tate, John; Teitelbaum, Jeremy On p-adic analogues of the conjectures of Birch and Swinnerton–Dyer, Invent. Math., Volume 84 (1986) no. 1, pp. 1-48 | Article | MR 830037 | Zbl 0699.14028

[22] Miyake, Toshitsune Modular forms, Springer, 1989, x+335 pages (Translated from the Japanese by Yoshitaka Maeda) | Article | MR 1021004

[23] Ochiai, Tadashi Control theorem for Bloch–Kato’s Selmer groups of p-adic representations, J. Number Theory, Volume 82 (2000) no. 1, pp. 69-90 | Article | MR 1755154

[24] the Sage Developers SageMath, the Sage Mathematics Software System, 2016 (http://www.sagemath.org/)

[25] Scholl, Anthony J. Motives for modular forms, Invent. Math., Volume 100 (1990) no. 2, pp. 419-430 | Article | MR 1047142

[26] Skinner, Christopher; Urban, Eric The Iwasawa main conjectures for GL 2 , Invent. Math., Volume 195 (2014) no. 1, pp. 1-277 | Article | MR 3148103

[27] Sprung, Florian The Iwasawa Main Conjecture for elliptic curves at odd supersingular primes (2016) (https://arxiv.org/abs/1610.10017)

[28] Vatsal, Vinayak Canonical periods and congruence formulae, Duke Math. J., Volume 98 (1999) no. 2, pp. 397-419 | Article | MR 1695203 | Zbl 0979.11027

[29] Wiles, Andrew On ordinary λ-adic representations associated to modular forms, Invent. Math., Volume 94 (1988) no. 3, pp. 529-573 | Article | MR 969243