On the theory of Kolyvagin systems of rank 0
Journal de théorie des nombres de Bordeaux, Tome 33 (2021) no. 3.2, pp. 1077-1102.

Dans cet article, nous définissons un système Kolyvagin de rang 0 et développons la théorie des systèmes Kolyvagin de rang 0. En particulier, nous prouvons que le module des systèmes Kolyvagin de rang 0 est libre de rang un sous les hypothèses standard.

In this paper, we define a Kolyvagin system of rank 0 and develop the theory of Kolyvagin systems of rank 0. In particular, we prove that the module of Kolyvagin systems of rank 0 is free of rank one under standard assumptions.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/jtnb.1189
Classification : 11F80, 11R34, 11R23
Mots clés : Kolyvagin system, Selmer group
Ryotaro Sakamoto 1

1 Department of Mathematics Faculty of Science and Technology Keio University, 3-14-1 Hiyoshi Kohoku-ku, Yokohama, 223-8522, Japan
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits
@article{JTNB_2021__33_3.2_1077_0,
     author = {Ryotaro Sakamoto},
     title = {On the theory of {Kolyvagin} systems of rank $0$},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {1077--1102},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {33},
     number = {3.2},
     year = {2021},
     doi = {10.5802/jtnb.1189},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1189/}
}
TY  - JOUR
AU  - Ryotaro Sakamoto
TI  - On the theory of Kolyvagin systems of rank $0$
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2021
SP  - 1077
EP  - 1102
VL  - 33
IS  - 3.2
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1189/
DO  - 10.5802/jtnb.1189
LA  - en
ID  - JTNB_2021__33_3.2_1077_0
ER  - 
%0 Journal Article
%A Ryotaro Sakamoto
%T On the theory of Kolyvagin systems of rank $0$
%J Journal de théorie des nombres de Bordeaux
%D 2021
%P 1077-1102
%V 33
%N 3.2
%I Société Arithmétique de Bordeaux
%U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1189/
%R 10.5802/jtnb.1189
%G en
%F JTNB_2021__33_3.2_1077_0
Ryotaro Sakamoto. On the theory of Kolyvagin systems of rank $0$. Journal de théorie des nombres de Bordeaux, Tome 33 (2021) no. 3.2, pp. 1077-1102. doi : 10.5802/jtnb.1189. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1189/

[1] David Burns; Ryotaro Sakamoto; Takamichi Sano On the theory of higher rank Euler, Kolyvagin and Stark systems, II (2018) (https://arxiv.org/abs/1805.08448, submitted)

[2] David Burns; Takamichi Sano On the Theory of Higher Rank Euler, Kolyvagin and Stark Systems, Int. Math. Res. Not., Volume 2021 (2021) no. 13, pp. 10118-10206 | DOI | MR

[3] Kâzım Büyükboduk Kolyvagin systems of Stark units, J. Reine Angew. Math., Volume 631 (2009), pp. 85-107 | MR | Zbl

[4] Kâzım Büyükboduk Stark units and the main conjectures for totally real fields, Compos. Math., Volume 145 (2009) no. 5, pp. 1163-1195 | DOI | MR | Zbl

[5] Kâzım Büyükboduk On Euler systems of rank r and their Kolyvagin systems, Indiana Univ. Math. J., Volume 59 (2010) no. 4, pp. 1277-1332 | DOI | MR | Zbl

[6] Kâzım Büyükboduk Λ-adic Kolyvagin systems, Int. Math. Res. Not., Volume 2011 (2011) no. 14, pp. 3141-3206 | MR | Zbl

[7] Kâzım Büyükboduk Stickelberger elements and Kolyvagin systems, Nagoya Math. J., Volume 203 (2011), pp. 123-173 | DOI | MR | Zbl

[8] Masato Kurihara Refined Iwasawa theory and Kolyvagin systems of Gauss sum type, Proc. Lond. Math. Soc., Volume 104 (2012) no. 4, pp. 728-769 | DOI | MR | Zbl

[9] Masato Kurihara The structure of Selmer groups of elliptic curves and modular symbols, Iwasawa theory 2012 (Contributions in Mathematical and Computational Sciences), Volume 7, Springer, 2012, pp. 317-356 | DOI | Zbl

[10] Masato Kurihara Refined Iwasawa theory for p-adic representations and the structure of Selmer groups, Münster J. Math., Volume 7 (2014) no. 1, pp. 149-223 | MR | Zbl

[11] Barry Mazur; Karl Rubin Kolyvagin systems, Memoirs of the American Mathematical Society, 799, American Mathematical Society, 2004

[12] Barry Mazur; Karl Rubin Controlling Selmer groups in the higher core rank case, J. Théor. Nombres Bordeaux, Volume 28 (2016) no. 1, pp. 145-183 | DOI | Numdam | MR | Zbl

[13] Ryotaro Sakamoto Stark systems over Gorenstein local rings, Algebra Number Theory, Volume 12 (2018) no. 10, pp. 2295-2326 | DOI | MR | Zbl

[14] Ryotaro Sakamoto A higher rank Euler system for 𝔾 m over a totally real field (2020) (https://arxiv.org/abs/2002.04871, to appear in Am. J. Math.)

[15] Ryotaro Sakamoto On the theory of higher rank Euler, Kolyvagin and Stark systems: a research announcement, RIMS Kôkyûroku Bessatsu, Volume B83 (2020), pp. 141-159 (Algebraic Number Theory and Related Topics 2017) | MR | Zbl

Cité par Sources :