×

Signed Selmer groups over \(p\)-adic Lie extensions. (English. French summary) Zbl 1283.11154

In this paper, the authors define signed Selmer groups \(\mathrm{Sel}^i (E/K_{\infty})\), \(i=1,2\), of an elliptic curve \(E\) over \(K_{\infty}\), a \(p\)–adic Lie extension of \({\mathbb Q}\) containing \({\mathbb Q}(\mu_{p^{ \infty}})\). This definition is analogous to that of S.-i. Kobayashi [Invent. Math. 152, No. 1, 1–36 (2003; Zbl 1047.11105)] of an elliptic curve over \({\mathbb Q}\) with good supersingular reduction at a prime \(p\geq 3\) and \(a_p=0\). The main idea is the use of L. Berger’s comparison isomorphism [Invent. Math. 148, No. 2, 219–284 (2002; Zbl 1113.14016)] and the theory of \((\varphi, \Gamma)\)-modules. The authors and Loeffler defined \(\mathrm{Sel}^i(E/{\mathbb Q} (\mu_{p^{\infty}}))\), \(i=1,2\) when \(E\) has good ordinary reduction at \(p\) in [A. Lei, Asian J. Math. 14, No. 4, 475–528 (2010; Zbl 1281.11095)] similarly to the good supersingular case. The definition of signed Selmer groups over \(K_{\infty}\) is justified showing that on extending the construction to the good ordinary case, \(\mathrm{Sel}^2(E/K_{\infty})\) agrees with the usual Selmer groups \(\mathrm{Sel}(E/K_{\infty})\) for any finite extension \(K\) of \({\mathbb Q}\).
In Section 4, a more explicit description of the local conditions is used to define the signed Selmer groups in the supersingular case. It is shown that if \(\mathrm{Sel}_N^i(E/K_n)\) is defined by replacing the local conditions at places above \(p\) in the definition of \(\mathrm{Sel}( E/K_n)\) by some “jumping conditions”, then \(\mathrm{Sel}_N^{(i)} (E/K_{\infty})\cong \mathrm{Sel}^i(E/K_{\infty})\) for \(i=1,2\).

MSC:

11R23 Iwasawa theory
11G05 Elliptic curves over global fields
11G07 Elliptic curves over local fields
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Laurent Berger, Représentations \(p\)-adiques et équations différentielles. Invent. Math. 148 (2002), no. 2, 219-284. · Zbl 1113.14016
[2] Laurent Berger, Bloch and Kato’s exponential map: three explicit formulas. Doc. Math. Extra Vol. 3 (2003), 99-129, Kazuya Kato’s fiftieth birthday. · Zbl 1064.11077
[3] Laurent Berger, Limites de représentations cristallines. Compos. Math. 140 (2004), no. 6, 1473-1498. · Zbl 1071.11067
[4] Laurent Berger, Représentations de de Rham et normes universelles. Bull. Soc. Math. France 133 (2005), no. 4, 601-618. · Zbl 1122.11036
[5] Spencer Bloch and Kazuya Kato, \(L\)-functions and Tamagawa numbers of motives. The Grothendieck Festschrift, Vol. I (Cartier et al, ed.), Progr. Math., vol. 86, Birkhäuser, Boston, MA, 1990, pp. 333-400. · Zbl 0768.14001
[6] Frédéric Cherbonnier and Pierre Colmez, Représentations \(p\)-adiques surconvergentes. Invent. Math. 133 (1998), no. 3, 581-611. · Zbl 0928.11051
[7] Frédéric Cherbonnier and Pierre Colmez, Théorie d’Iwasawa des représentations \(p\)-adiques d’un corps local. J. Amer. Math. Soc. 12 (1999), no. 1, 241-268. · Zbl 0933.11056
[8] John Coates and Ralph Greenberg, Kummer theory for abelian varieties over local fields. Invent. Math. 124 (1996), no. 1-3, 129-174. · Zbl 0858.11032
[9] John Coates and Ramdorai Sujatha, Galois cohomology of elliptic curves. Tata Institute of Fundamental Research Lectures on Mathematics, 88, Published by Narosa Publishing House, New Delhi, 2000. · Zbl 1213.11115
[10] John Coates and Susan Howson, Euler characteristics and elliptic curves. II, J. Math. Soc. Japan 53 (2001), no. 1, 175-235. · Zbl 1046.11079
[11] John Coates, Takako Fukaya, Kazuya Kato, Ramdorai Sujatha, and Otmar Venjakob, The \(\operatorname{GL}_2\) main conjecture for elliptic curves without complex multiplication. Pub. Math. IHÉS 101 (2005), 163-208. · Zbl 1108.11081
[12] Jean-Marc Fontaine, Le corps des périodes \(p\)-adiques (Bures-sur-Yvette, 1988). Astérisque No. 223 (1994), 59-111. · Zbl 0940.14012
[13] Laurent Herr, Sur la cohomologie galoisienne des corps \(p\)-adiques. Bull. Soc. Math. France 126 (1998), no. 4, 563-600. · Zbl 0967.11050
[14] Shinichi Kobayashi, Iwasawa theory for elliptic curves at supersingular primes. Invent. Math. 152 (2003), no. 1, 1-36. · Zbl 1047.11105
[15] Antonio Lei, David Loeffler, and Sarah Livia Zerbes, Wach modules and Iwasawa theory for modular forms. Asian J. Math. 14 (2010), no. 475-528. · Zbl 1281.11095
[16] Bernadette Perrin-Riou, Fonctions \(Lp\)-adiques des représentations \(p\)-adiques. Asté- risque No. 229 (1995), 1-198. · Zbl 0845.11040
[17] Bernadette Perrin-Riou, Représentations \(p\)-adiques et normes universelles. I. Le cas cristallin. J. Amer. Math. Soc. 13 (2000), no. 3, 533-551 (electronic). · Zbl 1024.11069
[18] Nathalie Wach, Représentations \(p\)-adiques potentiellement cristallines. Bull. Soc. Math. France 124 (1996), no. 3, 375-400. · Zbl 0887.11048
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.