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Controlling Selmer groups in the higher core rank case. (English. French summary) Zbl 1411.11065

Summary: We define Kolyvagin systems and Stark systems attached to \(p\)-adic representations in the case of arbitrary “core rank” (the core rank is a measure of the generic Selmer rank in a family of Selmer groups). Previous work dealt only with the case of core rank one, where the Kolyvagin and Stark systems are collections of cohomology classes. For general core rank, they are collections of elements of exterior powers of cohomology groups. We show under mild hypotheses that for general core rank these systems still control the size and structure of Selmer groups, and that the module of all Kolyvagin (or Stark) systems is free of rank one.

MSC:

11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11R23 Iwasawa theory
11R34 Galois cohomology
11F80 Galois representations
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References:

[1] B. Mazur & K. Rubin, « Kolyvagin systems », Mem. Amer. Math. Soc.168 (2004), no. 799, p. viii+96. · Zbl 1055.11041
[2] —, « Refined class number formulas for \(\mathbb{G}_m\) », J. Théor. Nombres Bordeaux28 (2016), no. 1, p. 185-211. · Zbl 1414.11155
[3] J. S. Milne, Arithmetic duality theorems, Perspectives in Mathematics, vol. 1, Academic Press, Inc., Boston, MA, 1986, x+421 pages. · Zbl 0613.14019
[4] B. Perrin-Riou, « Théorie d’Iwasawa et hauteurs \(p\)-adiques », Invent. Math.109 (1992), no. 1, p. 137-185. · Zbl 0781.14013
[5] —, « Systèmes d’Euler \(p\)-adiques et théorie d’Iwasawa », Ann. Inst. Fourier (Grenoble)48 (1998), no. 5, p. 1231-1307. · Zbl 0930.11078
[6] K. Rubin, « A Stark conjecture “over \(\mathbf{Z} \)” for abelian \(L\)-functions with multiple zeros », Ann. Inst. Fourier (Grenoble)46 (1996), no. 1, p. 33-62. · Zbl 0834.11044
[7] —, Euler systems, Annals of Mathematics Studies, vol. 147, Princeton University Press, Princeton, NJ, 2000, Hermann Weyl Lectures. The Institute for Advanced Study, xii+227 pages. · Zbl 0977.11001
[8] T. Sano, « A generalization of Darmon’s conjecture for Euler systems for general \(p\)-adic representations », J. Number Theory144 (2014), p. 281-324. · Zbl 1296.11143
[9] J. Tate, Les conjectures de Stark sur les fonctions \(L\) d’Artin en \(s=0\), Progress in Mathematics, vol. 47, Birkhäuser Boston, Inc., Boston, MA, 1984, Lecture notes edited by Dominique Bernardi and Norbert Schappacher, 143 pages. · Zbl 0545.12009
[10] A. Wiles, « Modular elliptic curves and Fermat’s last theorem », Ann. of Math. (2)141 (1995), no. 3, p. 443-551. · Zbl 0823.11029
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