A control theorem for the torsion Selmer pointed set
type: Item
instance of: scholarly article
title: A control theorem for the torsion Selmer pointed set
zbMATH Open document ID: 1445.11126
author: Kenji Sakugawa
published in: Tôhoku Mathematical Journal. Second Series
publication date: 2018-08-30T00:00:00Z
review text: Let \(F\) be a number field, let \(p\) be an odd prime, and let \(F_n\) be the \(n\)th level of the cyclotomic \(\mathbb Z_p\)-extension of \(F\), with \(\text{Gal}(F_{\infty}/F_n)=\Gamma_n\). \textit{B. Mazur} [Invent. Math. 18, 183--266 (1972; Zbl. 0245.14015)] showed that if \(E\) is an elliptic curve over \(F\) that is ordinary at all primes above \(p\), then the maps \[ \text{ Sel}_p(E, F_n) \to \text{Sel}_p(E, F_{\infty})^{\Gamma_n} \] have finite kernel and cokernel whose orders are bounded independently of \(n\). The main result of the present paper gives a non-abelian generalization of this theorem by defining a torsion analogue of the Selmer variety of \textit{M. Kim} [Publ. Res. Inst. Math. Sci. 45, No. 1, 89--133 (2009; Zbl. 1165.14020)]. Let \(R\) be a finite flat commutative \(\mathbb Z_p\)-algebra and let \(R^{\text{mon}}\) be the monoid associated to the multiplicative structure of \(R\). The author considers a subcategory \((R^{\text{mon}}\)-\(P\)-\(\text{Sets})\) of pointed sets and defines the notion of \(p\)-exponent of cokernels in this subcategory. Also, for a smooth curve \(X\) over \(F\), a torsion analogue \(H_f^1(F, \mathfrak{g}^{\le m}(X)_{R/(p^r),a})\) of the Selmer variety is defined in this subcategory, where \(\mathfrak{g}^{\le m}(X)_{R/(p^r),a}\) is the set of \(R/(p^r)\)-valued points of a certain algebraic group. Assume \(F\) is a finite abelian extension of \(\mathbb Q\) with Galois group \(\Delta\) of order prime to \(p\). Also, assume that the completion of \(F\) at each prime above \(p\) is linearly disjoint from \(\mathbb Q_p(\mu_p)\) over \(\mathbb Q_p\). Let \(X\) be a smooth curve over \(\mathbb Q\) that is isomorphic over \(\mathbb Q\) to the projective line minus finitely many \(\mathbb Q\)-rational points, or an elliptic curve with good ordinary reduction at \(p\), minus the origin, or a proper smooth curve whose Jacobian is isogenous to a product of elliptic curves with good ordinary reduction at \(p\) and satisfying a c…
reviewed by: Lawrence C. Washington
Mathematics Subject Classification ID: 11R23
Mathematics Subject Classification ID: 11R34
zbMATH DE Number: 6929333
zbMATH Keywords: Selmer variety
zbMATH Keywords: control theorem
zbMATH Keywords: Iwasawa theory
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1527904820
OpenAlex ID: W2805299878
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DOI: 10.2748/TMJ/1527904820
description: scientific article; zbMATH DE number 6929333