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Selmer groups over \({\mathbb {Z}}_p^d\)-extensions

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Abstract

Consider an abelian variety \(A\) defined over a global field \(K\) and let \(L/K\) be a \({\mathbb {Z}}_p^d\)-extension, unramified outside a finite set of places of \(K\), with \({{\mathrm{Gal}}}(L/K)=\Gamma \). Let \(\Lambda (\Gamma ):={\mathbb {Z}}_p[[\Gamma ]]\) denote the Iwasawa algebra. In this paper, we study how the characteristic ideal of the \(\Lambda (\Gamma )\)-module \(X_L\), the dual \(p\)-primary Selmer group, varies when \(L/K\) is replaced by a strict intermediate \({\mathbb {Z}}_p^e\)-extension.

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Acknowledgments

The author would like to thank King Fai Lai, Ignazio Longhi and Fabien Trihan for many valuable suggestions. This research was supported in part by the National Science Council of Taiwan, NSC97-2115-M-002-006-MY2, NSC99-2115-M-002-002-MY3.

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  1. Department of Mathematics, National Taiwan University, Taipei , 10764, Taiwan

    Ki-Seng Tan

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Tan, KS. Selmer groups over \({\mathbb {Z}}_p^d\)-extensions. Math. Ann. 359, 1025–1075 (2014). https://doi.org/10.1007/s00208-014-1023-9

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  • DOI: https://doi.org/10.1007/s00208-014-1023-9

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