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Galois cohomology of elliptic curves. (English) Zbl 0973.11059

Lectures on Mathematics and Physics. Mathematics. Tata Institute of Fundamental Research. 88. New Delhi: Narosa Publishing House. Mumbai: Tata Institute of Fundamental Research, x, 100 p. (2000).
This short well-written book offers theoretical background and several thorougly worked examples of the calculation of Selmer groups at infinite level as Iwasawa modules. More precisely: Let \(E\) be an elliptic curve over the number field \(F\), and let \(H_\infty\) be either the cyclotomic \(\mathbb{Z}_p\)-extension of \(F\) (“cyclotomic case”) or the extension obtained by adjoining all \(p\)-power torsion points of \(E\) to \(F\) (“division field case”). In the former case, \(G(H_\infty/F)\) is \(\Gamma\), a copy of \(\mathbb{Z}_p\). In both cases, \(S(E/H_\infty)\) is the inductive limit of the Selmer groups \(S(E/L)\), with \(L\) running through the finite extensions of \(F\) inside \(H_\infty\). The Iwasawa module \(X_E\) (reviewer’s notation) that one wants to study is the Pontryagin dual of the discrete module \(S(E/H_\infty)\).
We only give some highlights of the contents of this book. The main concern is with the cyclotomic case; here \(X_E\) is a module over \(\Lambda= \mathbb{Z}_p[[T]]\), and \(X_E\) is finitely generated over \(\Lambda\). For the elliptic curves of conductor 11 over \(\mathbb{Q}\) and every \(p\), the dual Selmer group \(X_E\) is calculated in detail for the cyclotomic case and all \(p\); curves of conductor 294 are treated as well. Fairly often but definitely not always, \(X_E\) is zero. It does occur that \(X_E\) is non-torsion, indeed this happens for \(p\) of supersingular reduction. The division field case, in which less is known, is treated more briefly.
The authors have taken great care in presenting the necessary theoretical framework; on the other hand, in the central parts of the book the reader is sometimes referred to other sources for proofs, in particular to: J. Coates and R. Greenberg, Invent. Math. 124, No. 1-3, 129-174 (1996; Zbl 0858.11032). The interested reader should also consult the related notes of J. Coates, and of R. Greenberg, in the Proceedings of the Cetraro conference, Springer Lect. Notes Math. 1716, 1-50 (1999; Zbl 1029.11016) and 51-144 (1999; Zbl 0946.11027).

MSC:

11G05 Elliptic curves over global fields
11R23 Iwasawa theory
11-02 Research exposition (monographs, survey articles) pertaining to number theory
11G07 Elliptic curves over local fields
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11R34 Galois cohomology
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