Integral and modular representations attached to the Selmer group of an elliptic curve with complex multiplication (Q1291043)

This article considers Galois \(p\)-extensions \(L/F\) and the behaviour of a certain arithmetic object \(S_{L_\infty}\) attached to \(L\) as a \({\mathbb Z}_p[G]\)-module. Here \(G=G(L/F)\); \(p\) is a fixed prime; and \(S_{L_\infty}\) is a certain Selmer group associated with an elliptic curve defined over \(F\), with complex multiplication by the integers in an imaginary quadratic field \(K\), in which \(p\) is split. The principal results, obtained for the cases where \(G\) is cyclic of order \(p\) or \(p^2\), are a kind of Riemann-Hurwitz formula; for the precise statements, we refer to the article. There is a standard technique for the case \(| G| =p\), used previously by several people including Kida, Nguyen Quang Do, Wingberg, and Madan-Rzedowski-Villa. It goes roughly as follows: one uses Reiner's theorem which classifies all indecomposable \({\mathbb Z}_p[G]\)-lattices. There are only three of them. Thus any \({\mathbb Z}_p[G]\)-lattice is given by three multiplicities, written \(a_1, a_{p-1},a_p\) in standard notation. One tries to detect these multiplicities by calculating cohomology, sometimes using Herbrand quotients. This is also the method of the paper under review; a key point here is a formula of Coates, expressing the Selmer group in terms of a standard Iwasawa module, which puts us on familiar territory. The required information about the Herbrand quotient is taken from an earlier paper of the author [ Ann. Inst. Fourier 43, 57-84 (1993; Zbl 0769.11027)]. In order to deal with the case \(| G| =p^2\), the author uses the following nice trick: he determines the representation of \(G\) on \(S_{L_\infty}[p]\) for any cyclic \(p\)-group \(G\), and then he invokes the (more cumbersome) classification of lattices for \(| G| =p^2\). As the author indicates himself, this is not the only possibility to prove the final result Theorem 6. A few misprints and potential ambiguities should be mentioned for the readers' sake: Four lines before Thm. 1 on p.452, read \(F(E_{p^\infty})\) (the \(\infty\) is missing); a little further down, \(\lambda_{F_\infty}\) is the rank of Hom\((S_{F_\infty},{\mathbb Q}_p/{\mathbb Z}_p)\), not Hom\((S_{L_\infty},{\mathbb Q}_p/{\mathbb Z}_p)\); in the second line after Thm. 2 on p.456, insert ``dual'' after ``Pontryagin'' ; in (ii) near top of p.457, replace \(H^2(G,{\mathbb Z}_p)\) by \(H^2(G,A_1)\); the phrase ``... satisfies Galois theory'' near the middle of this page means presumably ``... satisfies Galois descent''; on middle of p.459, the ``fixed'' divisors are (I guess) ``fixed under \(G\)''; read ``ambiguous'' for ``ambigous'' throughout; in second line of section 4 (p. 466) read ``points'' for ''point''; in the proof of Lemma 7, allusion is made to Lemma 1 which does not seem to exist: perhaps Lemma 3 is meant.