Euler characteristics as invariants of Iwasawa modules.

*(English)*Zbl 1036.11053This paper deals with generalised Iwasawa modules, that is, with modules \(M\) over the completed group algebra \(\Lambda(G)\) of a \(p\)-adic Lie group \(G\), both in general and in concrete arithmetical applications. The first main result is a formula which expresses the rank of \(M\) (which is well-defined if one assumes that \(G\) has no torsion elements) as an Euler characteristic involving the \(G\)-homology of \(M\). There is also a dual formulation involving cohomology. This is a very general result, whose proof actually is not that difficult. The author explains nicely how the old criterion for the case that \(G\) is the pro-\(p\)-cyclic group \(\Gamma\) comes out of this: \(M\) is \(\Lambda(\Gamma)\)-torsion iff \(M_\Gamma\) is finite; note \(M_\Gamma\) is the zeroth homology of \(\Gamma\) with coefficients in \(M\). The author gives a revealing example that shows a behaviour quite different from the case \(G=\Gamma\): a non-torsion module \(M\) with finite module of \(G\)-coinvariants (the only cohomology detecting that \(M\) is non-torsion is the second one). This is followed by a discussion of modules annihilated by \(p\) or by a power of \(p\) and a definition of the \(\mu\)-invariant in general.

The second section is occupied by a generalisation called “homological rank”: here \(G\) need not be pro-\(p\), and \(M\) need not be finitely generated. In particular a situation coming from elliptic curves is discussed, and a numerical example is provided. Here \(G\) is \(H\), the Galois group of \(F_\infty\) over the cyclotomic \({\mathbb Z}_p\)-extension of \(F\), with \(F_\infty\) gotten from \(F\) by adjoining all \(p\)-power torsion points of an elliptic curve \(E/F\) without CM. The main result is Theorem 2.8. In the numerical example, the module \(M\) (which is essentially the dual of a Selmer group) is in fact not finitely generated, but its homological rank is nevertheless calculated and found to agree with the “naive” rank.

The third and last section considers the change of the \(\mu\)-invariant for Selmer groups of elliptic curves under an isogeny \(\phi\). More precisely, \(\phi\) induces a map \(\phi_1\) between the Selmer groups, and Theorem 3.1 gives a formula for the difference \(\mu(\widehat{\text{ Coker}}(\phi_1)) - \mu(\widehat{\text{ Ker}}(\phi_1))\). (The author points out that Coates and Sujatha have an alternative way of obtaining such a result.) This is a quite interesting paper, and pleasant reading to boot.

The second section is occupied by a generalisation called “homological rank”: here \(G\) need not be pro-\(p\), and \(M\) need not be finitely generated. In particular a situation coming from elliptic curves is discussed, and a numerical example is provided. Here \(G\) is \(H\), the Galois group of \(F_\infty\) over the cyclotomic \({\mathbb Z}_p\)-extension of \(F\), with \(F_\infty\) gotten from \(F\) by adjoining all \(p\)-power torsion points of an elliptic curve \(E/F\) without CM. The main result is Theorem 2.8. In the numerical example, the module \(M\) (which is essentially the dual of a Selmer group) is in fact not finitely generated, but its homological rank is nevertheless calculated and found to agree with the “naive” rank.

The third and last section considers the change of the \(\mu\)-invariant for Selmer groups of elliptic curves under an isogeny \(\phi\). More precisely, \(\phi\) induces a map \(\phi_1\) between the Selmer groups, and Theorem 3.1 gives a formula for the difference \(\mu(\widehat{\text{ Coker}}(\phi_1)) - \mu(\widehat{\text{ Ker}}(\phi_1))\). (The author points out that Coates and Sujatha have an alternative way of obtaining such a result.) This is a quite interesting paper, and pleasant reading to boot.

Reviewer: Cornelius Greither (Neubiberg)

##### MSC:

11R23 | Iwasawa theory |

16E10 | Homological dimension in associative algebras |

20E18 | Limits, profinite groups |

20J06 | Cohomology of groups |

16E05 | Syzygies, resolutions, complexes in associative algebras |