The non-\(p\)-part of the fine Selmer group in a \(\mathbb{Z}_p\)-extension (Q6547204)

This article is about the growth of class groups and Selmer groups along \(\mathbb Z_p\)-towers \(K_\infty/K\) of number fields. (Recall that here \(K_\infty\) is the ascending union of fields \(K_n\), and \(K_n\) is called the \(n\)-th layer of the tower.) It is well known that \(\mu=0\) for the cyclotomic \(\mathbb Z_p\)-extension \(K^{cyc}/K\), which means that in the \(p\)-part, the class numbers only grow roughly like \(p^{\lambda n}\). In contrast with this, already Iwasawa knew that \(\mu>0\) is possible for non-cyclotomic \(\mathbb Z_p\)-extensions; this means that the \(p\)-parts of the class numbers grow at least like \(p^{p^n}\). There is the following interesting twist. One considers the \(\ell\)-part \(\ell^{e_n}\) of the class number of the layer \(K_n\), where \(\ell\) is a prime number distinct from \(p\). Washington proved that the sequence \((e_n)_n\) is actually bounded in the cyclotomic case, which looks much stronger than \(\mu=0\) (one has to be a little cautious as the quantity \(\mu\) does not make sense here, strictly speaking). This makes one wonder about the non-cyclotomic case, and there Washington showed that for any given \(N\) there are examples with \(e_n \ge Np^n\) for all \(n>>0\), and this looks like \(\mu>0\).\N\NThe article under review discusses analogs for Selmer groups attached to an elliptic curve over a number field \(F\). There are previous results running parallel to the above: Kundu showed that for \(\ell=p\) (the classical setting of Iwasawa theory) one may have \(\mu>0\), and Kundu and Lei showed that for \(\ell\not= p\) the analogously defined exponents \(e_n\) remain bounded for the cyclotomic extension, just as above. This begs the question what happens in the non-cyclotomic case for \(\ell\not=p\). \N\NTo make it short, the main result (Theorem 1.4) is: Again one can have exponential growth of the sequence \((e_n)_n\). Here it is possible to replace \(E\) by any given abelian variety \(A/\mathbb Q(\zeta_l)\). One has to enlarge the bottom field \(\mathbb Q(\zeta_\ell)\) to some large field \(L\), and pick a suitable \(\mathbb Z_p\)-extension of \(L\). Also, one needs to assume that \(A\) has nonzero \(\ell\)-torsion over \(\mathbb Q(\zeta_\ell)\). There is even a more general result (Theorem 1.6.) where the Galois group \(\mathbb Z_p\) is replaced by a uniform pro-\(p\)-group \(\Gamma\). Some not so intuitive hypotheses are needed there, but they can be fulfilled by making good choices.\N\NWe will not go into the details of the proofs. It must be said at the present stage that there are a few problems with the arguments. One is in the last line of section 3, on p. 120; here the field degree \([L_n:L]\) should be \(p^{nd}\), not \(dp^n\). Another is on page 124. In line 7 and 8, two inequalities are stated (separated by ``and''). They have the shape \((1): A\ge A'\) and \((2): B\ge B'\), in shortened notation. A few lines below, we find an inequality saying \(A-B \ge A'-B'\) in the same shortened notation. Of course this deduction is invalid without further knowledge on the involved four quantities.\N\NThe reviewer is in contact with the author, and a corrigendum has already been submitted (the reviewer has seen it). It appears that the problems are mended: as to the first problem, the correction is indeed as indicated above, and concerning the second problem, it seems that inequality (2) above is now replaced by some equality, and this of course makes the deduction correct. It is hoped that the corrigendum will appear in print soon.