On the group of modular units and the ideal class group
In 1991, \textit{J. M. Kim, S. Bae} and \textit{I.-S. Lee} [Isr. J. Math. 75, No. 2--3, 161--165 (1991; Zbl 0765.11042)] proved a remarkable isomorphism concerning cyclotomic fields: if \(A_n\) denotes the \(p\)-part of the class group of \(K_n=Q(\zeta_{p^{n+1}})^+\) and \(B_n\) denotes the quotient of the unit group by the subgroup of cyclotomic units attached to \(K_n\), then \(A_n \cong B_n\) for large \(n\), provided that the \(\lambda\)-invariant vanishes (as predicted by Greenberg's conjecture). This fact was independently rediscovered by Kraft and Schoof a few years later [see \textit{J. Kraft} and \textit{R. Schoof}, Compos. Math. 97, No. 1--2, 135--155 (1995; Zbl 0840.11043)]. In the present paper, the authors give an analogous result in the setting of complex multiplication: \(k\) is imaginary quadratic of conductor greater than 4, \(p\) is a prime that splits in \(k\), and \(K_n\) is now the ray class field \(k(P^{n+1})\), where \(P\) is one of the two prime divisors of \(p\) in \(k\). There is a close analogue to the group of cyclotomic units, afforded by the group of modular units, and this leads to a definition of a quotient \(B_n\) which is quite analogous to the above. The authors prove \(A_n \cong B_n\) for \(n\) large, again under the assumption that \(\lambda\) is zero, with the extra hypothesis that \(P\) does not split at all in the Hilbert class field of \(k\) (which is contained in \(K_n\)); so in particular the class group of \(k\) must be cyclic. The algebraic part of the argument, which uses a little group cohomology, is similar in style to the arguments of Kim, Bae and Lee. On the analytic side, of course more has to be done, but the existing work of Kubert and Lang on modular units is put to good use. Reviewer's remark: Komatsu informs me that the leftmost exponent in the statement of Lemma~3 should read \(h(p-1)/2-1\) (the last \(-1\) was omitted in print). Otherwise the rank of the module will not come out right.
scientific article; zbMATH DE number 5129754