×

Annihilators of the ideal class group of a cyclic extension of an imaginary quadratic field. (English) Zbl 1443.11224

Let \(K\) be an imaginary quadratic field and let \(L/K\) be a cyclic extension of degree \(p^k\) where \(p\) is an odd prime that does not divide the class number of \(K\) and \(k\) is a positive integer. Let \(\Gamma\) be the Galois group of \(L/K\), say, \(\Gamma=\langle \sigma \rangle \simeq {\mathbb Z}/p^k{\mathbb Z}\). Assume that there are exactly \(s\geq 2\) (pairwise distinct) prime ideals \({\wp}_1, \ldots, {\wp}_s\) of \(K\) that ramify in \(L/K\).
From a subgroup of elliptic units \(\mathcal{C}_L\) of finite index in the group of units \(\mathcal{O}_L^{\times}\) of \(L\) the authors obtain the extended group of elliptic units \(\overline{\mathcal{C}}_L\) of \(L\), a certain \({\mathbb {Z}[\Gamma]}\)-submodule of finite index in \(\mathcal{O}_L^{\times}\).
The main result of the paper under review identifies annihilators of the \(p\)-Sylow subgroup \(\mathrm{Cl}(L)_p\) of the ideal class group \(\mathrm{Cl}(L)\) of \(L\). These annihilators are obtained from annihilators of the \(p\)-Sylow subgroup \((\mathcal{O}_L^{\times}/\overline{\mathcal{C}}_L)_p\) of \(\mathcal{O}_L^{\times}/\overline{\mathcal{C}}_L\).
More precisely, for each \(j\in\{1, \ldots, s\}\) fix an arbitrarily chosen prime ideal \({\mathfrak P}_j\) of \(L\) above \({\wp}_j\). Let \(t_j\) be the ramification index of \({\mathfrak P}_j\) over \({\wp}_j\) and let \(n_j\) be the index of the ramification group of \({\mathfrak P}_j\) in \(\Gamma\). Without loss of generality we can assume that \(n_1\leq n_2\leq, \ldots , \leq n_s\). Label the subfields of \(L\) that contain \(K\) as follows:
\[K=L_0\subsetneq L_1\subsetneq L_2\subsetneq \cdots \subsetneq L_k=L.\]
In particular we must have \([L_i:K]=p^i\). For each \(i=1\ldots k\) define
\[M_i=\{j\in \{1,\ldots, s\}; t_j>p^{k-i}\}\]
and define \(\mu_i=n_{\mathrm {max}M_i}.\) Then \(\mu_i\leq \mu_{i+1}\) and we call an index \(i\in \{1,\ldots, k-1\}\) a \(\mathit {jump}\) if \(\mu_i<\mu_{i+1}\).
The main result of the paper under review is
Theorem 7.5. Let \(r\) be the highest jump less than k, i.e., \(\mu_r<\mu_{r+1}=n_s\). If \(\varkappa \in \mathrm{Ann}_{\mathbb {Z}[\Gamma]}((\mathcal{O}_L^{\times}/\overline{\mathcal{C}}_L)_p)\), then \((1-\sigma^{p^r})\varkappa\) annihilates \(\mathrm{Cl}(L)_p\). In other words, we have \[ \mathrm{Ann}_{\mathbb {Z}[\Gamma]}((\mathcal{O}_L^{\times}/\overline{\mathcal{C}}_L)_p)\subseteq \mathrm{Ann}_{\mathbb {Z}[\Gamma]}((1-\sigma^{p^r})\mathrm{Cl}(L)_p).\]
The number \(r\) can be characterized as follows: \(p^{k-r}=\mathrm{max}\{t_j;j\in J\}\), where \(J=\{j\in\{1,\ldots, s\};n_j=n_s\}\).
As the authors point out, the present paper is highly motivated by the series of papers [C. Greither and the second author, Acta Arith. 112, No. 2, 177–198 (2004; Zbl 1065.11089); Can. J. Math. 58, No. 3, 580–599 (2006; Zbl 1155.11054); Publ. Math. 86, No. 3–4, 401–421 (2015; Zbl 1374.11087)]. They emphasize that many of the techniques employed in the present paper borrow heavily from those introduced in the first and last of these.

MSC:

11R20 Other abelian and metabelian extensions
11R27 Units and factorization
11R29 Class numbers, class groups, discriminants
11R37 Class field theory
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] W.Bley, Wild Euler systems of elliptic units and the equivariant Tamagawa number conjecture. J. Reine Angew. Math.577(2004), 117-146. https://doi.org/10.1515/crll.2004.2004.577.117. · Zbl 1056.11035
[2] W.Bley, Equivariant Tamagawa number conjecture for abelian extensions of a quadratic imaginary field. Doc. Math.11(2006), 73-118. · Zbl 1178.11070
[3] D.Burns, Congruences between derivatives of abelian L-functions at s = 0. Invent. Math.169(2007), 451-499. https://doi.org/10.1007/s00222-007-0052-3. · Zbl 1133.11063
[4] G.Gras, Class field theory. From theory to practice. Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. · Zbl 1019.11032
[5] C.Greither and R.Kučera, Annihilators for the class group of a cyclic field of prime power degree. Acta Arith.112(2004), 177-198. https://doi.org/10.4064/aa112-2-6. · Zbl 1065.11089
[6] C.Greither and R.Kučera, Annihilators for the class group of a cyclic field of prime power degree II. Canad. J. Math.58(2006), 580-599. https://doi.org/10.4153/CJM-2006-024-2. · Zbl 1155.11054
[7] C.Greither and R.Kučera, Linear forms on Sinnott’s module. J. Number Theory141(2014), 324-342. https://doi.org/10.1016/j.jnt.2014.02.003. · Zbl 1309.11079
[8] C.Greither and R.Kučera, Eigenspaces of the ideal class group. Ann. Inst. Fourier (Grenoble)64(2014), 2165-2203. https://doi.org/10.5802/aif.2908. · Zbl 1315.11093
[9] C.Greither and R.Kučera, Annihilators for the class group of a cyclic field of prime power degree III. Publ. Math. Debrecen86(2015), no. 3-4, 401-421. · Zbl 1374.11087
[10] T.Ohshita, On higher Fitting ideals of Iwasawa modules of ideal class groups over imaginary quadratic fields and Euler systems of elliptic units. Kyoto J. Math.53(2013), 845-887. https://doi.org/10.1215/21562261-2366118. · Zbl 1372.11102
[11] H.Oukhaba, Index formulas for ramified elliptic units. Compositio Math.137(2003), 1-22. https://doi.org/10.1023/A:1023667807218. · Zbl 1045.11043
[12] K.Rubin, Global units and ideal class groups. Invent. Math.89(1987), 511-526. https://doi.org/10.1007/BF01388983. · Zbl 0628.12007
[13] K.Rubin, Stark units and Kolyvagin’s “Euler systems”. J. Reine Angew. Math.425(1992), 141-154. https://doi.org/10.1515/crll.1992.425.141. · Zbl 0752.11045
[14] W.Sinnott, On the Stickelberger ideal and the circular units of an abelian field. Invent. Math.62(1980), 181-234. https://doi.org/10.1007/BF01389158. · Zbl 0465.12001
[15] F.Thaine, On the ideal class groups of real abelian number fields. Ann. of Math. (2)128(1988), no. 1, 1-18. https://doi.org/10.2307/1971460. · Zbl 0665.12003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.