\(L\)-series and isomorphisms of number fields. (English) Zbl 1470.11294
From the Abstract: “Two number fields with equal Dedekind-zeta-function are not necessarily isomorphic. If the number field have equal sets of Dirichlet \(L\)-series, then they are isomorphic. [This result is extended] by showing that the isomorphisms between the number fields are in bijection with \(L\)-series preserving isomorphism [well described] between the character groups.”
Reviewer’s remarks: Indeed, it all started with the work by F. Gaßmann [Math. Z. 25, 661–675 (1926; JFM 52.0156.03)], that number fields with the same Dedekind-zeta-functions need not be isomorphic. In the Introduction the author describes research and developments up to these days, in particular around the analytic number theory problems concerned with that; his main result Theorem A (paraphrased) runs as follows:
Given number fields \(K\) and \(K'\), and denoting by \(\check{G}_K[\ell]\) and \(\check{G}_{K'}[\ell]\) the \(\ell\)-torsion of the character groups of their absolute Galois groups, and letting \(\text{Iso}_{L\text{-series}}(\check{G}_K[\ell],\check{G}_{K'}[\ell])\) being the set of isomorphisms \(\psi: \check{G}_K[\ell]\overset{\sim}{\rightarrow} \check{G}_{K'}[\ell]\) such that \(\chi\) and \(\psi(\chi)\) have the same \(L\)-series for any \(\chi\in\check{G}_K[\ell]\), where \(\ell\) is any prime number, it then holds that there exists a bijection \[ \Theta:\text{Iso}_{L\text{-series}}(\check{G}_K[\ell], \check{G}_{K'}[\ell])\to \text{Iso}(K,K'). \] Theorem A is the main result in the author’s recent PhD-thesis at the University of Utrecht. The case \(\ell=2\) has been shown by Gabriele Dalla Torre in his PhD-thesis at the University of Leiden.
The paper itself is reasonably technical but straightforward.
As to connections with group theory, it has also to do with the following: Given a finite group \(G\) with subgroups \(H\) and \(T\). Let \(1_H\) be the principal character of \(H\) and \(1_T\) that of \(T\). Consider the induced characters \((1_H)^G\) and \((1_T)^G\). Describe necessary and sufficient conditions on the structures of \(G\), \(H\) and \(T\), in order that \((1_H)^G= (1_T)^G\) is fulfilled.
As to those aspects, view the work of B. de Smit [Lect. Notes Comput. Sci. 1423, 392–399 (1998; Zbl 0914.11055)], W. Bosma and B. de Smit [Lect. Notes Comput. Sci. 2369, 67–79 (2002; Zbl 1068.11080)], B. de Smit and H. W. Lenstra jun. [J. Algebra 228, No. 1, 270–285 (2000; Zbl 0958.20011)], N. Gavioli [Trans. Am. Math. Soc. 349, No. 7, 2969–2980 (1997; Zbl 0872.20005)], A. Caranti et al. [Commun. Algebra 22, No. 3, 877–895 (1994; Zbl 0802.20007)].
All these papers just described do establish a welcome extra to the list of recent publications as printed in the paper under review. – All in all, the author’s work has to be praised as a very good piece of the work.
Reviewer’s remarks: Indeed, it all started with the work by F. Gaßmann [Math. Z. 25, 661–675 (1926; JFM 52.0156.03)], that number fields with the same Dedekind-zeta-functions need not be isomorphic. In the Introduction the author describes research and developments up to these days, in particular around the analytic number theory problems concerned with that; his main result Theorem A (paraphrased) runs as follows:
Given number fields \(K\) and \(K'\), and denoting by \(\check{G}_K[\ell]\) and \(\check{G}_{K'}[\ell]\) the \(\ell\)-torsion of the character groups of their absolute Galois groups, and letting \(\text{Iso}_{L\text{-series}}(\check{G}_K[\ell],\check{G}_{K'}[\ell])\) being the set of isomorphisms \(\psi: \check{G}_K[\ell]\overset{\sim}{\rightarrow} \check{G}_{K'}[\ell]\) such that \(\chi\) and \(\psi(\chi)\) have the same \(L\)-series for any \(\chi\in\check{G}_K[\ell]\), where \(\ell\) is any prime number, it then holds that there exists a bijection \[ \Theta:\text{Iso}_{L\text{-series}}(\check{G}_K[\ell], \check{G}_{K'}[\ell])\to \text{Iso}(K,K'). \] Theorem A is the main result in the author’s recent PhD-thesis at the University of Utrecht. The case \(\ell=2\) has been shown by Gabriele Dalla Torre in his PhD-thesis at the University of Leiden.
The paper itself is reasonably technical but straightforward.
As to connections with group theory, it has also to do with the following: Given a finite group \(G\) with subgroups \(H\) and \(T\). Let \(1_H\) be the principal character of \(H\) and \(1_T\) that of \(T\). Consider the induced characters \((1_H)^G\) and \((1_T)^G\). Describe necessary and sufficient conditions on the structures of \(G\), \(H\) and \(T\), in order that \((1_H)^G= (1_T)^G\) is fulfilled.
As to those aspects, view the work of B. de Smit [Lect. Notes Comput. Sci. 1423, 392–399 (1998; Zbl 0914.11055)], W. Bosma and B. de Smit [Lect. Notes Comput. Sci. 2369, 67–79 (2002; Zbl 1068.11080)], B. de Smit and H. W. Lenstra jun. [J. Algebra 228, No. 1, 270–285 (2000; Zbl 0958.20011)], N. Gavioli [Trans. Am. Math. Soc. 349, No. 7, 2969–2980 (1997; Zbl 0872.20005)], A. Caranti et al. [Commun. Algebra 22, No. 3, 877–895 (1994; Zbl 0802.20007)].
All these papers just described do establish a welcome extra to the list of recent publications as printed in the paper under review. – All in all, the author’s work has to be praised as a very good piece of the work.
Reviewer: Robert W. van der Waall (Amsterdam)
MSC:
| 11R37 | Class field theory |
| 11R42 | Zeta functions and \(L\)-functions of number fields |
| 11M41 | Other Dirichlet series and zeta functions |
| 20C15 | Ordinary representations and characters |
Keywords:
class field theory; \(L\)-series; arithmetic equivalence; Dedekind-zeta-function; permutation characters; absolute Galois groupCitations:
Zbl 0914.11055; Zbl 1068.11080; Zbl 0958.20011; Zbl 0872.20005; Zbl 0802.20007; JFM 52.0156.03References:
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