The Iwasawa main conjecture and Gauss sums
type: Item
instance of: scholarly article
title: The Iwasawa main conjecture and Gauss sums
zbMATH Open document ID: 0998.11055
author: Miho Aoki
published in: Journal of Number Theory
publication date: 2002-06-04T00:00:00Z
review text: The author gives a new proof of the main conjecture of Iwasawa theory for an imaginary abelian number field \(K\) and an odd prime number \(p\) in the semisimple case (i.e. when \(p\) does not divide \([K:\mathbb Q]\)). The technique is that of Kolyvagin's Euler systems. The novelty is the use of Euler systems of Gauss sums, whereas previous proofs, such as Rubin's [\textit{S. Lang}, Cyclotomic fields. I and II. Combined 2nd ed. Graduate Texts in Mathematics, 121. Springer-Verlag, New York (1990; Zbl 0704.11038)] and Greither's [\textit{C. Greither}, Ann. Inst. Fourier 42, 449-499 (1992; Zbl 0729.11053)], used Euler systems of cyclotomic units and Spiegelung (reflection theorems). But note that the author imposes the restrictive condition that \(p\)-adic primes do not split in \(K/K^+\) (the maximal real subfield of \(K\)), which eliminates the technical difficulties due to trivial zeros of \(p\)-adic \(L\)-functions.
reviewed by: A. Movahhedi
Mathematics Subject Classification ID: 11R23
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11R42
zbMATH DE Number: 1749028
zbMATH Keywords: Iwasawa main conjecture
zbMATH Keywords: Gauss sums
zbMATH Keywords: imaginary abelian number field
zbMATH Keywords: Euler systems
Wikidata QID: Q123189728
MaRDI profile type: Publication
OpenAlex ID: W2040519518
cites work: On p-adic L-functions and cyclotomic fields
cites work: Class groups of abelian fields, and the main conjecture
cites work: Lectures on P-Adic L-Functions. (AM-74)
cites work: Q3972742
cites work: Class fields of abelian extensions of \(\mathbb Q\)
cites work: Q3995821
cites work: Q3348987
cites work: Q5690489
cites work: The Iwasawa conjecture for totally real fields
DOI: 10.1006/JNTH.2001.2637
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description: scientific article; zbMATH DE number 1749028