The Iwasawa main conjecture and Gauss sums
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.
scientific article; zbMATH DE number 1749028
scientific article