The Chevalley-Herbrand formula and the real abelian Main Conjecture

DOI10.1016/J.JNT.2023.01.002arXiv2207.13911WikidataQ123265690 ScholiaQ123265690MaRDI QIDQ6406321

Publication date: 28 July 2022

Abstract: The Main Theorem for abelian fields (often called Main Conjecture despite proofs in most cases) has a long history which has found a solution by means of "elementary arithmetic", as detailed in Washington's book from Thaine's method having led to Kolyvagin's Euler systems. Analytic theory of real abelian fields

K

says (in the semi-simple case) that the order of the

p

-class group

m a t h c a l H_{K}

is equal to the

p

-index of cyclotomic units

(m a t h c a l E_{K} : m a t h c a l F_{K})

. We have conjectured (1977) the relations

for the isotypic

p

-adic components using the irreducible

p

-adic characters

v a r p h i

of

K

. We develop, in this article, new promising links between: (i) the Chevalley-Herbrand formula giving the number of ``ambiguous classes in $p$ -extensions $L / K$ , $L s u b s e t K (m u_{e} l l^{})$ for the auxiliary prime numbers $e l l e q u i v 1 p m o d 2 p^{N}$ inert in $K$ ; (ii) the phenomenon of capitulation of $m a t h c a l H_{K}$ in $L$ ; (iii) the real Main Conjecture for all~ $v a r p h i$ . We prove that the real Main Conjecture is trivially fulfilled as soon as $m a t h c a l H_{K}$ capitulates in $L$ (Theorem ef{thmppl}). Computations with PARI programs support this new philosophy of the Main Conjecture. The very frequent phenomenon of capitulation suggests Conjecture 1.2.

Mathematics Subject Classification ID

Algebraic number theory computations (11Y40) Class numbers, class groups, discriminants (11R29) Iwasawa theory (11R23) Other abelian and metabelian extensions (11R20)

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