The real Abelian main conjecture in the finite non semi-simple case
Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry - Trang 1-29 - 2023
Tóm tắt
Let
$$K/\mathbb {Q}$$
be a real cyclic extension of degree divisible by p. We analyze the statement of the “Real Abelian Main Conjecture”, for the p-class group
$${\mathscr {H}}_K$$
of K. The classical algebraic definition of the p-adic isotypic components,
$${\mathscr {H}}^\textrm{alg}_{K,\varphi }$$
, for p-adic characters
$$\varphi = \varphi _0^{} \varphi _p$$
(
$$\varphi _0$$
of prime-to-p order,
$$\varphi _p$$
of p-power order), is inappropriate with respect to analytic formulas, because of capitulation of p-classes in the p-sub-extension of
$$K/\mathbb {Q}$$
. In the 1970’s we have given an arithmetic definition,
$${\mathscr {H}}^{\textrm{ar}}_{K,\varphi }$$
, and formulated the conjecture, still unproven,
$$\#{\mathscr {H}}^{\textrm{ar}}_{K,\varphi } = \#({\mathscr {E}}_K / \widehat{{\mathscr {E}}}_K \, {\mathscr {F}}_{\!K})_{\varphi _{0}^{}}$$
, in terms of units
$${\mathscr {E}}_K$$
,
$$\widehat{{\mathscr {E}}}_K$$
(units of the strict subfields) and
$${\mathscr {F}}_{\!K}$$
(Leopoldt’s cyclotomic units). We prove here that the conjecture holds as soon as there exists a prime
$$\ell $$
, totally inert in K, such that
$${\mathscr {H}}_K$$
capitulates in
$$K(\mu _\ell ^{})$$
, existence having been checked, in various circumstances, as a promising new tool. An Appendix of numerical examples is given with PARI programs. A second Appendix deals with the special case
$$ K \cap \mathbb {Q}(\mu _{p^\infty }^{})^+ \ne \mathbb {Q}$$
.
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