The real Abelian main conjecture in the finite non semi-simple case

Abstract

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 $K \cap \mathbb {Q}(\mu _{p^\infty }^{})^+ \ne \mathbb {Q}$.

Appendices

Examples of $\#{\mathscr {H}}^\textrm{ar}_{K,\varphi } \!= \#({\mathscr {E}}_K / \widehat{\mathscr {E}}_K\, {\mathscr {F}}_{\!K})_{\varphi _0^{}}$

1.1 Introduction

Since for $K = K_\chi$ the existence of (infinitely many) extensions L/K in which ${\mathscr {H}}_K$ capitulates, is a conjecture largely tested in Gras (2022, 2023), we limit ourselves, in this Appendix, to an illustration of the formula ${\#{\mathscr {H}}^\textrm{ar}_{K,\chi } = \#({\mathscr {E}}_K/ \widehat{\mathscr {E}}_K\, {\mathscr {F}}_{\!K})}$ (Theorem 3.2), for $p=3$, k real quadratic and $K_0/\mathbb {Q}$ cubic cyclic (so that $\varphi = \chi$); then we only check this relation.

Many cases occur when the structure of the class groups varies as well as the relations $\eta _i = \prod _{j=1}^3 \varepsilon _j^{a_{i,j}}$, $1 \le i \le 3$, describing the relative cyclotomic units $\eta _i$ from the relative units $\varepsilon _j$ (relative units meaning of norm 1 over $K_0$, which allows to compute in direct products of groups of units). Once again, we will see that the two $\mathbb {Z}_p[{\mathscr {G}}_K]$-modules, of same order, ${\mathscr {E}}_K / \widehat{\mathscr {E}}_K {\mathscr {F}}_{\!K}$ and ${\mathscr {H}}^\textrm{ar}_{K,\chi }$, are not isomorphic in general.

The program computes $\textbf{H}_k$ and $\textbf{H}_K$; so ${\#{\mathscr {H}}^\textrm{ar}_{K,\chi } = \#{\mathscr {H}}_K \times \#{\mathscr {H}}_k^{-1}}$.

1.2 Computation of the group structure of ${\mathscr {E}}_K/\widehat{\mathscr {E}}_K {\mathscr {F}}_{\!K}$

To simplify (especially, for the computation of the cyclotomic units), we take $k = \mathbb {Q}(\sqrt{f})$, where f is a prime number congruent to 1 modulo 4 (hence the conductor of k) and various cubic fields $K_0$ of prime conductors $q \equiv 1 \pmod 3$.

In the program, the main parameters are the following:

$\textsf {Pk, PK}$: defining polynomials of $\textsf {k, K}$;

$\textsf {Ck, CK}$: class groups of $\textsf {k, K}$;

$\textsf {Ek, EK}$: unit groups of $\textsf {k, K}$;

$\textsf {LEtaK}$: list of the 3 independent relative cyclotomic units of $\textsf {K}$.

The conductor $\textsf {f}$ varies in $\textsf {[bf, Bf]}$; the prime $\textsf {q}$ varies in $\textsf {[bq, Bq]}$; then, $\textsf {kronecker(f,q)}$ gives the splitting of q in $k/\mathbb {Q}$, which may give larger 3-class groups for K if $\textsf {kronecker(f,q) = 1}$.

Then the computations of some indices of groups of units are done taking the logarithms of the units, to get easier linear relations. We do not write the units and their logarithms since they are oversized; but the running of the program gives rapidly a complete data (use large precision and memory); for instance, with $\textsf {f = 257}$ and $\textsf {q = 4597}$ a precision $\textsf {p \ 300}$ is necessary for the computation of some logarithms.

For some rare cases, the program gives 5 fundamental relative units $\varepsilon _i$ (instead of 3), but with two relations of dependence; this is due to PARI computation of the fundamental units in $\textsf {K.fu}$ where the units of the cubic field $K_0$ do not appear as a direct factor; a same remark may occur for the fundamental unit $\varepsilon _0$ of the quadratic field k (in general, $\varepsilon _0 = \varepsilon _1$, but for $k=\mathbb {Q}(\sqrt{1129})$ and $q=7$, $\varepsilon _0 = \varepsilon _1\varepsilon _2^{-1}\varepsilon _3^{-1}$.

The final matrix $\big ( a_{i,j}\big )_{i,j}$ is such that $\eta _i = \prod _{j=1}^3 \varepsilon _j^{a_{i,j}}$.

1.2.1 PARI program

The program uses the package PARI/GP (The PARI Group 2016):

1.2.2 Quadratic field k of conductor 229

In the following case, the program gives 5 relative units $\varepsilon '_i$ with relations $\varepsilon '_3=\varepsilon '_2$ and $\varepsilon '_5= \varepsilon '_2\cdot \varepsilon '^{-1}_4$, giving the basis $\varepsilon _1= \varepsilon '_1$, $\varepsilon _2= \varepsilon '_2$, $\varepsilon _3= \varepsilon '_4$:

1.2.3 Quadratic field k of conductor 1129

In the next example, the fundamental unit $\varepsilon _0$ of k is not $\varepsilon _1$ given by PARI; one gets $\varepsilon _0 = \varepsilon _1 \varepsilon _2^{-1}\varepsilon _3^{-1}$:

In the following case, the program gives 5 relative units $\varepsilon '_i$ with the relations $\varepsilon '_3=\varepsilon '_2$ and $\varepsilon '_5= \varepsilon '_4\cdot \varepsilon '^{-1}_1 \cdot \varepsilon '_2$, giving the basis $\varepsilon _1= \varepsilon '_1$, $\varepsilon _2= \varepsilon '_2$, $\varepsilon _3= \varepsilon '_4$.

1.3 Checking of the relations $\#{\mathscr {H}}^\textrm{ar}_{K,\chi } = \#({\mathscr {E}}_K / \widehat{\mathscr {E}}_K {\mathscr {F}}_{\!K})$

The quotient ${\mathscr {E}}_{K} / \widehat{\mathscr {E}}_{K} {\mathscr {F}}_{\!K}$, is monogenic as $\varphi$-object (whence as $\mathbb {Z}_3[j]$-module).

(a) Quadratic field k of conductor 229 (${\mathscr {H}}_k \simeq \mathbb {Z}/3\mathbb {Z}$):

Case (i): $\textsf {CH}_\textsf {K} = [\textsf {3,3,3}]$ and $\textsf {CH}_{\textsf {K},\chi } = [\textsf {3,3}]$. Since $\varepsilon _1 = \varepsilon _0$, the unit index is $\big (\langle \varepsilon _1,\varepsilon _2,\varepsilon _3 \rangle : \langle \varepsilon _1,\varepsilon _2^3,\varepsilon _3^{-3}, \varepsilon _2^{-3}\varepsilon _3^3 \rangle \big ) = \big (\langle \varepsilon _1, \varepsilon _2,\varepsilon _3 \rangle : \langle \varepsilon _1,\varepsilon _2^3,\varepsilon _3^3 \rangle \big ) = 9.$

Then ${\mathscr {E}}_{K} / \widehat{\mathscr {E}}_{K} {\mathscr {F}}_{\!K}$ is isomorphic to $\mathbb {Z}/9 \mathbb {Z}$, while ${\mathscr {H}}_{K,\chi } \simeq \mathbb {Z}/3\mathbb {Z}\times \mathbb {Z}/3\mathbb {Z}$.

Case (ii): $\textsf {CH}_\textsf {K} = [\textsf {9,3}]$ and $\textsf {CH}_{\textsf {K},\chi } = [\textsf {3,3}]$ or $\textsf {[9]}$. Once we are using the basis $\varepsilon _1= \varepsilon '_1$, $\varepsilon _2= \varepsilon '_2$, $\varepsilon _3= \varepsilon '_4$, the matrix is analogous to the previous one, whence the result with the index 9.

Case (iii): $\textsf {CH}_\textsf {K} = [\textsf {9,9}]$ and $\textsf {CH}_{\textsf {K},\chi } = [\textsf {9,3}]$. The unit index is:

$$\begin{aligned}{} & {} \big (\langle \varepsilon _1,\varepsilon _2,\varepsilon _3 \rangle : \langle \varepsilon _1, \varepsilon _1^{-2} \varepsilon _2^{-6}\varepsilon _3^{-3}, \varepsilon _1 \varepsilon _2^3 \varepsilon _3^{-3}, \varepsilon _1^4\varepsilon _2^{3}\varepsilon _3^6 \rangle \big ) \\{} & {} \quad = \big (\langle \varepsilon _1, \varepsilon _2, \varepsilon _3 \rangle : \langle \varepsilon _1,\varepsilon _2^3 \varepsilon _3^{-3}, \varepsilon _2^9, \varepsilon _3^9 \rangle \big ) = \big (\langle \varepsilon _1, \varepsilon _2, \varepsilon _3 \rangle : \langle \varepsilon _1,\varepsilon _2^3 \varepsilon _3^{-3}, \varepsilon _2^9 \rangle \big )= 27. \end{aligned}$$

Case (iv): $\textsf {CH}_\textsf {K} = [\textsf {9,3,3}]$ and $\textsf {CH}_{\textsf {K},\chi } = [\textsf {9,3}]$ or $\textsf {[3,3,3]}$:

$$\begin{aligned}{} & {} \big (\langle \varepsilon _1,\varepsilon _2,\varepsilon _3 \rangle : \langle \varepsilon _1, \varepsilon _1^{-3} \varepsilon _2^{6}\varepsilon _3^{3}, \varepsilon _1^{3} \varepsilon _2^{-3} \varepsilon _3^{-6}, \varepsilon _2^{-3}\varepsilon _3^3 \rangle \big ) \\{} & {} \quad = \big (\langle \varepsilon _1, \varepsilon _2, \varepsilon _3 \rangle : \langle \varepsilon _1,\varepsilon _2^{-3} \varepsilon _3^{3}, \varepsilon _2^9, \varepsilon _3^9 \rangle \big ) = \big (\langle \varepsilon _1, \varepsilon _2, \varepsilon _3 \rangle : \langle \varepsilon _1,\varepsilon _2^{-3} \varepsilon _3^{3}, \varepsilon _2^9 \rangle \big )= 27. \end{aligned}$$

(b) Quadratic field k of conductor 1129 (${\mathscr {H}}_k \simeq \mathbb {Z}/9\mathbb {Z}$):

Case (v): $\textsf {CH}_\textsf {K} = [\textsf {27}]$ and $\textsf {CH}_{\textsf {K},\chi } = [\textsf {3}]$. We have $\varepsilon _0 = \varepsilon _1 \varepsilon _2^{-1} \varepsilon _3^{-1}$:

$$\begin{aligned}{} & {} \big (\langle \varepsilon _1 \varepsilon _2^{-1} \varepsilon _3^{-1},\varepsilon _2,\varepsilon _3 \rangle : \langle \varepsilon _1 \varepsilon _2^{-1} \varepsilon _3^{-1}, \varepsilon _1^{-1} \varepsilon _2^{-1}, \varepsilon _2 \varepsilon _3^{-1}, \varepsilon _1 \varepsilon _3 \rangle \big ) \\{} & {} \quad = \big (\langle \varepsilon _1 \varepsilon _2^{-1} \varepsilon _3^{-1},\varepsilon _2,\varepsilon _3 \rangle : \langle \varepsilon _1 \varepsilon _2^{-1} \varepsilon _3^{-1}, \varepsilon _2 \varepsilon _3^{-1}, \varepsilon _1^{-1} \varepsilon _2^{-1}, \varepsilon _1 \varepsilon _3\rangle \big ) = \big (\langle \varepsilon _1 \varepsilon _2^{-1} \varepsilon _3^{-1}, \varepsilon _1 \varepsilon _2,\varepsilon _1 \varepsilon _3 \rangle \big )= 3. \end{aligned}$$

Case (vi): $\textsf {CH}_\textsf {K} = [\textsf {27,3}]$ and $\textsf {CH}_{\textsf {K},\chi } = [\textsf {3,3}]$:

$$\begin{aligned}{} & {} \big (\langle \varepsilon _1,\varepsilon _2,\varepsilon _3 \rangle : \langle \varepsilon _1, \varepsilon _1^{8} \varepsilon _2^{-6}\varepsilon _3^{9}, \varepsilon _1^{2} \varepsilon _2^{-3} \varepsilon _3^{-6}, \varepsilon _1^{-1}\varepsilon _2^{9}\varepsilon _3^{-3} \rangle \big ) \\{} & {} \quad = \big (\langle \varepsilon _1, \varepsilon _2, \varepsilon _3 \rangle : \langle \varepsilon _1, \varepsilon _2^{-6} \varepsilon _3^{9}, \varepsilon _2^{-3} \varepsilon _3^{-6},\varepsilon _2^9 \varepsilon _3^{-3} \rangle \big ) = \big (\langle \varepsilon _1, \varepsilon _2, \varepsilon _3 \rangle : \langle \varepsilon _1, \varepsilon _2^3 \varepsilon _3^6, \varepsilon _2^9 \varepsilon _3^{-3} \rangle \big ) = 9. \end{aligned}$$

Case (vii): $\textsf {CH}_\textsf {K} = [\textsf {9,9,9}]$ and $\textsf {CH}_{\textsf {K},\chi } = [\textsf {9,9}]$:

$$\begin{aligned} \big (\langle \varepsilon _1,\varepsilon _2,\varepsilon _3 \rangle : \langle \varepsilon _1, \varepsilon _3^{-9}, \varepsilon _1^{9} \varepsilon _2^{9}\varepsilon _3^{9}, \varepsilon _2^{-9} \rangle \big )= \big (\langle \varepsilon _1, \varepsilon _2, \varepsilon _3 \rangle : \langle \varepsilon _1, \varepsilon _2^{9}, \varepsilon _3^{9} \rangle \big ) = 81. \end{aligned}$$

Case (viii): $\textsf {CH}_\textsf {K} = [\textsf {27,9,3}]$ and $\textsf {CH}_{\textsf {K},\chi } = [\textsf {27,3}]$ or $\textsf {[3,9,3]}$:

$$\begin{aligned}{} & {} \big (\langle \varepsilon _1,\varepsilon _2,\varepsilon _3 \rangle : \langle \varepsilon _1, \varepsilon _1^{-3} \varepsilon _2^{-9}\varepsilon _3^{9}, \varepsilon _1^{6} \varepsilon _2^{9}, \varepsilon _1^{6} \varepsilon _3^{-9} \rangle \big ) \\{} & {} \quad = \big (\langle \varepsilon _1, \varepsilon _2, \varepsilon _3 \rangle : \langle \varepsilon _1, \varepsilon _2^{-9} \varepsilon _3^{9}, \varepsilon _2^{9}, \varepsilon _3^{9} \rangle \big ) = \big (\langle \varepsilon _1, \varepsilon _2, \varepsilon _3 \rangle : \langle \varepsilon _1, \varepsilon _2^9, \varepsilon _3^9 \rangle \big ) = 81. \end{aligned}$$

Cases where $K \cap \mathbb {Q}(\mu _{p^\infty }^{})^+ \ne \mathbb {Q}$

1.1 Analytic expression of ${\#{\mathscr {H}}^\textrm{ar}_{K,\chi }}$ and the FRAMC

As we have explained, the particular case $K \cap \mathbb {Q}(\mu _{p^\infty }^{})^+ \ne \mathbb {Q}$ must be examined separately; this is coherent with the specific properties of p-adic $\textrm{L}$-functions regarding characters of conductors powers of p.

Let ${\mathscr {C}}:= K \cap \mathbb {Q}(\mu _{p^\infty }^{})^+$ and let ${\mathscr {C}}_0 := k \cap \mathbb {Q}(\mu _p^{})^+$ with $d = [k : \mathbb {Q}] > 1$, ${\varphi _0^{}} \mid \chi _0^{} \ne 1$ and $[K : k] = p^e$, $e \ge 1$. Note that for $p \le 3$, ${\mathscr {C}}_0 = \mathbb {Q}$ and ${\mathscr {C}}$ is a layer of the cyclotomic $\mathbb {Z}_p$-extension $\mathbb {Q}_\infty$; since $k \ne \mathbb {Q}$ and ${\varphi _0^{}} \ne 1$, this case will be solved in Sect. B.3.2. So we can focus on the case $p > 3$ in the sequel.

We will analyze the FRAMC, under capitulation properties, in two steps corresponding to the following cases:

• (i) ${\mathscr {C}}= K \subset \mathbb {Q}(\mu _{p^\infty }^{})^+$ (Sect. B.2),

• (ii) $\mathbb {Q}\varsubsetneqq {\mathscr {C}}\varsubsetneqq K$, with two sub-cases ${\mathscr {C}}_0 \varsubsetneqq k$ and the “special case” ${\mathscr {C}}_0 = k$ (Sect. B.3.1).

1.2 Case $K \subset \mathbb {Q}(\mu _{p^\infty }^{})^+$ and Greenberg’s conjecture

Note that this framework makes sense only if Vandiver’s conjecture is false, otherwise all the subfields of $\mathbb {Q}(\mu _{p^\infty }^{})^+$ are p-principal. Thus, k is a subfield of $\mathbb {Q}(\mu _p^{})^+$ and K is the layer $k_e$ of the cyclotomic $\mathbb {Z}_p$-extension $k_\infty$ of k.

This case can not use an auxiliary totally inert prime $\ell \equiv 1 \pmod {2p^N}$ (since it is totally split in a non-trivial subfield of K), but assuming Greenberg’s conjecture (Greenberg 1976, Theorem 2), as we have recalled in Sect. 1.4, there exists $n_0 \ge 0$ such that, for all $m \ge n \ge n_0$, arithmetic norms $\textbf{N}_{k_m/k_n}: {\mathscr {H}}_{k_m} \rightarrow {\mathscr {H}}_{k_n}$ are isomorphisms and the p-class group of any layer $k_n$ capitulates in $k_\infty$; so, ${\mathscr {H}}_K$ capitulates in $k_\infty$ (in some sense, p replaces $\ell$ and $k_\infty$ replaces $L \subset K(\mu _\ell ^{})$).

Since p is totally ramified in $k_\infty /\mathbb {Q}$, the Chevalley–Herbrand formula yields ${\#{\mathscr {H}}_{k_n}^{\textrm{Gal}(k_n/k)} = \#{\mathscr {H}}_{k}}$, and in the same way, ${\mathscr {H}}_{k_n,\varphi _0^{}}^{\textrm{Gal}(k_n/k)} \simeq \big ( {\mathscr {E}}_{k}/ \textbf{N}_{k_n/k}({\mathscr {E}}_{k_n}) \big )_{\varphi _0^{}}$, of order ${\#{\mathscr {H}}_{k,\varphi _0^{}}}$, as soon as ${\mathscr {H}}_{k,\varphi _0^{}}$ capitulates in $k_n$.

In terms of cyclotomic units ${\mathscr {F}}_{\! k_n}$, we have the fundamental norm relation $\textbf{N}_{k_m/k_n}({\mathscr {F}}_{\!k_m}) = {\mathscr {F}}_{\!k_n}$, for all $m \ge n \ge 0$; the previous principle of computation holds, which leads to the proof of the non semi-simple FRAMC, for $K \subset \mathbb {Q}(\mu _{p^\infty }^{})^+$.

However, Greenberg’s conjecture is far to be proved and is perhaps more difficult than the capitulation conjecture using primes $\ell$ totally inert in $K/\mathbb {Q}$, since these are infinite in number and many numerical computations go in this direction.

1.3 Analysis of the FRAMC when $K \cap \mathbb {Q}(\mu _{p^\infty }^{})^+ \varsubsetneqq K$

1.3.1 Expression of ${\#{\mathscr {H}}^\textrm{ar}_{K,\varphi }}$

We will use the following definitions when $K = K_\chi$ (of degree $d p^e$) contains ${\mathscr {C}}:= K \cap \mathbb {Q}(\mu _{p^\infty }^{})^+$, such that $\mathbb {Q}\varsubsetneqq {\mathscr {C}}\varsubsetneqq K$. Let $\varphi = \varphi _0 \varphi _p \mid \chi$.

Let $M_0 := M(\ell ) {\mathscr {C}}$, where $M(\ell )$ is the subfield of degree $p^n$, $n \le N$, of $\mathbb {Q}(\mu _\ell ^{})$ for $\ell \equiv 1 \pmod {2p^N}$, $N \ge 1$, with $\ell$ prime totally inert in $K/{\mathscr {C}}$ (from the Chebotarev density theorem) and let $L= M_0 K$. So any prime ideal ${{\mathfrak {l}}} \mid \ell$ of K is totally ramified in L/K, then $\ell$ is (totally if N is large enough) split in ${\mathscr {C}}$ and totally inert in $L/M_0$:

We recall some notations; in particular, let $\widehat{K}$ and $\widehat{L}$ be the subfields of K and L such that $[K: \widehat{K}] = [L: \widehat{L}] = p$. Put $\widehat{\textbf{N}}:= \textbf{N}_{L/\widehat{L}}$, $\widehat{\textbf{J}}:= \textbf{J}_{L/\widehat{L}}$. If ${\mathscr {H}}_K$ capitulates in L, then ${\mathscr {H}}_{\widehat{K}}$ capitulates in $\widehat{L}$. By abuse, put ${({\mathscr {A}}: {\mathscr {B}})_{\varphi _0^{}} = \#({\mathscr {A}}/{\mathscr {B}})_{\varphi _0^{}}}$.

Under the above assumptions, the p-localized Chevalley–Herbrand formulas for p-class groups are the following ones [similar to Formula (9) using the properties of the maps $\omega _{L/K}$ and $\omega _{\widehat{L}/\widehat{K}}$, with $\Omega _{L/K} \simeq \Omega _{\widehat{L}/\widehat{K}}$ as $\mathbb {Z}_p[\gamma ]$-modules]:

$$\begin{aligned} \#{\mathscr {H}}_{L,\varphi _0^{}}^G = \displaystyle \#{\mathscr {H}}_{K,\varphi _0^{}} \times \frac{\#\Omega _{L/K, \varphi _0^{}}}{\#\omega _{L/K} ({\mathscr {E}}_{K,{\varphi _0^{}}})},\ \ \#{\mathscr {H}}_{\widehat{L},\varphi _0^{}}^G = \displaystyle \#{\mathscr {H}}_{\widehat{K},\varphi _0^{}} \times \frac{\#\Omega _{\widehat{L}/\widehat{K}, \varphi _0^{}}}{\#\omega _{\widehat{L}/\widehat{K}} ({\mathscr {E}}_{\widehat{K},\varphi _0^{}}) }.\nonumber \\ \end{aligned}$$

(B1)

Under capitulations in L/K and $\widehat{L}/\widehat{K}$, we have the exact sequences:

(B2)

where (for L and $\widehat{L}$) ${\mathscr {H}}^\textrm{ram}\subseteq {\mathscr {H}}$ is generated by the p-classes of the ramified prime ideals dividing $\ell$ (let ${{\mathfrak {A}}}$ be a representative of a G-invariant p-class of L, whence ${{\mathfrak {A}}}^{1 - \sigma } =: (\alpha )$, $\alpha \in L^\times$; then $\textbf{N}_{L/K} (\alpha ) =: \varepsilon \in {\mathscr {E}}_K \cap \textbf{N}_{L/K} (L^\times ) = {\mathscr {E}}_K \cap {\mathscr {N}}_{L/K}$ and straightforward computations lead to the image and kernel).

From (B1), (B2) giving two expressions of ${\mathscr {H}}_{L,\varphi _0^{}}^{G}$ and ${\mathscr {H}}_{\widehat{L},\varphi _0^{}}^{G}$, one gets:

$$\begin{aligned} {\#{\mathscr {H}}_{K,{\varphi _0^{}}} \!\! = \frac{\#{\mathscr {H}}_{L,{\varphi _0^{}}}^\textrm{ram}\times \big ({\mathscr {E}}_K: \textbf{N}({\mathscr {E}}_L)\big )_{\varphi _0^{}}}{\#\Omega _{{L/K},\varphi _0^{}}}, \ \#{\mathscr {H}}_{\widehat{K},{\varphi _0^{}}}\!\! = \frac{\#{\mathscr {H}}_{\widehat{L},{\varphi _0^{}}}^\textrm{ram}\times \big ({\mathscr {E}}_{\widehat{K}}: \textbf{N}({\mathscr {E}}_{\widehat{L}})\big )_{\varphi _0^{}}}{\#\Omega _{\widehat{L}/\widehat{K}, \varphi _0^{}}}.}\nonumber \\ \end{aligned}$$

(B3)

One sees easily that ${\mathscr {H}}_{L,{\varphi _0^{}}}^\textrm{ram}= \widehat{\textbf{J}}\big ({\mathscr {H}}_{\widehat{L},{\varphi _0^{}}}^\textrm{ram}\big )$ since prime ideals over $\ell$ are inert in $L/\widehat{L}$. We have the exact sequence defining ${\mathscr {H}}^\textrm{ar}_{K,\varphi }$:

then, from (B3), and the fact that ${\#\Omega _{L/K,{\varphi _0^{}}} = \#\Omega _{\widehat{L}/\widehat{K},{\varphi _0^{}}}}$, we get, under capitulations, the fundamental relation:

$$\begin{aligned} {\#{\mathscr {H}}^\textrm{ar}_{K,\varphi } = \frac{\#\widehat{\textbf{J}}\big ({\mathscr {H}}_{\widehat{L},{\varphi _0^{}}}^\textrm{ram}\big )}{\#{\mathscr {H}}_{\widehat{L},{\varphi _0^{}}}^\textrm{ram}} \times \frac{ \big ( {\mathscr {E}}_K: \textbf{N}({\mathscr {E}}_L)\big )_{\varphi _0^{}}}{\big ({\mathscr {E}}_{\widehat{K}}: \textbf{N}({\mathscr {E}}_{\widehat{L}})\big )_{\varphi _0^{}}},} \end{aligned}$$

(B4)

to be compared with $X_{\varphi _0^{}} := \big ({\mathscr {E}}_K : {\mathscr {E}}_{\widehat{K}} {\mathscr {F}}_{\!K}\big )_{\varphi _0^{}}$. So, the proof of the FRAMC is essentially based on $\textbf{N}({\mathscr {F}}_{\!L})$ regarding $\textbf{N}({\mathscr {E}}_L)$.

We have $\textbf{N}({\mathscr {F}}_{\!L}) = {\mathscr {F}}_{\!K}^{1-\tau _\ell ^{-1}}$, where $\tau _\ell ^{}$ is the Artin automorphism of $\ell$ in $K/\mathbb {Q}$; it generates $\textrm{Gal}(K/{\mathscr {C}})$ and is of p-power order if and only if ${\mathscr {C}}_0 = k$, which defines the “special case”.

1.3.2 Proof of the FRAMC when ${\mathscr {C}}_0 \varsubsetneqq k$

In this case, $(1-\tau _\ell ^{-1}) e_{\varphi _0^{}}$ is invertible in $\mathbb {Z}_p[{\mathscr {G}}_K] e_{\varphi _0^{}}$. Recall that this applies for $p \le 3$ since ${\mathscr {C}}_0 = \mathbb {Q}$ and $k \ne \mathbb {Q}$.

The computations leads in this case to the relation:

$$\begin{aligned} X_{\varphi _0^{}} = \frac{\big ({\mathscr {E}}_K: \textbf{N}({\mathscr {E}}_L)\big )_{\varphi _0^{}}}{\big ({\mathscr {E}}_{\widehat{K}}: \textbf{N}({\mathscr {E}}_{\widehat{L}})\big )_{\varphi _0^{}}} \times Z_{\varphi _0^{}}, \end{aligned}$$

(B5)

where $Z_{\varphi _0^{}} := \displaystyle \frac{\big ({\mathscr {E}}_{\widehat{K}} \textbf{N}({\mathscr {E}}_L) : \textbf{N}({\mathscr {E}}_{\widehat{L}}) {\mathscr {F}}_{\!K})_{\varphi _0^{}}}{({\mathscr {E}}_{\widehat{K}} \textbf{N}({\mathscr {E}}_L) : \textbf{N}({\mathscr {E}}_L)\big )_{\varphi _0^{}}} \, \big (\textbf{N}({\mathscr {E}}_{\widehat{L}}) \widehat{\mathscr {F}}_{\!K} : \textbf{N}({\mathscr {E}}_{\widehat{L}})\big )_{\varphi _0^{}}$ and $\widehat{\mathscr {F}}_{\!K}:= {\mathscr {F}}_{\!K} \cap {\mathscr {E}}_{\widehat{K}}$.

Since $\textbf{N}({\mathscr {E}}_{\widehat{L}}) {\mathscr {F}}_{\!K} \subseteq \textbf{N}({\mathscr {E}}_L)$, $Z_{\varphi _0^{}}$ is an integer and we obtain from the relations (B4) and (B5) under capitulations ${\displaystyle X_{\varphi _0^{}} = \#{\mathscr {H}}^\textrm{ar}_{K,\varphi } \times \frac{\#{\mathscr {H}}_{\widehat{L},{\varphi _0^{}}}^\textrm{ram}}{\#\widehat{\textbf{J}}\big ({\mathscr {H}}_{\widehat{L},{\varphi _0^{}}}^\textrm{ram}\big )} \times Z_{\varphi _0^{}} \ge \#{\mathscr {H}}^\textrm{ar}_{K,\varphi }}$; in other words, one gets the supplementary factor ${\#\textrm{Ker}(\widehat{\textbf{J}})_{\varphi _0^{}}}$ which was trivial in the case $\ell$ totally inert in K and $\varphi _0^{} \ne 1$. Whence the equality as usual implying the supplementary interesting relations:

$$\begin{aligned} {\mathscr {F}}_{\!K,\varphi _0^{}} \cap {\mathscr {E}}_{\widehat{K},\varphi _0^{}} \subseteq \textbf{N}({\mathscr {E}}_{\widehat{L},\varphi _0^{}}), \ \ \textbf{N}({\mathscr {E}}_{\widehat{L},\varphi _0^{}}) {\mathscr {F}}_{\!K,\varphi _0^{}} = \textbf{N}({\mathscr {E}}_{L,\varphi _0^{}}), \ \ \widehat{\textbf{J}}\big ({\mathscr {H}}_{\widehat{L},\varphi _0^{}}^\textrm{ram}\big ) \simeq {\mathscr {H}}_{\widehat{L},\varphi _0^{}}^\textrm{ram}. \end{aligned}$$

1.3.3 The special case ${\mathscr {C}}_0 = k$

So, $\tau _\ell ^{}$ is of p-power order, and $(1-\tau _\ell ^{-1}) e_{\varphi _0^{}}$ is not invertible in $\mathbb {Z}_p[{\mathscr {G}}_K] e_{\varphi _0^{}}$.

Under capitulations, we still have the fundamental relation (B4) for which we have to prove the inequality $\displaystyle \big ({\mathscr {E}}_K : {\mathscr {E}}_{\widehat{K}} {\mathscr {F}}_{\!K}\big )_{\varphi _0^{}} \ge \#{\mathscr {H}}^\textrm{ar}_{K,\varphi } = \frac{\#\widehat{\textbf{J}}\big ({\mathscr {H}}_{\widehat{L},{\varphi _0^{}}}^\textrm{ram}\big )}{\#{\mathscr {H}}_{\widehat{L},{\varphi _0^{}}}^\textrm{ram}} \times \frac{\big ({\mathscr {E}}_K : \textbf{N}({\mathscr {E}}_L)\big )_{\varphi _0^{}}}{\big ({\mathscr {E}}_{\widehat{K}} : \textbf{N}({\mathscr {E}}_{\widehat{L}})\big )_{\varphi _0^{}}}$,

whence to prove, since the first factor is ${\big (\#\textrm{Ker}(\widehat{\textbf{J}})_{\varphi _0^{}}\big )^{-1}}$, that:

$$\begin{aligned} {\#\textrm{Ker}(\widehat{\textbf{J}})_{\varphi _0^{}} \times \frac{\big ( {\mathscr {E}}_{\widehat{K}}: \textbf{N}({\mathscr {E}}_{\widehat{L}})\big )_{\varphi _0^{}} \big ({\mathscr {E}}_K: {\mathscr {E}}_{\widehat{K}} {\mathscr {F}}_{\!K}\big )_{\varphi _0^{}}}{\big ({\mathscr {E}}_K: \textbf{N}({\mathscr {E}}_L)\big )_{\varphi _0^{}}} \ge 1.} \end{aligned}$$

(B6)

We have the exact sequence and inclusions $\textbf{N}({\mathscr {E}}_{\widehat{L}}) \subseteq {\mathscr {E}}_{\widehat{K}}\cap \textbf{N}({\mathscr {E}}_L) \subseteq {\mathscr {E}}_{\widehat{K}}$, giving:

$$\begin{aligned} \begin{aligned} \frac{\big ( {\mathscr {E}}_{\widehat{K}}: \textbf{N}({\mathscr {E}}_{\widehat{L}})\big )_{\varphi _0^{}} \big ({\mathscr {E}}_K: {\mathscr {E}}_{\widehat{K}} {\mathscr {F}}_{\!K}\big )_{\varphi _0^{}}}{\big ({\mathscr {E}}_K: \textbf{N}({\mathscr {E}}_L)\big )_{\varphi _0^{}}}&= \frac{\big ({\mathscr {E}}_{\widehat{K}}: \textbf{N}({\mathscr {E}}_{\widehat{L}})\big )_{\varphi _0^{}} \big ({\mathscr {E}}_K: {\mathscr {E}}_{\widehat{K}} {\mathscr {F}}_{\!K}\big )_{\varphi _0^{}}}{\big ({\mathscr {E}}_{\widehat{K}}: {\mathscr {E}}_{\widehat{K}} \cap \textbf{N}({\mathscr {E}}_L)\big )_{\varphi _0^{}}\, \big ({\mathscr {E}}_K: {\mathscr {E}}_{\widehat{K}} \textbf{N}({\mathscr {E}}_L)\big )_{\varphi _0^{}}} \\&= \frac{\big ({\mathscr {E}}_{\widehat{K}}\cap \textbf{N}({\mathscr {E}}_L): \textbf{N}({\mathscr {E}}_{\widehat{L}})\big )_{\varphi _0^{}} \big ({\mathscr {E}}_K: {\mathscr {E}}_{\widehat{K}} {\mathscr {F}}_{\!K}\big )_{\varphi _0^{}} }{\big ({\mathscr {E}}_K: {\mathscr {E}}_{\widehat{K}} \textbf{N}({\mathscr {E}}_L) \big )_{\varphi _0^{}}}\, \cdot \end{aligned} \end{aligned}$$

We have ${\mathscr {F}}_{\!K}^{1-\tau _\ell ^{-1}} = \textbf{N}({\mathscr {F}}_{\!L}) \subseteq \textbf{N}({\mathscr {E}}_L)$, which is not necessarily true for ${\mathscr {F}}_{\!K}$ and prevents to conclude, contrary to the case $\ell$ totally inert.

If this case of FRAMC is true, because of the semi-simple formula of Theorem (2.4), over $\varphi \mid \chi$, one should obtain, under capitulation of ${\mathscr {H}}_K$ in L, from (B6):

$$\begin{aligned} \#\textrm{Ker}(\widehat{\textbf{J}})_{\varphi _0^{}} \! \times \! \frac{\big ({\mathscr {E}}_{\widehat{K}}\cap \textbf{N}({\mathscr {E}}_L): \textbf{N}({\mathscr {E}}_{\widehat{L}})\big )_{\varphi _0^{}} \big ({\mathscr {E}}_K: {\mathscr {E}}_{\widehat{K}} {\mathscr {F}}_{\!K}\big )_{\varphi _0^{}} }{\big ({\mathscr {E}}_K: {\mathscr {E}}_{\widehat{K}} \textbf{N}({\mathscr {E}}_L) \big )_{\varphi _0^{}}} = 1. \end{aligned}$$

(B7)

The relations obtained for the case ${\mathscr {C}}_0 \varsubsetneqq k$ (Sect. B.3.2), $\textbf{N}\big ({\mathscr {E}}_{\widehat{L},\varphi _0^{}}\big ) {\mathscr {F}}_{\!K,\varphi _0^{}} = \textbf{N}\big ({\mathscr {E}}_{L,\varphi _0^{}}\big )$, $\textrm{Ker}(\widehat{\textbf{J}})_{\varphi _0^{}} \!= 1$, with the supplementary one ${\mathscr {E}}_{\widehat{K},{\varphi _0^{}}} \! \cap \textbf{N}\big ({\mathscr {E}}_{L,{\varphi _0^{}}}\big )\! = \textbf{N}\big ({\mathscr {E}}_{\widehat{L},{\varphi _0^{}}}\big )$, satisfies the relation (B7), but this is only speculation.