\usepackage{amsfonts,amsthm,amsmath,amssymb,mathtools}
\usepackage[all,cmtip]{xy}
\usepackage[english]{babel}
\usepackage[utf8]{inputenc}
\usepackage[OT1]{fontenc}
\usepackage{mathrsfs}
\usepackage{xpunctuate}
\usepackage[dvipsnames]{xcolor}
\usepackage{enumitem}
\usepackage[colorlinks=true,linkcolor=Emerald,citecolor=Emerald, backref]{hyperref}  % for hyper-referencing 
\usepackage{geometry}
\numberwithin{equation}{section}
\mathtoolsset{showonlyrefs=true}

\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
\newtheorem*{theorem*}{Theorem}
\newtheorem*{conjecture}{Conjecture}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem*{defthm*}{Definition/Theorem}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem*{lemma*}{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\makeatletter
\newtheorem*{intr@thm}{\intr@thmname}
\newenvironment{introtheorem}[1]{%
	\def\intr@thmname{Theorem #1}
	\begin{intr@thm}}
	{\end{intr@thm}}
\newtheorem*{intr@crl}{\intr@crlname}
\newenvironment{introcorollary}[1]{%
	\def\intr@crlname{Corollary #1}
	\begin{intr@crl}}
	{\end{intr@crl}}
\newtheorem*{intrd@f}{\intrd@fname}
\newenvironment{introdefinition}[1]{%
	\def\intrd@fname{Definition #1}
	\begin{intrd@f}}
	{\end{intrd@f}}
\makeatother


\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{example}[theorem]{Example}
\newtheorem*{acknowledgment}{Acknowledgment}


\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem*{definition*}{Definition}
\newtheorem{assumption}{Assumption}
\newtheorem{notation}[theorem]{Notation}

%%%%%%%%%%%%%%Mathematical symbols
\newcommand{\expp}[1]{p^{#1}}
\newcommand*{\Gal}[2]{\mathrm{Gal}(#1/#2)} % Galois group
\newcommand*{\splitG}[1][0]{D_{#1}} %The extension of \Gamma by \splitG_0
\newcommand*{\filindex}[3][,]{h_{#2#1#3}}
\newcommand*{\subsplit}[1][0]{\Sigma_{#1}} %The subgroup of \splitG isomorphic to \splitG_0
\newcommand*{\catMod}[2][{[\splitG]}]{\mathbf{Mod}^{\mathrm{#2}}_{\ZZ_p#1}} %The category of #1=Z_p[G]-modules, with #2-property.
\newcommand*{\cofg}{co\textrm{-}f.g.} %Property of being co-finitely generated torsion
\newcommand*{\ZZ}{\mathbb{Z}} % rational integers
\newcommand*{\NN}{\mathbb{N}} % positive integers
\newcommand*{\QQ}{\mathbb{Q}} % rational numbers
\DeclareMathOperator{\pr}{pr} %A projection
\makeatletter %%The definition of \cev, a vector going from right to left
\DeclareRobustCommand{\cev}[1]{%
  {\mathpalette\do@cev{#1}}%
}
\newcommand{\do@cev}[2]{%
  \vbox{\offinterlineskip
    \sbox\z@{$\m@th#1 x$}%
    \ialign{##\cr
      \hidewidth\reflectbox{$\m@th#1\vec{}\mkern4mu$}\hidewidth\cr
      \noalign{\kern-\ht\z@}
      $\m@th#1#2$\cr
    }%
  }%
}
\makeatother

%%%%%%%%%%Typical number theoretical objects
\newcommand{\field}[1]{L_{#1}} %the fields in the Z_p-extensions
\newcommand{\rint}[1]{\mathcal{O}_{#1}} % ring of integers
\newcommand{\unit}[1]{\rint{#1}^\times} % units
\newcommand*{\realemb}[1]{r_1(#1)} %number of real embeddings
\newcommand*{\cpxemb}[1]{r_2(#1)} %number of conjugate pairs of complex embeddings
\newcommand{\clgp}[1]{Cl_{#1}} % class group
\newcommand*{\clsyl}[1]{A_{#1}} % p-Sylow of the class group
\newcommand*{\hsyl}[1]{{\varepsilon_{#1}}} % order of the above p-Sylow of the class group
\newcommand*{\algnm}[1]{\mathrm{Nm}_{#1}}% Algebraic norm for the group #1
\newcommand*{\arnm}[2]{\mathrm{N}_{#1/#2}}% Arithmetic norm in the extension #1/#2 (old \nu)
\newcommand*{\arextsymb}{\iota} %The symbol for arrow extension
\newcommand*{\arext}[2]{\arextsymb_{#1/#2}}% Extension of ide(a)l(e)s induced by inclusion of subfields in the extension #1/#2
\newcommand*{\princ}[1]{\mathrm{Pr}_{#1}} % principal ideals
\newcommand*{\ideal}[1]{\mathrm{Id}_{#1}} % ideals
\newcommand*{\aboveT}[1]{\Pi_{#1}} % Classes generated by ramified primes
\newcommand*{\Ram}[2]{\operatorname{T}(#1/#2)} %The primes in #2 which ramify in #1
%\newcommand*{\RamC}[2]{\mathrm{T}'(#1/#2)} %Conjugacy classes of primes in #2 which ramify in #1
\newcommand*{\oram}[2]{\tau(#1/#2)} %The order of the set Ram
\newcommand*{\idele}[1]{\mathbb{A}^\times_{#1}} % ideles
\newcommand*{\idunit}[1]{\mathcal{U}_{#1}} % unit ideles
\newcommand*{\idclg}[1]{\mathcal{C}_{#1}} % ideles class group
\newcommand*{\q}[1]{Q_{#1}} % Schoof's Q
\def\primen@me{p}
\def\conjprimen@me{q}
\def\oth@rprime{l}
\newcommand*{\primeid}{\mathfrak{\primen@me}}
\newcommand*{\conjprimeid}{\mathfrak{\conjprimen@me}}
\newcommand*{\otherprime}{\mathfrak{\oth@rprime}}
\newcommand*{\Primeid}[1]{\mathfrak{\MakeUppercase{\primen@me}}_{#1}}
\newcommand*{\Conjprimeid}[1]{\mathfrak{\MakeUppercase{\conjprimen@me}}_{#1}}
\newcommand*{\Otherprime}[1][\empty]{\mathfrak{\MakeUppercase{\oth@rprime}}_{#1}}
\newcommand*{\locunits}[2][\empty]{\mathbb{E}_{#2}^{#1}} %Local units
\newcommand*{\semiloc}[2][\empty]{\mathcal{V}^{#1}_{#2}} %The semi-local units
\newcommand*{\ramdeg}[2]{e_{#1/#2}} % The ramification index for the _local_ extension (so, no prime needs to be specified) #1/#2
\newcommand*{\indeg}[3][/]{f_{#2#1#3}} % The inertia degree for the _local_ extension (so, no prime needs to be specified) #1/#2
\newcommand*{\logindeg}[1][\empty]{\ell_\infty^{#1}} % The logarithm of the inertia degree for the _local_ Z_p-extension
\newcommand*{\locdeg}[2]{d_{#1/#2}} %The local degree of the local extension #1/#2
\newcommand*{\nprimes}[1][\primeid]{g_{#1}} %Number of primes in L_cyc above #1=\mathfrak{p}
\newcommand*{\fundclass}[2]{u_{#1/#2}} % fundamental class of the extension #1/#2
\newcommand*{\loclev}[1]{E_{#1}} % light notation for the level #1 of the local Z_p-extension
\newcommand*{\invJP}[1][\loclev{\empty}]{\operatorname{inv}_{#1}} %JPS's homomorphism inv

%%%%%%%%%%%%Cohomology%%%%%%%%%%%%%%%
\newcommand{\cch}[1]{\delta^{(#1)}} % cohomological connecting homomorphism H^{#1}(G,C) -> H^{#1 + 1}(G,A) 
\newcommand{\cchT}[1]{\widehat{\delta}^{(#1)}} %The same thing, but leaving from Tate cohomology groups
\newcommand*{\tateiso}[3][\empty]{(\cup\varkappa_{#2}^{#1})^{#3}} % cup product with \chi (#3-#2)/2 times
\DeclarePairedDelimiterXPP{\hh}[3]{\widehat{h}^{#1}}{(}{)}{}{#2,#3} %h^i as order of H^i (both Tate)
\DeclarePairedDelimiterXPP{\hhN}[3]{h^{#1}}{(}{)}{}{#2,#3} %h^i as order of H^i (not Tate)
\DeclarePairedDelimiterXPP{\HH}[3]{\widehat{H}^{#1}}{(}{)}{}{#2,#3} %H^i (Tate)
\DeclarePairedDelimiterXPP{\HHm}[2]{\widehat{H}^{#1}}{(}{)}{}{#2} %H^i (Tate) for a map
\DeclarePairedDelimiterXPP{\HHN}[3]{H^{#1}}{(}{)}{}{#2,#3} %H^i (not Tate) for a module
\DeclareMathOperator{\res}{Res} %res=restriction
\DeclareMathOperator{\defl}{Defl} %Defletion map
\DeclareMathOperator{\infl}{Inf} % Inflation maps
\renewcommand{\inf}{__} % to get an error with AMS definition of inf
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%% Normic and double systems %%%%%%%%%%%
\newcommand*{\pallino}[1][{\field}]{{#1{\bullet}}} %The "bullet" as index for systems. It defaults to L_{\bullet}, but can be modified to be \mathfrak{P}_{\bullet} (setting [1]=[\Primeid]) or p^\bullet (setting [1]=[\expp])
\newcommand*{\nsu}{j} %The "going up" morphism in normic systems
\newcommand*{\nsd}{k} %The "going up" morphism in normic systems
\newcommand*{\normsys}{\mathbf{NS}_{\DD}} %The category of normic systems
\newcommand*{\hatnsu}[2][\jmath]{\widehat{#1}_{#2}} % The "going up" morphism in 0th Tate cohomology of normic systems with surjective going down
\makeatletter
\DeclarePairedDelimiterXPP{\s@sHH}[2]{\p@rH^{#1}}{(}{)}{}{#2} %The double system of H^i (Tate)
\newcommand{\sysHH}[3][\empty]{%
\def\p@rH{\widehat{\calH}_{#1}}
\s@sHH{#2}{#3}%
%\relax
}
\makeatother

\newcommand*{\sysnorm}[1]{\overline{\mathcal{N}#1}} %The double systems of the norms down to F, modulo p^n-th powers
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%Other arithmetic operators %%%%%%%%%%%%%%%%
%%The command \roots: called without option, as in \roots{M}, gives \mu_M; with an option, as in \roots[3]{M} gives \mu_M(3)
\makeatletter
\def\r@otsl#1#2#3#4{\mu_{#4}(#2)}
\def\r@ots#1{\mu_{#1}}
\def\roots{\futurelet\radici\r@otschose}
\def\r@otschose{%
	\ifx\radici[
		\expandafter\r@otsl
	\else
		\expandafter\r@ots
\fi}
\makeatother %End of def. for roots
\newcommand{\defect}[2]{\beta_{#1}} % "q-defect" of K, defined as \mu_K/\mu_K^q.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%Iwasawa-theoretic objects %%%%%%%%%%%%%%%%
\newcommand*{\iwLambda}[1][]{\Lambda#1} %The Iwasawa algebra
\newcommand*{\topgen}{\gamma_0} %A topological generator of Gamma
\newcommand*{\iwmu}[2][]{\mu_{#2}^{#1}} % Iwasawa mu
\newcommand*{\iwlambda}[2][]{\lambda_{#2}^{#1}} % Iwasawa lambda
\newcommand*{\lambdacohom}[2][\unit{\pallino}]{\lambda(#2,#1)}
\newcommand*{\iwnu}[2][]{\nu_{#2#1}} % Iwasawa nu 
\makeatletter
\def\th@X{X} % Iwasawa X
\newcommand*{\iwXL}{\th@X_{\iw}} % Iwasawa XL
\newcommand*{\iwXK}{\th@X_{\fk}} % Iwasawa XK
\makeatother
\newcommand*{\indextr}[1][\fromkprimeid]{{n_\mathrm{tr}^{#1}}} %The index starting from which the Zp_extension is totally ramified; either locally or globally
\newcommand*{\Lcyc}{L_\mathrm{cyc}} %The cyclotomic Z_p extension of F
\newcommand*{\Ldue}{\widetilde{L}} %The compositum of Lcyc and Linfty
\newcommand*{\DD}{\mathcal{D}} %The Galois group L_\infty/k
\newcommand*{\GG}{\mathcal{G}} %The Galois group L_\infty/Q
\newcommand*{\Minfty}{M_\infty} %The maximal unramified ext's of the anticyclotomic of F=im.quadratic
\newcommand*{\Frob}[2]{\operatorname{Frob}(#1,#2)} %The Frobenius element at the prime #1 in the extension #2
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%Paper specific%%%%%%%%%
\newcommand*{\nuceta}[1]{\eta_{#1}} %Nuccio's eta
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%Algebraic objects %%%%%%%%%%%%%%%%%
\DeclareMathOperator{\rank}{rk} %rank
\DeclareMathOperator{\corank}{cork} %corank
\DeclarePairedDelimiterX\gindex[2]{(}{)}{#1:#2} % index of a subgroup
\DeclarePairedDelimiter\gorder{\lvert}{\rvert} % order of a group
\renewcommand{\ker}{__} % to get an error with AMS definition of kernel
\newcommand*{\kernel}[1]{\operatorname{Ker} #1} % kernel
\newcommand*{\cokernel}[1]{\operatorname{Coker} #1} % cokernel
\newcommand*{\im}[1]{\operatorname{Im} #1}%image
\newcommand*{\tors}[2][p]{{{#2}[#1]}}%q-torsion
\renewcommand*{\div}[1]{\bigl(#1\bigr)_{\mathrm{div}}} %maximal divisible subgroup
\newcommand*{\id}[1]{\mathrm{id}_{#1}} %identity isomorphism
\DeclareMathOperator{\Hom}{Hom}
\newcommand*{\sch}{\delta} % snake connecting homomorphism
\makeatletter
\newcommand{\Ind}[2]{%
\def\gr@up{#1}%
\def\subgr@up{#2}%
\operatorname{Ind}_{\gr@up}^{\subgr@up} 
					} %Induction from a subgroup
\makeatother

%%%%%%%%Categories%%%%%%%%%%%%%%%
\makeatletter
\newcommand*{\dblsys}[2][\empty]{\mathbf{DS}^{\mathrm{#1}}_{#2}}%_{\ZZ_p}} %Double systems
\newcommand*{\dsu}{\varphi} %The going up morphism for double systems
\newcommand*{\dsd}{\pi}  %The going down morphism for double systems
\newcommand*{\capzero}[2][\scriptscriptstyle\vartriangle]{{^{#1}{#2}}} % Caputo's kernel
\newcommand*{\capbar}[2][\scriptscriptstyle\blacktriangle]{{_{#1}{#2}}} %Caputo's quotient
\newcommand*{\nuczero}[2][\dagger]{{^{#1}{#2}}} %Nuccio's (other) kernel
\newcommand*{\nucbar}[2][\ddagger]{{_{#1}{#2}}} %Nuccio's (other) quotient
\newcommand*{\bddsys}[2][\empty]{\mathbf{B}^{\mathrm{#1}}_{#2}} %Bounded systems for generic \splitG
\newcommand*{\quotsys}[2][\empty]{{\dblsys[#1]{#2}}/{\bddsys[#1]{#2}}} %The quotient category (double)/(bounded)
\newcommand*{\natr}{\Phi} %The natural transformation between id and limemb
\newcommand*{\bddcong}[1][\bddsys{\DD}]{\cong_{#1}} %Isomorphism in the quotient category
\newcommand*{\bddeq}{\propto} %equality up to a bounded error
\newcommand*{\longdashrightarrow}{\dashrightarrow} 
\makeatother
\newcommand*{\stepfunctor}{\mathscr{S}}
\newcommand*{\limemb}{\mathscr{L}} %The composition of varinjlim and \embup
\newcommand*{\calH}{\mathcal{H}}
\newcommand*{\calX}{\mathcal{X}}
\newcommand*{\calY}{\mathcal{Y}}
\newcommand*{\calZ}{\mathcal{Z}}
\newcommand*{\calB}{\mathcal{B}}
\newcommand*{\calC}{\mathcal{C}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%Abreviations and other typography%%%%%%%%%%%%%%
\newcommand*{\longhookrightarrow}{\lhook\joinrel\relbar\joinrel\rightarrow} %Long injective arrow
\newcommand*{\longhookdashedarrow}{\lhook\dashrightarrow} %Long injective dashed arrow
\newcommand*{\longtwoheadrightarrow}{\joinrel\relbar\joinrel\twoheadrightarrow}%Long surjective arrow
\newcommand*{\longtwoheaddashedarrow}{\dashrightarrow\mathrel{\mkern-19mu}\dashrightarrow}%Long surjective dashed arrow
\newcommand{\ie}{\textit{i.~e.~}} % id est
\newcommand*{\rsp}{{resp.~}} % respectively: not \resp to search it in the file
\newcommand{\loccit}{\emph{loc.\kern3pt cit}\xperiod} % loco citato
\newcommand{\ibid}{\emph{ibid}\xperiod} % ibidem
\newcommand{\fk}{\mathrm{fake}} %Roman "fake"
\newcommand{\cyc}{\mathrm{cyc}} %Roman "cyc"
\newcommand{\iw}{\mathrm{Iw}} %For the "classical" Iwasawa invariants
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%