The universal ordinary deformation ring associated to a real quadratic field

Abstract

We study ring structure of the big ordinary Hecke algebra ${{\mathbb {T}}}$ with the modular deformation $\rho _{{\mathbb {T}}}:{\mathrm {Gal}}({\bar{{{\mathbb {Q}}}}}/{{\mathbb {Q}}})\rightarrow {\mathrm {GL}}_2({{\mathbb {T}}})$ of an induced Artin representation ${\text {Ind}}_F^{{\mathbb {Q}}}\varphi$ from a real quadratic field F with a fundamental unit $\varepsilon$, varying a prime $p\ge 3$ split in F. Under mild assumptions (H0)–(H3) given in the Introduction (on the prime p), we prove that ${{\mathbb {T}}}$ is an integral domain free of even rank $e>0$ over $\Lambda$ for the weight Iwasawa algebra $\Lambda$ étale outside ${\mathrm {Spec}}(\Lambda /p(\langle \varepsilon \rangle -1))$ for $\langle \varepsilon \rangle {:}{=}(1+T)^{\log _p(\varepsilon )/\log _p(1+p)}\in {{\mathbb {Z}}}_p[[T]]\subset \Lambda$. If $p\not \mid e$, ${{\mathbb {T}}}$ is shown to be a normal noetherian domain of dimension 2 with ramification locus exactly given by $(\langle \varepsilon \rangle -1)$. Moreover, only under p-distinguishedness (H0), we prove that any modular specialization of weight $\ge 2$ of $\rho _{{\mathbb {T}}}$ is indecomposable over the inertia group at p (solving a conjecture of Greenberg without exception).

Appendices

Appendices

Appendix A. Control of Selmer group

We used in the main text a precise control result and determination of the size (i.e., the characteristic ideal and Fitting ideal) of the adjoint Selmer group, though perhaps it is well known to specialists. We present a detailed exposition of the control in this first appendix. The result is valid for any odd absolutely irreducible p-ordinary p-distinguished representation ${{\bar{\rho }}}:G\rightarrow {\mathrm {GL}}_2({{\mathbb {F}}})$ not necessarily induced from F. We only assume that the ramification index of ${{\bar{\rho }}}$ at primes outside p is prime to p and write G for the Galois group over ${{\mathbb {Q}}}$ of the maximal p-profinite extension unramified outside p of the splitting field of ${{\bar{\rho }}}$. Write S for the set of primes $\ne p$ ramified in $F({{\bar{\rho }}})/{{\mathbb {Q}}}$ such that ${{\bar{\rho }}}|_{I_l}=\bar{\epsilon }_l\oplus \bar{\delta }_l$. By (l) in Section 2, if $l\ne p$ outside S ramifies in $F({{\bar{\rho }}})/{{\mathbb {Q}}}$, ${{\bar{\rho }}}|_{D_l}$ is irreducible.

Let ${\varvec{\kappa }}{:}{=}\det ({\varvec{\rho }}):G\rightarrow \Lambda ^\times$. Then $(\Lambda ,{\varvec{\kappa }})$ represents the deformation functor

$$\begin{aligned} A\mapsto \{\xi :G\rightarrow A^\times |\xi \ {\hbox {mod }}{{\mathfrak {m}}}_A=\det ({{\bar{\rho }}})\}. \end{aligned}$$

Consider the following deformation functor $D_{\varvec{\kappa }}:CL_{/\Lambda }\rightarrow SETS$ slightly different from the one ${{\mathcal {D}}}$ defined in (s6) (or (7.1)):

$$\begin{aligned} D_{\varvec{\kappa }}(A)=\{\rho \in {{\mathcal {D}}}(A)|\det (\rho )=i_A\circ {\varvec{\kappa }}\}/\Gamma ({{\mathfrak {m}}}_A), \end{aligned}$$

where writing $i_A:\Lambda \rightarrow A$ for $\Lambda$-algebra structure of A. This functor is again represented by $(R=R_{{\mathbb {Q}}},{\varvec{\rho }})$ regarding R as a $\Lambda$-algebra by the W-algebra homomorphism induced by $\det ({\varvec{\rho }}):G\rightarrow R^\times$. Indeed, if $\rho \in D_{\varvec{\kappa }}(A)$, we have $i_A\circ {\varvec{\kappa }}=\det (\rho )$. Regarding $\rho \in {{\mathcal {D}}}(A)$, we have a unique W-algebra homomorphism $R\xrightarrow {\phi }A$ such that $\phi \circ {\varvec{\rho }}\approx \rho$, where “$\approx$” is conjugation by $\Gamma ({{\mathfrak {m}}}_A)$. Taking determinant, we get $\phi \circ {\varvec{\kappa }}=\det (\rho )$ showing that $\phi$ is compatible with $i_R$ and $i_A$; so, it is a $\Lambda$-algebra homomorphism, showing ${\mathrm {Hom}}_\Lambda ( R,A)\cong D_{\varvec{\kappa }}(A)$ by $\phi \leftrightarrow \rho$.

By the condition (l) for $l\in S$, the universal representation ${\varvec{\rho }}$ is equipped with a basis $({{\mathbf {v}}}_l,{{\mathbf {w}}}_l)$ so that ${\varvec{\rho }}(g){{\mathbf {v}}}_l=\epsilon _l(g){{\mathbf {v}}}_l$ and ${\varvec{\rho }}(g){{\mathbf {w}}}_l=\delta _l(g){{\mathbf {w}}}_l$ for $g\in I_l$ for the Teichimüller lift $\epsilon _l$ and $\delta _l$ of $\bar{\epsilon }_l$ and $\bar{\delta }_l$, respectively. At p, for $g\in D_p$, the matrix form of ${\varvec{\rho }}|_{D_p}$ for this basis is $\left( {\begin{matrix}{\varvec{\epsilon }}&{}*\\ 0&{}{\varvec{\delta }}\end{matrix}}\right)$ with ${\varvec{\delta }}$ unramified. By representability, each class $c\in D_{\varvec{\kappa }}(A)$ has $\rho$ such that $V(\rho )=V({\varvec{\rho }})\otimes _{R,\iota }A$ for a unique $\iota \in {\mathrm {Hom}}_\text {B-alg}( R,A)$, we can choose a unique $\rho \in c$ is equipped with a basis

$$\begin{aligned} \{(v_l={{\mathbf {v}}}_l\otimes 1,w_l={{\mathbf {w}}}_l\otimes 1)\}_{l\in S\cup \{p\}} \end{aligned}$$

compatible with specialization. We always choose such a specific representative $\rho$ for each class in $D_{\varvec{\kappa }}(A)$ hereafter.

Fix $\rho _0\in D_{\varvec{\kappa }}(A)$. We study ${\mathrm {Sel}}(Ad(\rho _0)$. Take an A-module M with finite order and consider the ring $A[M]=A\oplus M$ with $M^2=0$. Then A[M] is still p-profinite. Pick $\rho \in D_{\varvec{\kappa }}(A[M])$ such that $\rho \mod M=\rho _0$. By our choice of representative $\rho$ and $\rho _0$ as above, we may (and do) assume $\rho \mod M=\rho _0$.

Let $\rho _0$ act on $M_2(A)$ and ${{\mathfrak {s}}}{{\mathfrak {l}}}_2(A)=\{x\in M_2(A)|{\mathrm {Tr}}(x)=0\}$ by conjugation. Write this representation $ad(\rho )$ and $Ad(\rho )$ as before. Let $ad(M)=ad(A)\otimes _AM$ and $Ad(M)=Ad(A)\otimes _AM$ and regard them as G-modules by the action on ad(A) and Ad(A). Then we define

$$\begin{aligned} {{\mathcal {F}}}(A[M])=\frac{\{\rho :G\rightarrow {\mathrm {GL}}_2(A[M])|(\rho \ {\hbox {mod }}M)=\rho _0, [\rho ]\in D_{\varvec{\kappa }}(A[M])\}}{\Gamma (M)}, \end{aligned}$$

(A.1)

where $[\rho ]$ is the isomorphism class in $D_{\varvec{\kappa }}(A)$ containing $\rho$ and $\Gamma (M){:}{=}{\text {Ker}}({\mathrm {GL}}_2(A[M])\rightarrow {\mathrm {GL}}_2(A))$ acts on $\rho$ by conjugation.

Take M finite as above. For $\rho \in {{\mathcal {F}}}(M)$, we can write $\rho =\rho _0\oplus u'_\rho$ letting $\rho _0$ acts on $M_2(M)$ by matrix multiplication from the right. Then as before

$$\begin{aligned} \rho _0(gh)\oplus u'_\rho (gh)= & {} (\rho _0(g)\oplus u'_\rho (g))(\rho _0(h)\oplus u'_\rho (h))\\= & {} \rho _0(gh)\oplus (u'_\rho (g)\rho _0(h)+ \rho _0(g)u'_\rho (h)) \end{aligned}$$

produces $u'_\rho (gh)=u'_\rho (g)\rho _0(h)+ \rho _0(g)u'_\rho (h)$ and multiplying by $\rho _0(gh)^{-1}$ from the right, we get the cocycle relation for $u_\rho (g)=u'_\rho (g)\rho _0(g)^{-1}$:

$$\begin{aligned} u_\rho (gh)=u_\rho (g)+gu_\rho (h)\ \text { for } gu_\rho (h)=\rho (g)u_\rho (h)\rho _0(g)^{-1}, \end{aligned}$$

getting the map ${{\mathcal {F}}}(A[M])\rightarrow H^1(G,ad(M))$ which factors through $H^1(G,Ad(M))$. By computation, we can easily verify that $\rho$ is $(1+M_2(M))$-conjugate to $\rho '$ if and only if $u_\rho$ is cohomologous to $u_{\rho '}$; so, this map is injective A-linear map identifying ${{\mathcal {F}}}(A[M])$ with

$$\begin{aligned} {\mathrm {Sel}}(Ad(M)){:}{=}{\text {Ker}}(H^1(G,Ad(M))\xrightarrow {{\mathrm {Res}}} \frac{H^1({{\mathbb {Q}}}_p,Ad(M))}{F^+_-H^1({{\mathbb {Q}}}_p,Ad(M))}), \end{aligned}$$

where $F^+_-H^1({{\mathbb {Q}}}_p,Ad(M))\subset H^1({{\mathbb {Q}}}_p,Ad(M))$ is a A-submodule spanned by cohomology classes of cocycles $u:G\rightarrow Ad(M)$ upper triangular over $D_p$ and upper nilpotent over $I_p$.

If $M=\varinjlim _iM_i$ for finite A-modules $M_i$, we just define

$$\begin{aligned} {\mathrm {Sel}}(Ad(M))=\varinjlim _i{\mathrm {Sel}}(Ad(M_i)). \end{aligned}$$

Then for finite $M_i$, ${{\mathcal {F}}}(A[M_i])={\mathrm {Sel}}(Ad(M_i))$ and $\varinjlim _i{{\mathcal {F}}}(M_i)={\mathrm {Sel}}(\varinjlim _iAd(M_i)).$

For each $[\rho _0]\in D_{\varvec{\kappa }}(A)$, choose a representative $\rho _0=\iota \circ {\varvec{\rho }}$. Then we have a map ${{\mathcal {F}}}(A[M])\rightarrow D_{\varvec{\kappa }}(A[M])$ for each finite A-module M sending $\rho \in {{\mathcal {F}}}(A[M])$ to the class $[\rho ]\in D_{\varvec{\kappa }}(A[M])$. By our choice of $\rho$, this map is injective.

Conversely pick a class $c\in D_{\varvec{\kappa }}(A[M])$ over $[\rho _0]\in D_{\varvec{\kappa }}(A)$. Then for $\rho \in c$, we have $x\in 1+M_2({{\mathfrak {m}}}_{A[M]})$ such that $x\rho x^{-1}\mod M=\rho _0$. By replacing $\rho$ by $x\rho x^{-1}$ and choosing the lifted base, we conclude ${{\mathcal {F}}}(A[M])\cong \{[\rho ]\in D_{\varvec{\kappa }}(A[M])|\rho \mod M\sim \rho _0\}$; so, for finite M,

$$\begin{aligned} {\mathrm {Sel}}(Ad(M))= & {} {{\mathcal {F}}}(A[M])= \{\phi \in {\mathrm {Hom}}_\Lambda -\text {alg}( R,A[M]):\phi \ {\hbox {mod }}M=\iota \}\\= & {} Der_\Lambda ( R,M)\cong {\mathrm {Hom}}_A(\Omega _{ R/\Lambda }\otimes _{ R,\iota }A,M). \end{aligned}$$

Thus

$$\begin{aligned} {\mathrm {Sel}}(Ad(M))\cong {\mathrm {Hom}}_A(\Omega _{ R/\Lambda }\otimes _{ R,\iota }A,M). \end{aligned}$$

(A.2)

Theorem A.1

(B. Mazur). For any $A\in CL_\Lambda ,$ we have a canonical isomorphism: ${\mathrm {Sel}}(Ad(\rho _0))^\vee \cong \Omega _{R/\Lambda }\otimes _{ R,\iota }A$. In particular,  if $\rho _0$ is modular with $\iota$ factoring through ${{\mathbb {T}}},$ we have ${\mathrm {Sel}}(Ad(\rho _0))^\vee ={\mathrm {Sel}}(Ad(\rho _{{\mathbb {T}}}))^\vee \otimes _{{\mathbb {T}}}A$.

The last assertion follows from the transitivity of tensor product:

$$\begin{aligned} {\mathrm {Sel}}(Ad(\rho _{{\mathbb {T}}}))^\vee \otimes _{{\mathbb {T}}}A=\Omega _{R/\Lambda }\otimes _R{{\mathbb {T}}}\otimes _{{\mathbb {T}}}A=\Omega _{R/\Lambda }\otimes _RA={\mathrm {Sel}}(Ad(\rho _0))^\vee . \end{aligned}$$

We only prove the first assertion.

Proof

Take the Pontryagin dual $A^\vee {:}{=}{\mathrm {Hom}}_B(A,\Lambda ^\vee )={\mathrm {Hom}}_{{{\mathbb {Z}}}_p}(A\otimes _\Lambda \Lambda ,{{\mathbb {Q}}}_p/{{\mathbb {Z}}}_p)={\mathrm {Hom}}(A,{{\mathbb {Q}}}_p/{{\mathbb {Z}}}_p).$ Since $A=\varprojlim _iA_i$ for finite rings $A_i$ and ${{\mathbb {Q}}}_p/{{\mathbb {Z}}}_p=\varinjlim _jp^{-1}{{\mathbb {Z}}}/{{\mathbb {Z}}}$, $A^\vee =\varinjlim _i{\mathrm {Hom}}(A_i,{{\mathbb {Q}}}_p/{{\mathbb {Z}}}_p)=\varinjlim _iA_i^\vee$ is a union of the finite modules $A_i^\vee$. We define ${\mathrm {Sel}}(Ad(\rho _0)){:}{=}\varinjlim _j{\mathrm {Sel}}(Ad(A_i^\vee ))$. Defining ${{\mathcal {F}}}(A[A^\vee ])=\varinjlim _i{{\mathcal {F}}}_l(A[A_i^\vee ])$, we see from compatibility of cohomology with injective limit

$$\begin{aligned} {\mathrm {Sel}}(Ad(\rho _0))= & {} \varinjlim _i{\mathrm {Sel}}(Ad(A_i^\vee ))\\= & {} \varinjlim _j{\text {Ker}}\left( H^1(G,Ad(A_i^\vee ))\rightarrow \frac{H^1({{\mathbb {Q}}}_p,Ad(A_i^\vee ))}{F^+_-H^1({{\mathbb {Q}}}_p,Ad(A_i^\vee ))}\right) \end{aligned}$$

By the formula (A.2),

$$\begin{aligned} {\mathrm {Sel}}(Ad(\rho _0))= & {} \varinjlim _i{\mathrm {Sel}}(Ad(A_i^\vee ))=\varinjlim _i{\mathrm {Hom}}_{ R}(\Omega _{ R/\Lambda }\otimes _{ R}A,A_i^\vee )\\= & {} {\mathrm {Hom}}_A(\Omega _{ R/\Lambda }\otimes _{ R}A,A^\vee )={\mathrm {Hom}}_A(\Omega _{ R/\Lambda }\otimes _{ R}A,{\mathrm {Hom}}_{{{\mathbb {Z}}}_p}(A,{{\mathbb {Z}}}_p))\\= & {} {\mathrm {Hom}}_{{{\mathbb {Z}}}_p}(\Omega _{ R/\Lambda }\otimes _{ R}A,{{\mathbb {Q}}}_p/{{\mathbb {Z}}}_p)=(\Omega _{ R/\Lambda }\otimes _{ R}A)^\vee . \end{aligned}$$

Taking Pontryagin dual back, we finally get

$$\begin{aligned} {\mathrm {Sel}}(Ad(\rho _0))^\vee \cong \Omega _{ R/\Lambda }\otimes _{ R,\iota }A\text { and }{\mathrm {Sel}}(Ad({{\bar{\rho }}}))^\vee \cong \Omega _{ R/\Lambda }\otimes _{ R}{{\mathbb {F}}}\end{aligned}$$

as desired. In particular, we have ${\mathrm {Sel}}(Ad({\varvec{\rho }}))^\vee =\Omega _{ R/\Lambda }$. $\square$

Appendix B. p-Adic adjoint L-function

In [24, §6.5.5], we constructed a p-adic L-function interpolating the size of the adjoint Selmer group ${\mathrm {Sel}}(Ad(\rho ))$ for each specialization $\rho _P=\rho _{{\mathbb {T}}}\ {\hbox {mod }}P$ with $P\in {\mathrm {Spec}}(\Lambda )$ for each irreducible component of the form ${\mathrm {Spec}}(\Lambda )$ of ${\mathrm {Spec}}({{\mathbb {T}}})$. Here we generalize this construction of the L-function $L^\mathrm{mod}=L^\mathrm{mod}_{{\mathbb {I}}}\in {{\mathbb {I}}}$ to general irreducible components ${\mathrm {Spec}}({{\mathbb {I}}})$ of ${\mathrm {Spec}}({{\mathbb {T}}})$ and glue them together to obtain $L_{{\mathbb {T}}}\in {{\mathbb {T}}}$ having the interpolation property all over ${\mathrm {Spec}}({{\mathbb {T}}})$ (not just over ${\mathrm {Spec}}({{\mathbb {I}}})$). Here we keep the notation introduced in Section 10, in particular, $(R,{\varvec{\rho }})$ is the minimal universal deformation ring representing $D_{\varvec{\kappa }}$ and we have a canonical surjective morphism $\iota _{{\mathbb {T}}}:R\rightarrow {{\mathbb {T}}}$ such that $\rho _{{\mathbb {T}}}\approx \iota _{{\mathbb {T}}}\circ {\varvec{\rho }}$.

We assume $R={{\mathbb {T}}}$ and (2.3): ${{\mathbb {T}}}$ has a presentation ${{\mathbb {T}}}=\Lambda [[X_1,\dots ,X_r]/(S_1,\dots ,S_r)$ for $r=\dim _{{\mathbb {F}}}{\mathrm {Sel}}(Ad({{\bar{\rho }}}))$. Let $\rho :G\rightarrow {\mathrm {GL}}_2(A)\in D_{\varvec{\kappa }}(A)$ be a deformation of ${{\bar{\rho }}}$ such that $\rho \cong P\circ \rho _{{\mathbb {T}}}$. We have an exact sequence for $(P:{{\mathbb {T}}}\rightarrow A)\in {\mathrm {Hom}}_\Lambda -\text {alg}({{\mathbb {T}}},A)$

Then we define

$$\begin{aligned} L_{{\mathbb {T}}}{:}{=}\det \left( \bigoplus _{j=1}^rA\cdot dS_j\xrightarrow {\ell } \bigoplus _{j=1}^r {{\mathbb {T}}}\cdot dX_j\right) . \end{aligned}$$

(B.1)

The element $L_{{\mathbb {T}}}$ gives rise to a p-adic L-function with

$$\begin{aligned} {\mathrm {Spec}}({{\mathbb {T}}})(W)\ni P\mapsto |L_{{\mathbb {T}}}(P)|_p^{-1}=|{\mathrm {Sel}}(Ad( P\circ {\varvec{\rho }}))|. \end{aligned}$$

Let $\lambda :{{\mathbb {T}}}={{\mathbb {T}}}\twoheadrightarrow {{\mathbb {I}}}$ be a $\Lambda$-algebra surjective homomorphism for an integral domain ${{\mathbb {I}}}$ finite torsion-free over $\Lambda$. Let ${{\mathbb {T}}}_{{\mathbb {I}}}{:}{=}{{\mathbb {T}}}\otimes _\Lambda {{\mathbb {I}}}$ and $\tilde{\lambda }$ be the composite ${{\mathbb {T}}}_{{\mathbb {I}}}\twoheadrightarrow {{\mathbb {I}}}\otimes _\Lambda {{\mathbb {I}}}\xrightarrow [\twoheadrightarrow ]{a\otimes b\mapsto ab}{{\mathbb {I}}}$. Then for each $P\in {\mathrm {Spec}}({{\mathbb {I}}})(W)={\mathrm {Hom}}_\text {W-alg}({{\mathbb {I}}},W)$, $\tilde{\lambda }$ induces $\Lambda \hookrightarrow {{\mathbb {T}}}_{{\mathbb {I}}}\xrightarrow {\tilde{\lambda }}{{\mathbb {I}}}\xrightarrow {P} W$ by composition.

Writing $\rho _P{:}{=}P\circ \lambda \circ {\varvec{\rho }}$. Then $\det \rho _P$ is a deformation of $\det {{\bar{\rho }}}$; so, we have a unique morphism $\iota _P:\Lambda \rightarrow W$ such that $\iota _P\circ \kappa =\det (\rho _P)$. Since the $\Lambda$-algebra structure $\iota :\Lambda \rightarrow {{\mathbb {T}}}$ of ${{\mathbb {T}}}={{\mathbb {T}}}$ is given by $\det ({\varvec{\rho }})=\det (\rho _{{\mathbb {T}}})=\iota \circ \kappa$, we find out that the above composite is just $\iota _P$.

Let ${{\mathbb {T}}}_P={{\mathbb {T}}}_{{\mathbb {I}}}\otimes _{{{\mathbb {I}}},P}W$ under the above algebra homomorphism. Note that

$$\begin{aligned} {{\mathbb {T}}}_P={{\mathbb {T}}}\otimes _\Lambda {{\mathbb {I}}}\otimes _{{{\mathbb {I}}},P} W\cong {{\mathbb {T}}}\otimes _{\Lambda ,\iota _P} W \end{aligned}$$

by associativity of tensor product.

By construction, we have $\lambda _P:{{\mathbb {T}}}_P\rightarrow W$ induced by $\lambda$. Even if $\iota _P=\iota _{P'}$, $\lambda _P$ may be different from $\lambda _{P'}$. If $\lambda _P$ is associated to a Hecke eigenform of weight $\ge 2$, we call P a arithmetic point. If ${{\mathbb {T}}}_P\otimes _W{\text {Frac}}(W)={\text {Frac}}(W)\oplus ({\text {Ker}}(\lambda _P)\otimes _W{\text {Frac}}(W))$ as algebra direct sum, we call P admissible. If P is admissible, write $S_P$ for the image of ${{\mathbb {T}}}_P$ in $({\text {Ker}}(\lambda _P)\otimes _W{\text {Frac}}(W))$. Then define the congruence module $C_0(\lambda _P){:}{=}S_P\otimes _{{{\mathbb {T}}}_P,\lambda _P}W$. If P is arithmetic, it is admissible. Since ${{\mathbb {T}}}_{{\mathbb {I}}}$ is a complete intersection over ${{\mathbb {I}}}$, by a theorem of Tate (see [33, Appendix] or [24, Theorem 6.8]),

$$\begin{aligned} |C_0(\lambda _P)|=|\Omega _{{{\mathbb {T}}}_{{\mathbb {I}}}/{{\mathbb {I}}}}\otimes _{{{\mathbb {T}}}_{{\mathbb {I}}}}{{\mathbb {I}}}/P|=|\Omega _{{{\mathbb {T}}}/\Lambda }\otimes _{{\mathbb {T}}}{{\mathbb {I}}}/P|. \end{aligned}$$

(B.2)

If $\rho \in D_{\varvec{\kappa }}(A)$ for W-valued $\kappa =\det (\rho _P)$, then $\rho \in D_{\varvec{\kappa }}(A)$ and hence $\rho =\phi \circ {\varvec{\rho }}$ for $\phi :{{\mathbb {T}}}\rightarrow A$. By definition, $\phi$ factors through

$$\begin{aligned} {{\mathbb {T}}}/R(\det ({\varvec{\rho }})(g)-\kappa (g))_gR={{\mathbb {T}}}/R(\kappa (g)-\kappa (g))_gR=R\otimes _{\Lambda ,\kappa }W. \end{aligned}$$

This shows that ${{\mathbb {T}}}=R\otimes _{\Lambda ,\kappa }W$ for $\kappa :\Lambda =W[[\Gamma ]]\rightarrow W$ induced by $\kappa$. Applying this to ${{\mathbb {T}}}_P$, we get $R_{\det (\rho _P)}={{\mathbb {T}}}_P$.

Here is a well known theorem (a combination of (2.3) and an old result of mine [21]) which shows non-vanishing at arithmetic P of the p-adic L-function $L_{{\mathbb {T}}}$ interpolating the size of adjoint Selmer group over ${\mathrm {Spec}}({{\mathbb {T}}})$ (e.g., [19, §5.3.6]) for canonical periods $\Omega _{f,\pm }$ of f:

Theorem B.1

Assume $R={{\mathbb {T}}}$ and (2.3). Let $\lambda :{{\mathbb {T}}}\twoheadrightarrow {{\mathbb {I}}}$ be a surjective $\Lambda$-algebra homomorphism for a domain ${{\mathbb {I}}}$ containing $\Lambda$ and $\tilde{\lambda }:{{\mathbb {T}}}_{{\mathbb {I}}}\rightarrow {{\mathbb {I}}}$ be its scalar extension to ${{\mathbb {I}}}$. Then there exists $L_{{\mathbb {I}}}\in {{\mathbb {I}}}$ such that $C_0(\lambda )={{\mathbb {I}}}/(L_{{\mathbb {I}}})$ and for each admissible $P\in {\mathrm {Spec}}({{\mathbb {I}}}),$ $C_0(\lambda _P)=W/P(\lambda (L_{{\mathbb {I}}}))$ and if $P\circ \lambda \circ \rho _{{\mathbb {T}}}\cong \rho _f$ for a modular form of weight $\ge 2,$ we have $|{\mathrm {Sel}}(Ad(\rho _P)|=|C_0(\lambda _P)|=|W/L_{{\mathbb {I}}}(P)|=|\frac{L(1,Ad(\rho _f))}{\Omega _{f,+}\Omega _{f,-}}|_p^{-1}<\infty$ (so, $L_{{\mathbb {I}}}(P)\ne 0$ $),$ where $L_{{\mathbb {I}}}(P){:}{=}P(\lambda (L_{{\mathbb {I}}}))$.

By (B.2), we need to prove only $|C_0(\lambda _P)|=|W/P(\lambda (L_{{\mathbb {I}}}))|=|\frac{L(1,Ad(\rho _f))}{\Omega _{f,+}\Omega _{f,-}}|_p^{-1}$. If f is of weight 2 on a modular curve X, for ${{\mathcal {W}}}=W\cap {\bar{{{\mathbb {Q}}}}}$, we have $H^1(X,{{\mathcal {W}}})[\lambda _P]={{\mathcal {W}}}\omega _+(f)\oplus {{\mathcal {W}}}\omega _-(f)$ (±-eigenspace under the pull-back action of $z\mapsto -\bar{z}$ on the upper half complex plane) and $H^1(X,{{\mathbb {C}}})={{\mathbb {C}}}\delta _+(f)+{{\mathbb {C}}}\delta _-(f)$ for $\delta _\pm f=f(z)dz\mp f(-\bar{z})d\bar{z}$. Then $\Omega _{f,\pm }\omega _\pm (f)=\delta _\pm (f)$. We use Eichler–Shimura isomorphism to define $\Omega _{f,\pm }$ for higher weight (see [24, Theorem 6.28] for details).

Since $L_{{\mathbb {I}}}(P)\ne 0$ is a key to show local indecomposability of modular Galois representation (Theorem C), we give a sketch of a proof of the non-vanishing in two steps.

Proof

Step 1: Existence of $L_{{\mathbb {I}}}$. Write $M^*{:}{=}{\mathrm {Hom}}_{{\mathbb {I}}}(M,{{\mathbb {I}}})$ for an ${{\mathbb {I}}}$-module M. Let S be the image of ${{\mathbb {T}}}_{{\mathbb {I}}}$ in ${{\mathfrak {B}}}\otimes _{{\mathbb {I}}}{\text {Frac}}({{\mathbb {I}}})$ for ${{\mathfrak {B}}}={\text {Ker}}(\tilde{\lambda })$ in the decomposition ${{\mathbb {T}}}\otimes _\Lambda {\text {Frac}}({{\mathbb {I}}})={\text {Frac}}({{\mathbb {I}}})\oplus ({{\mathfrak {B}}}\otimes _{{\mathbb {I}}}{\text {Frac}}({{\mathbb {I}}}))$. Let $\mu :{{\mathbb {T}}}_{{\mathbb {I}}}\rightarrow S$ be the projection and put ${{\mathfrak {A}}}={\text {Ker}}(\mu )$. So we have a split exact sequence ${{\mathfrak {B}}}\hookrightarrow {{\mathbb {T}}}_{{\mathbb {I}}}\twoheadrightarrow {{\mathbb {I}}}$. A local complete intersection ${{\mathbb {T}}}_{{\mathbb {I}}}$ over ${{\mathbb {I}}}$ has such a self-dual pairing $(\cdot ,\cdot )$ with values in ${{\mathbb {I}}}$ such that $(xy,z)=(x,yz)$ for $x,y,z\in {{\mathbb {T}}}_{{\mathbb {I}}}$. Thus ${{\mathfrak {B}}}^*\cong {{\mathbb {T}}}_{{\mathbb {I}}}^*/{{\mathbb {I}}}^*$, and ${{\mathbb {I}}}^*\subset {{\mathbb {T}}}_{{\mathbb {I}}}={{\mathbb {T}}}_{{\mathbb {I}}}^*$ is a maximal submodule of ${{\mathbb {T}}}_{{\mathbb {I}}}$ on which ${{\mathbb {T}}}_{{\mathbb {I}}}$ acts through $\tilde{\lambda }$; so, ${{\mathbb {I}}}^*={{\mathfrak {A}}}$ inside ${{\mathbb {T}}}_{{\mathbb {I}}}$. This implies ${{\mathfrak {B}}}^*\cong S$; so, S is ${{\mathbb {I}}}$-free. In other words, applying ${{\mathbb {I}}}$-dual, we get a reverse exact sequence

This shows $?={{\mathfrak {A}}}\cong {{\mathbb {I}}}^*\cong {{\mathbb {I}}}$; so, ${{\mathfrak {A}}}$ is principal. Define $L_{{\mathbb {I}}}\in {{\mathbb {I}}}$ by ${{\mathfrak {A}}}=(L_{{\mathbb {I}}})$. Note that $C_0(\tilde{\lambda })={{\mathbb {I}}}/{{\mathfrak {A}}}$.

Step 2: Specialization property. We have ${{\mathfrak {B}}}^*=S$ and a split exact sequence ${{\mathfrak {B}}}\rightarrow {{\mathbb {T}}}_{{\mathbb {I}}}\rightarrow {{\mathbb {I}}}$; so, ${{\mathfrak {B}}}$ is an ${{\mathbb {I}}}$-direct summand of ${{\mathbb {T}}}_{{\mathbb {I}}}$. Tensoring W over ${{\mathbb {I}}}$ via P, ${{\mathfrak {B}}}\otimes _{{{\mathbb {I}}},P}W\rightarrow {{\mathbb {T}}}_P\rightarrow W$ is exact, and we get ${{\mathfrak {B}}}_P={{\mathfrak {B}}}\otimes _{{{\mathbb {I}}},P}W={\text {Ker}}(\lambda _P)$. Since ${{\mathbb {T}}}$ is $\Lambda$-free of finite rank, ${{\mathbb {T}}}_{{\mathbb {I}}}$ is ${{\mathbb {I}}}$-free of finite rank. Thus ${{\mathfrak {B}}}$ is ${{\mathbb {I}}}$-projective and hence ${{\mathbb {I}}}$-free; so, $S\cong {{\mathfrak {B}}}^*$ is ${{\mathbb {I}}}$-free. Tensoring W over ${{\mathbb {I}}}$ via P, we get

$$\begin{aligned} 0\rightarrow {{\mathfrak {A}}}\otimes _{{{\mathbb {I}}},P}W\rightarrow {{\mathbb {T}}}_P\rightarrow S\otimes _{{{\mathbb {I}}},P}W\rightarrow 0. \end{aligned}$$

Thus if P is admissible, $S_P{:}{=}S\otimes _{{{\mathbb {I}}},\lambda _P}W$ gives rise to the decomposition: ${{\mathbb {T}}}_P\otimes _W{\text {Frac}}(W)={\text {Frac}}(W)\oplus (S_P\otimes _W{\text {Frac}}(W))$. By ${{\mathfrak {B}}}_P={{\mathfrak {B}}}_P\otimes _{{{\mathbb {I}}},P}W={\text {Ker}}(\lambda _P)$, we get $C_0(\lambda _P)=S_P/{{\mathfrak {B}}}_P=(S/{{\mathfrak {B}}})\otimes _{{{\mathbb {I}}},P}W=C_0(\tilde{\lambda })\otimes _{{{\mathbb {I}}},P}W$, as desired. $\square$

Tensoring ${{\mathbb {I}}}$ with the exact sequence of ${{\mathbb {T}}}$-modules:

$$\begin{aligned} (S_1,\dots ,S_r)/(S_1,\dots ,S_r)^2\xrightarrow {f\mapsto df}\Omega _{\Lambda [[X_1,\dots ,X_r]]/\Lambda }\otimes _{\Lambda [[X_1,\dots ,X_r]]}{{\mathbb {T}}}\twoheadrightarrow \Omega _{{{\mathbb {T}}}/\Lambda } \end{aligned}$$

over ${{\mathbb {T}}}$, we get an exact sequence $\bigoplus _j{{\mathbb {I}}}dS_j\xrightarrow {d\otimes 1=\lambda (d)}\bigoplus _j{{\mathbb {I}}}dX_j\rightarrow \Omega _{{{\mathbb {T}}}/\Lambda }\otimes _{{{\mathbb {T}}},\lambda }{{\mathbb {I}}}\rightarrow 0.$ Since ${{\mathbb {T}}}_{{\mathbb {I}}}={{\mathbb {I}}}[[X_1,\dots ,X_r]]/(S_1,\dots ,S_r)_{{\mathbb {I}}}$, we have

$$\begin{aligned} \Omega _{{{\mathbb {T}}}_{{\mathbb {I}}}/{{\mathbb {I}}}}\otimes _{{{\mathbb {T}}}_{{\mathbb {I}}},\tilde{\lambda }}{{\mathbb {I}}}=\bigoplus _j{{\mathbb {I}}}dX_j/\bigoplus _j{{\mathbb {I}}}dS_j=\Omega _{{{\mathbb {T}}}/\Lambda }\otimes _{{{\mathbb {T}}},\lambda }{{\mathbb {I}}}. \end{aligned}$$

They have the same characteristic ideals (and Fitting ideals) by Tate’s theorem [24, Theorem 6.8]. Thus in general, we get

$$\begin{aligned} (\lambda (L_{{\mathbb {T}}}))&=(\lambda (\det (d)))=(\det (d\otimes 1))={\text {char}}(\Omega _{{{\mathbb {T}}}_{{\mathbb {I}}}/{{\mathbb {I}}}}\otimes _{{{\mathbb {T}}}_{{\mathbb {I}}},\tilde{\lambda }}{{\mathbb {I}}})\\&{\mathop {=}\limits ^{\text {Tate}}}{\text {char}}(C_0(\tilde{\lambda }))=(L_{{\mathbb {I}}}). \end{aligned}$$

Thus we obtain the following.

COROLLARY B.2

Let the notation and assumption be as in Theorem B.1. Then $\lambda (L_{{\mathbb {T}}})=L_{{\mathbb {I}}}$ up to units in ${{\mathbb {I}}},$ and if $P\in {\mathrm {Spec}}({{\mathbb {T}}})({\bar{{{\mathbb {Q}}}}}_p)$ satisfies $P\circ \lambda \circ \rho _{{\mathbb {T}}}\cong \rho _f$ for a modular form f of weight $\ge 2,$ we have $|{\mathrm {Sel}}(Ad(\rho _P)|=|C_0(\lambda _P)|=|W/L_{{\mathbb {T}}}(P)| =|\frac{L(1,Ad(\rho _f))}{\Omega _{f,+}\Omega _{f,-}}|_p^{-1}<\infty$ (so, $L_{{\mathbb {T}}}(P)\ne 0$), where $L_{{\mathbb {T}}}(P){:}{=}P(\lambda (L_{{\mathbb {T}}}))$.

The corollary tells us that $L_{\mathrm{mod}}\in {{\mathbb {I}}}$ glues (up to units) well to $L_{{\mathbb {T}}}$ so that the image $\lambda (L_{{\mathbb {T}}})$ of $L_{{\mathbb {T}}}$ in ${{\mathbb {I}}}$ is equal to $L_{{\mathbb {I}}}$ of ${{\mathbb {I}}}$ up to units, and in particular, $L_{{\mathbb {T}}}(P)\ne 0$ if P is an arithmetic point.

Appendix C. Action of $\sigma$ on Selmer groups

In this last section, we assume that ${{\bar{\rho }}}={\text {Ind}}_F^{{\mathbb {Q}}}\varphi$ for a real quadratic field F. We identify the “$+$”-eigenspace ${\mathrm {Sel}}^+(Ad(\rho _{{\mathbb {T}}}))^\vee$ under the action of $\sigma$ with $\Omega _{{{\mathbb {T}}}_+/\Lambda }$. We need a general facts on relative dualizing modules. Let B be a commutative p-profinite local ring for a prime $p>2$. Consider a local B-algebra A finite over B with $B\hookrightarrow A$. Write $\omega _{A/B}$ for the dualizing module for the finite (hence proper) morphism $X{:}{=}{\mathrm {Spec}}(A)\xrightarrow {f}{\mathrm {Spec}}(B){=}{:}Y$ if it exists (in the sense of [29, (6)]). For the dualizing functor $f^!$ from quasi coherent Y-sheaves into quasi coherent X-sheaves defined in [29, (2)], we have ${\mathrm {Hom}}_A(F,f^!N)={\mathrm {Hom}}_B(f_*F,N)$ for any quasi-coherent sheaves F over X and N over Y; so, if $\omega _{A/B}$ exists (i.e., $f^!(N)=N\otimes _B\omega _{A/B}$), taking $F=A$ and $N=B$, we have $\omega _{A/B}=f^!({{\mathcal {O}}}_Y)={\mathrm {Hom}}_B(A,B)$ as A-modules. As shown in [29, (21)], ${\mathrm {Spec}}(A)\xrightarrow {f}{\mathrm {Spec}}(B)$ has dualizing module if and only if f is Cohen Macaulay (e.g., if B is regular and A is free of finite rank over B). Even if we do not have dualizing module $\omega _{A/B}$, we just define $\omega _{A/B}{:}{=}{\mathrm {Hom}}_B(A,B)$ generally.

Suppose that we have an involution $\sigma \in {\text {Aut}}(A/B)$. Let $A_+=A^{{\mathcal {G}}}$ for the order 2 subgroup ${{\mathcal {G}}}$ of ${\text {Aut}}(A/B)$ generated by $\sigma$. Under the following four conditions:

• (1) B is a regular local ring,

• (2) A is free of finite rank over B,

• (3) A and $A_+$ are Gorenstein ring,

• (4) A/B is generically étale (i.e., ${\text {Frac}}(A)$ is reduced separable over ${\text {Frac}}(B)$),

in [31, §3.5.a], the module of regular differentials $\omega _{\square /\triangle }$ for $(\square ,\triangle )=(A,B), (A,A_+), (A_+,B)$ is defined as fractional ideals in ${\text {Frac}}(\square )$. By (1) and (2), A/B and $A_+/B$ are Cohen Macaulay; so, $\omega _{A/B}$ and $\omega _{A_+/B}$ as above are the dualizing modules.

We now identify the dualizing module with classical “inverse different”. Let $C\supset B$ be reduced algebras. By abusing notation, write $\omega _{C/B}{:}{=}{\mathrm {Hom}}_B(C,B)$ in general. Suppose that ${\text {Frac}}(C)/{\text {Frac}}(B)$ is étale to have a trace map ${\mathrm {Tr}}:{\text {Frac}}(C)\rightarrow {\text {Frac}}(B)$, and $\omega _{{\text {Frac}}(C)/{\text {Frac}}(B)}={\text {Frac}}(C){\mathrm {Tr}}$ by the trace pairing $(x,y)\mapsto {\mathrm {Tr}}(xy)$. We define a C-fractional ideal (called the inverse different for C/B) by

$$\begin{aligned} {{\mathfrak {d}}}^{-1}_{C/B}{:}{=}\{x\in C |{\mathrm {Tr}}(xC)\subset B\}. \end{aligned}$$

In other words, $\omega _{C/B}={\mathrm {Hom}}_B(C,B)\hookrightarrow {\mathrm {Hom}}_{{\text {Frac}}(B})({\text {Frac}}(C),{\text {Frac}}(B))={\text {Frac}}(C){\mathrm {Tr}}$ has image ${{\mathfrak {d}}}_{C/B}^{-1}{\mathrm {Tr}}$. Thus we have ${{\mathfrak {d}}}^{-1}_{C/B}\cong \omega _{C/B}$. If $C=B[\delta ]$ is free of rank 2 over B with an B-basis $1,\delta$ with $\delta ^2\in B$, we have ${{\mathfrak {d}}}^{-1}_{C/B}=\delta ^{-1}C$ for $\delta ^{-1}\in {\text {Frac}}(C)$. Here is a version of Dedekind’s formula of transitivity of inverse differents proven in [30, Proposition G.13] (see also [29, (26) (vii)]).

PROPOSITION C.1

Let B be a regular p-profinite local ring. Suppose that D/C/B is generically étale finite extensions of reduced algebras such that D and C are B-flat, $\omega _{C/B}\cong C$ as C-modules (i.e., C is Gorenstein) and that ${\text {Frac}}(D)$ is ${\text {Frac}}(C)$-free. Then we have ${{\mathfrak {d}}}^{-1}_{D/C}{{\mathfrak {d}}}^{-1}_{C/B}={{\mathfrak {d}}}^{-1}_{D/B}$ and $\omega _{D/C}\otimes _C\omega _{C/B}\cong \omega _{D/B}.$

We now prove the following.

Theorem C.2

Let ${\mathrm {Sel}}^+(Ad(\rho _{{\mathbb {T}}})){:}{=}\{x\in {\mathrm {Sel}}(Ad(\rho _{{\mathbb {T}}}))|\sigma (x)=x\}$, and suppose (H0)–(H2). Then we have a canonical isomorphism $\Omega _{{{\mathbb {T}}}_+/\Lambda }\cong {\mathrm {Sel}}^+(Ad(\rho _{{\mathbb {T}}}))^\vee$.

Proof

By Corollary 3.7(1), we can apply Proposition C.1 to $B=\Lambda ,$ $D={{\mathbb {T}}}$ and $C={{\mathbb {T}}}_+$. We have the first fundamental exact sequence [32, Theorem 25.1]:

$$\begin{aligned} \Omega _{{{\mathbb {T}}}_+/\Lambda }\otimes _{{{\mathbb {T}}}_+}{{\mathbb {T}}}\xrightarrow {i}\Omega _{{{\mathbb {T}}}/\Lambda }\rightarrow \Omega _{{{\mathbb {T}}}/{{\mathbb {T}}}_+}\rightarrow 0. \end{aligned}$$

(C.1)

We can choose $\Theta \in {{\mathbb {T}}}$ which is the image in ${{\mathbb {T}}}$ of X in Theorem 2.2 so that $\sigma (\Theta )=-\Theta$ as ${{\mathbb {T}}}/I$ is a surjective image of $\Lambda$ and I is generated by ${{\mathbb {T}}}_-$. Then ${{\mathbb {T}}}_+=\Lambda [\Theta ^2]$, which is a local complete intersection (so, Gorenstein). Thus we get from Proposition C.1, ${{\mathfrak {d}}}_{{{\mathbb {T}}}/{{\mathbb {T}}}_+}{{\mathfrak {d}}}_{{{\mathbb {T}}}_+/\Lambda }={{\mathfrak {d}}}_{{{\mathbb {T}}}/\Lambda }$. By Tate’s theorem [33, Appendix, (A.3)] or [24, §6.3.3], ${{\mathfrak {d}}}_{X/Y}={\text {Fitt}}_X(\Omega _{X/Y})$ for any subset $\{X,Y\}\subset \{{{\mathbb {T}}},{{\mathbb {T}}}_+,\Lambda \}$ with $X\supset Y$ and ${{\mathfrak {d}}}_{{{\mathbb {T}}}_+/\Lambda }{{\mathbb {T}}}={\text {Fitt}}_{{\mathbb {T}}}(\Omega _{{{\mathbb {T}}}_+/\Lambda }\otimes _{{{\mathbb {T}}}_+}{{\mathbb {T}}})$, writing ${\text {Fitt}}(M)$ for the Fitting ideal of a ${{\mathbb {T}}}$-module M of finite type (see [34, Appendix] for a concise description of the theory of Fitting ideal). Since $X_{/Y}$ is a relative local complete intersection, ${{\mathfrak {d}}}_{X/Y}$ is a principal ideal generated by a non-zero divisor again by Tate’s theorem.

Suppose relative complete intersection property: $X=Y[[(T)]]/(S)$ for a set of variable $(T){:}{=}(T_1,\dots ,T_r)$ and a regular sequence $(S){:}{=}(S_1,\dots ,S_r)\subset {{\mathfrak {m}}}_{Y[[(T)]]}$ with X free of finite rank over Y. Then we have the following commutative diagram with exact rows:

Thus ${\text {Fitt}}(\Omega _{X/Y})=(\det (\ell ))$ and hence ${\text {Fitt}}(\Omega _{X/Y})$ kills $\Omega _{X/Y}$. If $M\hookrightarrow L\twoheadrightarrow N$ is an exact sequence of ${{\mathbb {T}}}$-modules, we have ${\text {Fitt}}(M){\text {Fitt}}(N)\subset {\text {Fitt}}(L)$ and ${\text {Fitt}}(N)\supset {\text {Fitt}}(L)$ [34, Appendix, 1, 9]. By (C.1), ${\text {Fitt}}({\text {Im}}(i)){\text {Fitt}}(\Omega _{{{\mathbb {T}}}/{{\mathbb {T}}}_+})\subset {\text {Fitt}}(\Omega _{{{\mathbb {T}}}/\Lambda })$ and ${\text {Fitt}}({\text {Ker}}(i)){\text {Fitt}}({\text {Im}}(i))\subset {\text {Fitt}}(\Omega _{{{\mathbb {T}}}_+/\Lambda }\otimes _{{{\mathbb {T}}}_+}{{\mathbb {T}}})$. Since ${{\mathfrak {d}}}_{X/Y}$ is a principal ideal generated by a non-zero divisor (as already remarked), the identity ${{\mathfrak {d}}}_{{{\mathbb {T}}}/{{\mathbb {T}}}_+}{{\mathfrak {d}}}_{{{\mathbb {T}}}_+/\Lambda }={{\mathfrak {d}}}_{{{\mathbb {T}}}/\Lambda }$ then implies ${\text {Fitt}}({\text {Im}}(i))={\text {Fitt}}(\Omega _{{{\mathbb {T}}}_+/\Lambda }\otimes _{{{\mathbb {T}}}_+}{{\mathbb {T}}})$. Therefore ${\text {Fitt}}({\text {Ker}}(i))={{\mathbb {T}}}$; so, ${\text {Ker}}(i)$ is killed by $1\in {{\mathbb {T}}}$, and i is an injection, and (C.1) is a short exact sequence.

For a module M with $\sigma$-action, we write $M^+{:}{=}\{x\in M|\sigma (x)=x\}$. Since ${{\mathbb {T}}}\cong {{\mathbb {T}}}_+[X]/(X^2-\theta )$ for $\theta {:}{=}\Theta ^2$ by Corollary 3.7(1), $\sigma$ acts on $\Omega _{{{\mathbb {T}}}/{{\mathbb {T}}}_+}\cong ({{\mathbb {T}}}/(\Theta )) d\theta$ by $-1$. Thus by taking “$+$”-eigenspace of the exact sequence $\Omega _{{{\mathbb {T}}}_+/\Lambda }\otimes _{{{\mathbb {T}}}_+}{{\mathbb {T}}}\hookrightarrow \Omega _{{{\mathbb {T}}}/\Lambda }\twoheadrightarrow \Omega _{{{\mathbb {T}}}/{{\mathbb {T}}}_+}$, we get an isomorphism: $\Omega _{{{\mathbb {T}}}_+/\Lambda }\cong \Omega _{{{\mathbb {T}}}/\Lambda }^+$, because $\Omega _{{{\mathbb {T}}}_+/\Lambda }\otimes _{{{\mathbb {T}}}_+}{{\mathbb {T}}}=\Omega _{{{\mathbb {T}}}_+/\Lambda }\otimes _{{{\mathbb {T}}}_+}{{\mathbb {T}}}_+\oplus \Omega _{{{\mathbb {T}}}_+/\Lambda }\otimes _{{{\mathbb {T}}}_+}{{\mathbb {T}}}_-$ with $\Omega _{{{\mathbb {T}}}_+/\Lambda }\otimes _{{{\mathbb {T}}}_+}{{\mathbb {T}}}_+=\Omega _{{{\mathbb {T}}}_+/\Lambda }=(\Omega _{{{\mathbb {T}}}_+/\Lambda }\otimes _{{{\mathbb {T}}}_+}{{\mathbb {T}}})^+$. Since ${\mathrm {Sel}}(Ad(\rho _{{\mathbb {T}}}))^\vee \cong \Omega _{{{\mathbb {T}}}/\Lambda }$ by $R\cong {{\mathbb {T}}}$ (see Theorem A.1), we get the desired identity by taking “$+$” eigenspace of $\sigma$. $\square$