Assumption 3.1 Given an even weight $k \geq 12$ and prime $\ell $ , assume that $E/\mathbf {Q}_\ell $ is sufficiently large to contain the fields F from Theorem 2.1 for all forms $f \in S_k(\Gamma _2)$ . We also assume that for every eigenform $\phi \in S_k(\Gamma _1),$ the field E contains all the Hecke eigenvalues of $\phi $ as well as the value $L_{\mathrm {alg}}(2k-2, {\operatorname {Sym}^2}\phi )$ (see (3.1) for the definition). In addition, we suppose that E contains a primitive cube root of unity.

Proof We know via [Reference Weissauer59] that if $\rho _{f}$ is reducible, then the automorphic representation associated with f is either CAP or a weak endoscopic lift. Moreover, by [Reference Pitale and Schmidt42, Corollary 4.5] since $f \in S_{k}(\Gamma _2)$ and $k>2$ , the automorphic representation attached to f can be CAP only with respect to the Siegel parabolic, i.e., f is a classical Saito–Kurokawa lift. Suppose that f is a Saito–Kurokawa lift of $\psi \in S_{2k-2}(\Gamma _1)$ . Then, we have ${\overline {\rho }}_{f}^{\operatorname {\mathrm {ss}}} = {\overline {\rho }}_{\psi } \oplus \overline {\epsilon }^{k-1} \oplus \overline {\epsilon }^{k-2}$ . Using the fact that $f \equiv _{\operatorname {\mathrm {ev}}} E_{\phi }^{2,1}\ \pmod {\lambda }$ and that the eigenvalues of $E_{\phi }^{2,1}$ are given by $\lambda (p;E_{\phi }^{2,1}) = a(p;\phi ) + p^{k-2}a(p;\phi )$ , the Brauer–Nesbitt and Chebotarev Theorems give that ${\overline {\rho }}_{f}^{\operatorname {\mathrm {ss}}} = {\overline {\rho }}_{\phi } \oplus {\overline {\rho }}_{\phi }(k-2)$ , where recall that we write ${\overline {\rho }}_{\phi }(k-2)$ for ${\overline {\rho }}_{\phi } \otimes \overline {\epsilon }^{k-2}$ . This is a contradiction if ${\overline {\rho }}_{\phi }$ is irreducible. Thus, f cannot be a Saito–Kurokawa lift. It remains to show that the automorphic representation associated with f is not a weak endoscopic lift. The possible decompositions of $\rho _{f}$ are given in [Reference Skinner and Urban48, Theorem 3.2.1] under the assumption that $\ell> 4k-7$ . Of these, the only case remaining to check is Case B(v), which states if $\rho _{f} = \sigma \oplus \sigma '$ with $\sigma $ and $\sigma '$ both two-dimensional, then $\det (\sigma ) = \det (\sigma ')$ . In our case, this would require $\det (\rho _{\phi }) = \det (\rho _{\phi }(k-2))$ , i.e., $\overline {\epsilon }^{k-1} = \overline {\epsilon }^{2k-3}$ , which is impossible by our assumption that $\ell>4k-7$ . Thus, $\rho _{f}$ is irreducible.

Proof Set $H_{\phi }^{2,1}(z) = L_{\operatorname {\mathrm {alg}}}(2k-2,{\operatorname {Sym}^2} \phi ) E^{2,1}_{\phi }(z)$ . For $T \geq 0$ , define $c(T) = \operatorname {\mathrm {val}}_{\lambda }(a(T;H_{\phi }^{2,1})). $ Let $c = \min _{T \geq 0} c(T)$ . Since $H_{\phi }^{2,1} \in M_k(\Gamma _2)$ , the Fourier coefficients $a(T;H_{\phi }^{2,1})$ have bounded denominators so c is well-defined [Reference Shimura46]. Moreover, our assumption that there is a $T_0> 0$ with $\operatorname {\mathrm {val}}_{\lambda }(a(T_0;H_{\phi }^{2,1})) = \operatorname {\mathrm {val}}_{\lambda }\left (L_{\operatorname {\mathrm {alg}}}(2k-2, {\operatorname {Sym}^2} \phi )a(T_0,E^{2,1}_{\phi })\right ) \leq 0$ gives that $c \leq 0$ . Set

$$ \begin{align*} G^{2,1}_{\phi}(z) = \lambda^{-c} H_{\phi}^{2,1}(z). \end{align*} $$

We have $a(T;G_{\phi }^{2,1}) \in \mathcal {O}$ for all $T \geq 0$ since $c(T) - c \geq 0$ for all $T \geq 0$ . Observe that for T with $\det T=0$ , we have $a(T;G^{2,1}_{\phi }) = \lambda ^{-c}L_{\operatorname {\mathrm {alg}}}(2k-2,{\operatorname {Sym}^2} \phi ) a(n;\phi )$ for some $n \in \mathbf {Z}_{\geq 0}$ . Since $a(n;\phi ) \in \mathcal {O}$ by assumption and $-c\geq 0$ , this gives $\lambda \mid a(T;G^{2,1}_{\phi })$ for all T with $\det T = 0$ , i.e., all the Fourier coefficients indexed by singular T vanish modulo $\lambda $ . Moreover, since $c = c(\widetilde {T})$ for some $\widetilde {T}$ , we have $a(\widetilde {T};G_{\phi }^{2,1}) \in \mathcal {O}^{\times }$ for some $\widetilde {T}$ . Since $c\leq 0$ and $\lambda \mid a(T;G_{\phi }^{2,1})$ for all singular T, we have $\widetilde {T}>0$ . Thus, Corollary 3.2 and the fact that $G^{2,1}_{\phi }$ and $E^{2,1}_{\phi }$ have the same eigenvalues gives an eigenform $f \in S_{k}(\Gamma _2;\mathcal {O})$ so that $ E^{2,1}_{\phi } \equiv _{\operatorname {\mathrm {ev}}} f \not \equiv 0\ \pmod {\lambda }. $ By Lemma 3.4, we get that $\rho _{f}$ is irreducible.

Proof For a positive integer $k,$ let $I_k$ be the ideal of $\mathbf {F}[[X_1,\dots , X_k]]$ generated by all the monomials of degree at least 2. Set $S_k:=\mathbf {F}[[X_1,\dots , X_k]]/I_k$ and write $\phi _k: \mathbf {F}[[X_1,\dots , X_k]]\to S_k$ for the canonical $\mathbf {F}$ -algebra surjection. If $\phi _n(J)=0$ , then composing $\phi _n$ with the map $S_n\to \mathbf {F}[[T]]/T^2$ sending $X_1$ to T and $X_i$ for $i>1$ to zero gives the desired surjection.

Now suppose $\phi _n(J)\neq 0$ . Without loss of generality (renumbering the variables if necessary), we may assume then that J contains an element of the form $u:=X_n+f(X_1,\dots , X_{n-1})+g(X_1,\dots , X_n)$ , where f is homogeneous of degree one and all the terms in g have degree at least 2. Note that we can assume without loss of generality that some power of $X_n$ appears in g. (Indeed, if g contains no $X_n$ then we replace u by $u+u^2 \in J$ .) By Theorem 7.16(a) in [Reference Eisenbud23] there is a unique $\mathbf {F}$ -algebra map from $\mathbf {F}[[X_1,\dots , X_n]]$ to itself sending $X_n$ to $-f-g$ and $X_i$ to itself for $i<n$ . In other words, for any power series $h(X_1,\dots , X_n)$ , the element $h(X_1,\dots , X_{n-1}, -f-g)$ also lives in $\mathbf {F}[[X_1, \dots , X_n]]$ and we denote it by $h'(X_1, \dots , X_n)$ . Clearly, $h-h'\in J$ .

Thus, for any power series $h,$ where the smallest total degree of any term containing $X_n$ is $s,$ we have

$$ \begin{align*}h\equiv h' \quad\pmod{J}\end{align*} $$

for some power series $h'$ with the smallest total degree of any term containing $X_n$ equal to $s'>s$ . By the same process, we get an $h"$ such that $h' \equiv h"\ \mod {J}$ and the smallest total degree of any term $X_n$ in $h"$ is strictly greater than $s'$ . This way, we can construct a sequence of power series $h_s$ where for every $s,$ we have the smallest total degree of any term containing $X_n$ being greater than or equal to s and such that $h- h_s\in J$ for every s. We note that $h_s$ is a Cauchy sequence with respect to the $(X_1,\dots ,X_n)$ -adic topology (indeed, for $t,u>s,$ we see that $h_t-h_u$ lies in $(X_1,\dots ,X_n)^s$ ). Set $h_0=\lim _{s\to \infty }h_s$ . As J is a closed ideal, we get that $h_0-h\in J$ . For every $s,$ we have

$$ \begin{align*}h_0 \equiv h_s \equiv w_s \quad\mod{X_n^s},\end{align*} $$

for some $w_s \in \mathbf {F}[[X_1, \dots , X_{n-1}]]$ . Note that the $w_s$ also form a Cauchy sequence since $h_s$ does. Set $w:= \lim _{s \to \infty } w_s \in \mathbf {F}[[X_1, \dots , X_{n-1}]]$ . Thus, $h_0 \equiv w$ modulo $\bigcap _s (X_n^s) \subset \bigcap _s (X_1, \ldots , X_n)^s=0$ , so $h_0 \in \mathbf {F}[[X_1, \dots , X_{n-1}]]$ .

Hence, the natural $\mathbf {F}$ -algebra map $\psi _{n-1}:\mathbf {F}[[X_1,\dots , X_{n-1}]]\to \mathbf {F}[[X_1, \dots , X_n]]/J$ given by $h_0 \mapsto h_0+J$ is surjective. Thus, we get an $\mathbf {F}$ -algebra isomorphism $\mathbf {F}[[X_1,\dots , X_n]]/J \to \mathbf {F}[[X_1, \dots , X_{n-1}]]/J_{n-1}$ , where $J_{n-1}=\ker \psi _{n-1}$ .

If $\phi _{n-1}(J_{n-1})\neq 0$ , continue this way obtaining a sequence of ideals $J_{n-2},J_{n-3},\ldots $ . If at any stage ( $1\leq r\leq n-2$ ), we get $\phi _{n-r}(J_{n-r})=0$ , then we are done. Otherwise, we can eliminate all but one variable and get $\mathbf {F}[[X_1,\dots , X_n]]/J\cong \mathbf {F}[[X_1]]/J_1$ and now we must have $\phi _1(J_1)=0$ as otherwise $J_1$ and hence J is maximal.

Assumption 6.2 For k and $\phi $ as above, we assume that:

1. (i) there exists $f \in S_k(\Gamma _2)$ such that $f\equiv _{\mathrm {ev}}E^{2,1}_{\phi }$ (mod $\lambda $ ), and

2. (ii) $\#H^1_f(\mathbf {Q}, \mathrm {ad}^0\rho _{\phi }(2-k)\otimes _{\mathcal {O}}E/\mathcal {O})=\#\mathcal {O}/\lambda $ (recall that the left-hand side is independent of the choice of lattice, see Remark 5.4), and

3. (iii) $H^1_f(\mathbf {Q}, \operatorname {\mathrm {ad}}^0{\overline {\rho }}_{\phi })=0.$

Proof We have

(6.1) $$ \begin{align} \#\operatorname{\mathrm{Hom}}_{\mathcal{O}-\mathrm{alg}}(R_{{\overline{\rho}}_{\phi}}, \mathbf{F}[X]/X^2)=\#H^1_f(\mathbf{Q}, \operatorname{\mathrm{ad}}{\overline{\rho}}_{\phi})=0,\end{align} $$

where the first equality follows from the fact that our deformation condition is the property of being Fontaine–Laffaille (see, e.g., [Reference Clozel, Harris and Taylor18, Section 2.4.1]), and the second one holds since we have $H^1_f(\mathbf {Q}, \operatorname {\mathrm {ad}}{\overline {\rho }}_{\phi })=H^1_f(\mathbf {Q}, \operatorname {\mathrm {ad}}^0{\overline {\rho }}_{\phi }) \oplus H^1_f(\mathbf {Q}, \mathbf {F})=0$ and $H^1_f(\mathbf {Q}, \mathbf {F})=0$ by Lemma 5.6 as we have imposed Assumption 6.2(iii).

By Theorem 7.16 in [Reference Eisenbud23] we know that any local complete Noetherian $\mathcal {O}$ -algebra with residue field $\mathbf {F}$ is a quotient of $\mathcal {O}[[X_1, \dots , X_n]]$ for some positive integer n. Hence, $S:=R_{{\overline {\rho }}_{\phi }}/(\lambda R_{{\overline {\rho }}_{\phi }})\cong \mathbf {F}[[X_1, \dots , X_n]]/J$ for some ideal J. Suppose first that J is not maximal. Then, by Lemma 6.1, we know that S admits a surjection $\varphi $ to $\mathbf {F}[T]/T^2$ . This contradicts (6.1), hence $S=\mathbf {F}$ . We now use the complete version of Nakayama’s Lemma to conclude that the structure map $\mathcal {O}\to R_{{\overline {\rho }}_{\phi }}$ is surjective (cf. [Reference Eisenbud23, Exercise 7.2] or [Reference Matsumura37, Theorem 8.4]). Let us briefly explain why this version applies here. As $R_{{\overline {\rho }}_{\phi }}\otimes _{\mathcal {O}}\mathbf {F}\neq 0$ , we see that $\lambda \in \mathfrak {m}$ , where $\mathfrak {m}$ is the maximal ideal of $R_{{\overline {\rho }}_{\phi }}$ . Hence,

(6.2) $$ \begin{align} \bigcap_n \lambda^n R_{{\overline{\rho}}_{\phi}} \subset \bigcap_n\mathfrak{m}^n.\end{align} $$

The latter intersection is zero, since $R_{{\overline {\rho }}_{\phi }}$ is complete, so separated with respect to $\mathfrak {m}$ . Hence, (6.2) implies that $R_{{\overline {\rho }}_{\phi }}$ is separated with respect to $\lambda R_{{\overline {\rho }}_{\phi }}$ allowing for the application of the complete version of Nakayama’s Lemma.

As $\rho _{\phi }$ is a deformation to $\mathcal {O}$ , we conclude that $R_{{\overline {\rho }}_{\phi }}=\mathcal {O}$ . This implies that if $\rho : G_{\{\ell \}}\to \operatorname {\mathrm {GL}}_2(\mathcal {O})$ is any deformation of ${\overline {\rho }}_\phi $ , one has $\rho \cong \rho _{\phi }$ . Similarly, if $\rho : G_{\{\ell \}} \to \operatorname {\mathrm {GL}}_2(\mathcal {O})$ is a deformation of ${\overline {\rho }}_{\phi }(k-2)$ then $\rho (2-k)$ is a deformation of ${\overline {\rho }}_{\phi }$ . Note that our choice of $I=[3-2k, 2k-3]$ means that this twisting stays inside our category of Fontaine–Laffaille representations. Hence, we get that $\rho (2-k)\cong \rho _{\phi }$ , and so we are done.

Proof This is a simple consequence of Schur’s Lemma and the fact that $\tilde {a}: g \to \rho _2(g)^{-1}a(g)$ defines a cocycle from G to $\operatorname {\mathrm {Hom}}(\rho _2, \rho _1)$ which is not a coboundary.

Proof We use Proposition 7.2 in [Reference Berger and Klosin6] noting that Assumption 6.1(i) in [loc.cit.] is satisfied because we impose the current Assumption 6.2(ii). On the other hand, Assumption 6.1(ii) in [loc.cit.] is satisfied because of Lemma 6.6.

Proof This is proved as Corollary 7.8 in [Reference Berger and Klosin6].

Proof By Theorem 7.16 in [Reference Eisenbud23] we know that any local complete Noetherian $\mathcal {O}$ -algebra with residue field $\mathbf {F}$ is a quotient of $\mathcal {O}[[X_1, \dots , X_n]]$ for some positive integer n. Hence, $S:=R/(I_{\mathrm {re}}+\lambda R)\cong \mathbf {F}[[X_1, \dots , X_n]]/J$ for some ideal J. Suppose first that J is not maximal. Then, by Lemma 6.1, we know that S admits a surjection $\varphi $ to $\mathbf {F}[T]/T^2$ . This means that there exists a non-trivial (because the image of $\varphi $ is not contained in $\mathbf {F}$ ) deformation of $\rho $ to $\mathbf {F}[T]/T^2$ which is upper-triangular (by Proposition 6.12), which contradicts Lemma 6.10. Thus, indeed, $S=\mathbf {F}$ .

Hence, the structure map $\mathcal {O}\to R/I_{\mathrm {re}}$ is surjective by the complete version of Nakayama’s Lemma (see the proof of Lemma 6.6). So, $R/I_{\mathrm {re}}\cong \mathcal {O}/\lambda ^s$ for some $s\in \mathbf {Z}_{\geq 0}\cup \{\infty \}$ .

The composition of $\rho ^{\mathrm {univ}}$ with the map $R\to R/I_{\mathrm {re}}$ gives rise to a deformation $\rho _{\mathrm {re}}: G_{\{\ell \}}\to \operatorname {\mathrm {GL}}_4(R/I_{\mathrm {re}})=\operatorname {\mathrm {GL}}_4(\mathcal {O}/\lambda ^s)$ . By Proposition 6.12, this deformation is upper triangular, i.e., one has $\rho _{\mathrm {re}}=\left [ \begin {matrix} *_1&*_2\\ &*_3\end {matrix} \right ].$ As the property of being Fontaine–Laffaille is preserved by subobjects and quotients, we see that $*_1$ and $*_3$ are Fontaine–Laffaille representations with values in $\operatorname {\mathrm {GL}}_2(R/I_{\mathrm { re}})=\operatorname {\mathrm {GL}}_2(\mathcal {O}/\lambda ^s)$ . Thus, by Lemma 6.6, we can conclude that $*_1=\rho _{\phi }$ , $*_3=\rho _{\phi }(k-2)$ mod $\lambda ^s$ . Hence, by (5.4) and Proposition 5.1, $*_2$ gives rise to a class in $H_f^1(\mathbf {Q},\mathrm {ad}^0\rho _{\phi }(2-k)\otimes _{\mathcal {O}}E/\mathcal {O})$ as $\rho _{\mathrm {re}}$ is Fontaine–Laffaille. As $\rho $ is non-semi-simple, we conclude that $*_2$ is not annihilated by $\lambda ^{s-1}$ , i.e., the class of $*_2$ gives rise to a subgroup of $H^1_f(\mathbf {Q},\mathrm { ad}^0\rho _{\phi }(2-k)\otimes _{\mathcal {O}}E/\mathcal {O})$ isomorphic to $\mathcal {O}/\lambda ^s$ . Thus, $s\leq 1$ as $\#H^1_f(\mathbf {Q}, \mathrm { ad}^0\rho _{\phi }(2-k)\otimes _{\mathcal {O}}E/\mathcal {O})\leq \#\mathcal {O}/\lambda $ by Assumption 6.2(ii). Finally, $s>0$ as ${\overline {\rho }}_{f}$ itself is reducible. This concludes the proof.

Proof Since $\rho ^{\mathrm {univ}}$ is a trace representation in the sense of Section 1.3.3 of [Reference Bellaïche and Chenevier4] Lemma 1.3.7 in [loc.cit.] tells us that we can conjugate $\rho ^{\mathrm {univ}}$ by a matrix $P \in \operatorname {\mathrm {GL}}_2(R)$ (here we use that every finite type projective R-module is free since R is local) to get $\rho ^{\mathrm {univ}}$ adapted to a data of GMA idempotents for $R[G_{\{\ell \}}]/\ker \rho ^{\mathrm {univ}}$ . By [Reference Bellaïche and Chenevier4, Lemma 1.3.8] we then get an isomorphism of R-modules

$$ \begin{align*}R[G_{\{\ell\}}]/\ker \rho^{\mathrm{univ}}\cong\left[ \begin{matrix} \operatorname{\mathrm{Mat}}_2(R)&\operatorname{\mathrm{Mat}}_2(B)\\ \operatorname{\mathrm{Mat}}_2(C)&\operatorname{\mathrm{Mat}}_2(R) \end{matrix} \right]\end{align*} $$

for ideals $B, C \subset R$ . By [Reference Bellaïche and Chenevier4, Proposition 1.5.1] we further know that $I_{\mathrm {re}}=BC$ .

[Reference Bellaïche and Chenevier4, Theorem 1.5.5] proves that there are injections $\operatorname {\mathrm {Hom}}_R(B, \mathbf {F}) \hookrightarrow H^1(G_{\{\ell \}}, \operatorname {\mathrm {ad}}{\overline {\rho }}_{\phi }(2-k))$ and $\operatorname {\mathrm {Hom}}_R(C, \mathbf {F}) \hookrightarrow H^1(G_{\{\ell \}}, \operatorname {\mathrm {ad}}{\overline {\rho }}_{\phi }(k-2)).$ Arguing as in [Reference Akers1, Proposition 4.2] (see also [Reference Wake and Wang-Erickson55, Theorem 4.3.5 and Remark 4.3.6] one sees that the images are contained in the Selmer groups $H^1_f(\mathbf {Q}, \operatorname {\mathrm {ad}}{\overline {\rho }}_{\phi }(2-k))$ and $H^1_f(\mathbf {Q}, \operatorname {\mathrm {ad}}{\overline {\rho }}_{\phi }(k-2))$ , respectively. From Assumption 6.2 (ii) and Proposition 5.1, we see that $H^1(\mathbf {Q}, \operatorname {\mathrm {ad}}{\overline {\rho }}_{\phi }(2-k)) \cong \mathbf {F}$ . Together with the assumption $\dim H^1_f(\mathbf {Q}, \operatorname {\mathrm {ad}}{\overline {\rho }}_{\phi }(k-2))\leq 1,$ we deduce by Nakayama’s Lemma that both B and C, and therefore also $I_{\mathrm {re}}$ are principal ideals of R. Note that Nakayama’s Lemma applies since B and C are ideals in R, which is Noetherian, hence they are finitely generated over R.