Determinant of a Frobenius-normalised stable plane is integrally cyclotomic

Fix a nonzero natural number $M$ and a prime $\lambda$. Let $O''$ be a complete discrete valuation ring which is a domain of characteristic zero with finite residue field and is a $\mathbb{Z}_\lambda$-algebra, and let $K$ be its fraction field. Work with the degree-zero divisor class group $J =$ JZero M of the full modular function field of level $M$ over $\overline{\mathbb{Q}}$, equipped with an action of the Hecke polynomial ring, and with its $\lambda$-adic Tate module TateModule lam $J$, the submodule of sequences $x : \mathbb{N} \to J$ with $x_0 = 0$ and $\lambda \cdot x_{n+1} = x_n$, carrying a $\mathbb{Z}_\lambda$-module structure assumed to act levelwise: $(a \cdot x)_n = (a \bmod \lambda^n)\, x_n$. Let $S$ be a finite set of naturals, and let $\rho_M$ be a monoid homomorphism from $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ to the $O''$-endomorphisms of $O'' \otimes_{\mathbb{Z}_\lambda} T_\lambda J$ which is compatible with the Galois action on the Tate module, in the sense that whenever $y$ is the componentwise image $\sigma \cdot x$ one has $\rho_M(\sigma)(b \otimes x) = b \otimes y$ for all $b \in O''$, and which is adically continuous: for each $n$ there is a finite extension $L/\mathbb{Q}$ inside $\overline{\mathbb{Q}}$ such that every $\sigma$ fixing $L$ pointwise satisfies $\rho_M(\sigma)v - v \in \mathfrak{m}_{O''}^n \cdot (O'' \otimes T_\lambda J)$ for all $v$. Let $W$ be a $K$-submodule of $K \otimes_{O''} (O'' \otimes_{\mathbb{Z}_\lambda} T_\lambda J)$ of $K$-dimension $2$, stable under the base change of every $\rho_M(\sigma)$, and assume that for every prime $\ell \nmid M$ with $\ell \notin S$, every valuation subring $B$ of $\overline{\mathbb{Q}}$ with $\ell$ a nonunit of $B$, and every $\sigma$ lying in the decomposition subgroup of $B$ and inducing $x \mapsto x^{\ell}$ on its residue field, the determinant of $\rho_M(\sigma)$ restricted to $W$ equals $\ell$ in $K$. The conclusion is that for every $\sigma$ and all naturals $n, a$ such that $\sigma\mu = \mu^a$ for every $\mu \in \overline{\mathbb{Q}}$ with $\mu^{\lambda^n} = 1$, there is $d \in O''$ whose image in $K$ is the determinant of $\sigma$ on $W$ and with $d - a$ in the ideal generated by $\lambda^n$.

This is the statement that the determinant of the two-dimensional $\lambda$-adic Galois representation carried by $W$ is the $\lambda$-adic cyclotomic character, in integral form: knowing the determinant on Frobenius elements outside a finite set forces the congruence $\det\sigma \equiv a \pmod{\lambda^n}$ whenever $\sigma$ acts on $\lambda^n$-th roots of unity by the exponent $a$. It is used when newform eigenplanes and ordinary lines inside the Tate module of $J_0(M)$ are produced, where ramification of the cyclotomic character at $\lambda$ is needed to control inertia.