$\DeclareMathOperator{\PGL}{PGL} \DeclareMathOperator{\GL}{GL} \newcommand{\F}{\mathbb{F}} \newcommand{\p}{\mathfrak{p}} \DeclareMathOperator{\Sym}{Sym}$ Let $F$ be a number field, and let $\Gamma$ be a congruence subgroup of $\PGL_2(\mathbb{Z}_F)$ with level supported at some finite set of primes $S$. To a Hecke-eigenclass in $H^*(\Gamma,\F_p)$, we expect to be able to attach a semisimple continuous Galois representation $$ \rho \colon G_F \to \GL_2(\bar{\F}_p), $$ unramified outside $S\cup p$, and with characteristic polynomial of Frobenius determined by the Hecke eigenvalues in the usual way.

Let $\p \mid p$ be a prime ideal not in $S$ and unramified in $F$, let $F_\p$ be the completion of $F$ at $\p$ and $I_\p$ be the inertia group of $F_\p$. Let $k$ be the quadratic extension of the residue field of $F_\p$. For each embedding $\tau$ of the residue field $k \hookrightarrow \bar{\F}_p$, let $\omega_\tau \colon I_\p \to \bar{\F}_\p^\times$ be the associated tame character.

Assume that $\rho|I_\p$ is tamely ramified. Is it expected that $\rho|I_\p$ is then of the form $$ \rho|I_\p \cong \prod_{\tau}\omega_\tau^{a_\tau} \oplus \prod_\tau\omega_\tau^{b_\tau} $$ with $\{a_\tau,b_\tau\} = \{0,1\}$ for all $\tau$?

If that is the correct expectation, could you point me to a reference stating such a conjecture?

When $F$ is totally real or CM, the existence of $\rho$ was proved by Scholze.

In the totally real or CM case, is it known that $\rho|I_\p$ has the form described above whenever it is tamely ramified?

EDIT Let me try to provide a little more context.

If $\rho\colon G_F \to \GL_2(\bar{\F}_p)$ is a continuous representation, $\p\mid p$ is an unramified prime and $\rho|I_\p$ is tame, then this restriction (if I got this right) is of the form $$ \rho|I_\p = \prod_{\tau}(\omega_\tau\omega_{\tau\circ\sigma})^{d_\tau}(\prod_{\tau}\omega_\tau^{a_\tau} \oplus \prod_\tau\omega_\tau^{b_\tau}) $$ where $\sigma$ is the order $2$ automorphism of $k$, and we have $d_\tau\in\{0,\dots,p-1\}$, $a_\tau,b_\tau\in\{0,\dots,p\}$ and $\{0\}\subsetneq\{a_\tau,b_\tau\}$ for all $\tau$.

In papers about generalisations of Serre's conjecture, this form is used to define the set of Serre weights attached to the representation (the non-tame case is more complicated but I don't need it), and if I understand correctly the weights are $\bigotimes_{\tau}(\det^{d_\tau} \otimes \Sym^{\max(a_\tau,b_\tau)-1}\F_\p^2)\otimes_\tau\bar{\F}_p$ and sometimes some other weights. In the references I looked at, everything was formulated for $F$ totally real and for actual modular forms. Since the weight recipe is completely local, my naive expectation would be that it should remain the same in general. In addition, the direction I need is only that the Galois representations attached to cohomology with trivial coefficients have the shape above with $d_\tau = 0$ and $\max(a_\tau,b_\tau)=1$.

(One other less important point I am not sure about is whether this needs modification if $F$ is ramified at $\p$.)


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