modular forms - Free subquotient of Galois representations coming from Hida theory

$newcommandT{mathbf{T}_{mathfrak{m}}}$
$newcommandQ{mathbf{Q}}$
$newcommandm{mathfrak{m}}$
$newcommandF{mathbf{F}}$
$newcommandFrob{mathrm{Frob}}$
$newcommandrhobar{overline{rho}}$
$newcommandeps{epsilon}$

First, as Professor Emerton mentions, the construction of $L^{+}$ you gave
is not necessarily free over $T$. Thus, I will interpret your question
as asking the following: does there exist an exact sequence:

$$0 rightarrow L^{+} rightarrow (T)^2 rightarrow L^{-} rightarrow 0$$
of $T[G_{Q_p}]$-modules where $L^{+}$ and $L^{-}$ are free
$T$-modules of rank one.

( Edit Perhaps this extra remark might be useful.
Suppose that $L = (T)^2$ admits a free rank one quotient
$L^{-}$. Since $L^{-}$ is free, it admits a section
$L^{-} rightarrow L$, and hence the kernel
$L^{+}$ of $L rightarrow L^{-}$ is also free. Thus
the existence of a free rank one quotient asked
for in the question is equivalent
to the existence of the exact sequence above.)

The answer to this question, in general, is no. The following argument
is implicitly contained in papers of Wiese on the failure of multiplicity
one and weight one forms.

The action of $G_{Q_p}$ on $L^{+}$ is unramified and so acts
via $G_{mathbf{F}_p}$. Thus $Frob_p$ acts on a basis vector
as multiplication by some element of $T$. Since $T$ is determined
by its action on classical eigenforms,
one may identify this element with the Hecke operator $U$. In particular,
$U in T$ (it wasn't clear whether your $T$ included $U$ or not).

The exact sequence remains exact after tensoring with $T/m$,
for dimension reasons. It follows that the sequence is split as a sequence
of $T$-modules. Hence it remains exact after quotienting out by
any ideal of $T$.

Suppose that $rhobar: G_{Q} rightarrow mathrm{GL}_2(F_p)$ is
irreducible and modular (mod-$p$) of weight $1$. Suppose, moreover, that
$rhobar(Frob_p)$ acts by a scalar $lambda$.
Associated to $rhobar$ is a mod-$p$ weight $1$ form
$f = sum a_n q^n in F_p[[q]]$. If $A$ is the Hasse invariant, then
then $Af$ and $f^p$ are both mod-$p$ modular forms of weight $p$. One can check
that all elements of the $F_p$-vector space ${Af,f^p}$ are eigenvalues for all the Hecke operators
$T_l$ for $(l,p) = 1$, but the operator $T$ (and so $U$, which is the
same as $T$ in weight $> 1$) satisfies
$(U - lambda)^2 = 0$ but does not act by a scalar. Since $U$ acts
invertibly on this vector space, it gives rise to a surjective map:
$$T rightarrow F_p[eps]/eps^2,$$
where the image of $T_l$ lands in $F_p$ for all $(l,p) = 1$, but $U$
does not act by a scalar. Let $I$ be the kernel.

The Galois representation on $(T)^2/I simeq (F_p[eps]/eps^2)^2$
is equal to $rhobar oplus rhobar$. This follows from a result of Boston-Lenstra-Ribet,
since $T_l$ is acting by a scalar for each $(l,p) = 1$. It follows, by assumption, that
the action of $G_{Q_p}$ on $L^{+}/I L^{+} simeq T/I$ must also be trivial,
because this is a sub-representation of $rhobar oplus rhobar$. On the other hand, as we have seen, the action of Frobenius on $L^{+}$ and thus $L^{+}/I L^{+} = T/I$ is given
by $U$, which is acting non-trivially $T/I$ by the construction of $I$. This is a contradiction.

Such representations $rhobar$ exist (for example, with $p = 2$, and level $Gamma_0(431)$) as mentioned
in Professor Emerton's answer.

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