Question

I think this may be a silly question, but here goes. Let $\rho:\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})\to \mathrm{GL}_2(\overline{\mathbf{F}_p})$ be a representation; say $\rho$ is of S-type if it is continuous, unramified almost everywhere, and the determinant of complex conjugation is $-1$. Serre's conjecture, now a theorem of Khare-Wintenberger, states that every $\rho$ of S-type arises from some modular form $f=\sum a_n e(nz)$ in the sense that $\mathrm{tr}\rho(\mathrm{Frob}_l)=a_l\;( \mathrm{mod}\;p)$ for (almost all) primes $l$.

Question: Are S-type representations $\rho:\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})\to \mathrm{GL}_2(\mathbf{Z}/p^n\mathbf{Z})$ for $n\geq 2$ also expected/known to be modular?

Answer 1

In what sense? If you mean "come from the reduction of $\rho_f$ for some Hecke eigenform $f$'', no, they are not.

If you mean "come from the reduction of $\rho$ where $\rho:G_{\mathbb Q} \to GL_2(\mathbb T)$ is the Galois rep'n attached to the Hecke algebra $\mathbb T$ acting on modular forms of some sufficiently large level, then the answer is known to be yes in most cases (i.e. with comparitively minor technical restrictions on $\rho$). This is the content of so-called big $R = $ big $\mathbb T$ theorems, due to Gouvea--Mazur, Boeckle, and others (combined with Serre's conjecture to know that $\overline{\rho}$ is modular).