The semistable modular lifting conjecture

Let $G_{\bold Q}=\operatorname{Gal}(\bar{\bold Q}/\bold Q)$, let $p$ be an odd prime, and let $\bar{\varepsilon}_p:G_{\bold Q}\to\bold F_p^\times$ be the character describing the action on the $p$-th roots of unity. For a semistable elliptic curve $E$ over $\bold Q$, let $\bar\rho_p:G_{\bold Q}\to\operatorname{GL}_2(\bold F_p)$ be the representation on $E[p]$; call such a representation modular when its Frobenius traces agree, outside finitely many primes, with the coefficients of an eigenform. Semistable modular lifting conjecture. Suppose $p$ is an odd prime and $E$ is a semistable elliptic curve over $\bold Q$ satisfying (a) $\bar\rho_p$ is irreducible and (b) there are an eigenform $f(z)=\sum_{n=1}^{\infty}a_ne^{2\pi inz}$ and a prime ideal $\lambda$ of its coefficient ring $\mathcal O_f$ such that $p\in\lambda$ and, for all but finitely many primes $q$,

$$a_q\equiv q+1-\#(E(\bold F_q))\pmod{\lambda}.$$

Then $E$ is modular. The conjecture is a modularity-lifting assertion and is presented as an ingredient in the route from Galois representations to Fermat's Last Theorem.