Serre's modularity conjecture

Fix a prime $\ell$ and let $G_{\mathbb{Q}}=\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ be the absolute Galois group of $\mathbb{Q}$. Let $F=\mathbb{F}_{\ell^{r}}$ be a finite field and let

$$\rho\colon G_{\mathbb{Q}}\longrightarrow \mathrm{GL}_{2}(F)$$

be a continuous, absolutely irreducible representation which is odd, i.e. $\det\rho(c)=-1$ for $c\in G_{\mathbb{Q}}$ the image of complex conjugation under some embedding $\overline{\mathbb{Q}}\hookrightarrow\mathbb{C}$.

Recall the representations attached to modular forms: if

$$f=q+a_{2}q^{2}+a_{3}q^{3}+\cdots$$

is a normalized cuspidal Hecke eigenform of level $N\geq 1$, weight $k\geq 2$ and nebentype character $\chi$ modulo $N$, then there are a finite extension of $\mathbb{Q}_{\ell}$ with ring of integers $\mathcal{O}$, maximal ideal $\lambda$ and residue field $\mathcal{O}/\lambda$, an embedding of the field generated by the $a_{n}$ into the fraction field of $\mathcal{O}$, and a continuous representation

$$\rho_{f}\colon G_{\mathbb{Q}}\longrightarrow \mathrm{GL}_{2}(\mathcal{O})$$

characterized up to isomorphism by

$$\operatorname{Trace}\bigl(\rho_{f}(\operatorname{Frob}_{p})\bigr)=a_{p}\qquad\text{for all primes } p\nmid N\ell ,$$

where $\operatorname{Frob}_{p}$ denotes a Frobenius element at $p$. Write $\overline{\rho}_{f}$ for the semisimplification of the reduction of $\rho_{f}$ modulo $\lambda$, a representation $G_{\mathbb{Q}}\to \mathrm{GL}_{2}(\mathcal{O}/\lambda)$.

Then for every such $\rho$ there exist $N$, $k$, $\chi$ and a normalized cuspidal Hecke eigenform $f$ of level $N$, weight $k$ and nebentype $\chi$, together with embeddings of $F$ and of $\mathcal{O}/\lambda$ into $\overline{\mathbb{F}}_{\ell}$, such that

$$\rho\otimes_{F}\overline{\mathbb{F}}_{\ell}\;\cong\;\overline{\rho}_{f}\otimes_{\mathcal{O}/\lambda}\overline{\mathbb{F}}_{\ell};$$

equivalently,

$$\operatorname{Trace}\bigl(\rho(\operatorname{Frob}_{p})\bigr)\equiv a_{p} \pmod{\lambda}\qquad\text{for all primes } p\nmid N\ell .$$

Moreover such an $f$ exists with $N=N(\rho)$ equal to the Artin conductor of $\rho$ with the power of $\ell$ removed, and with $k=k(\rho)$ the weight determined by the restriction of $\rho$ to a decomposition group at $\ell$ by Serre's recipe.