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2 Added a couple of dollar signs.

Let $\bar{\rho} : Gal(\bar{\mathbb{Q}}/ \mathbb{Q} ) \rightarrow GL_2(\bar{\mathbb{F}}_p)$ be an odd, irreducible Galois representation mod p $p$ which is unramified outside $S$, where $S$ is a finite set of primes which contains $p$. Fix an integer $k \geq 2$ and a local Galois representation $\rho _p ' : Gal(\bar{\mathbb{Q}} _p/ \mathbb{Q} _p ) \rightarrow GL_2(\bar{\mathbb{Q}} _p)$

Question: Is there a way to compute precisely a number (which is finite) of modular lifts $\rho : Gal(\bar{\mathbb{Q}}/ \mathbb{Q} ) \rightarrow GL_2(\bar{\mathbb{Q}}_p)$ of $\bar{\rho}$, such that

1) $\rho _{|Gal(\bar{\mathbb{Q}} _p/ \mathbb{Q} _p )} \simeq \rho _p '$

2) $\rho$ comes from a modular form of weight $k$.

3) $\rho$ is unramified outside $S$.

I'd like to understand it even in the simplest (?) case, when $S =$ {$p$ }, $k=2$ so that $\bar{\rho}$ actually comes from a reduction of some level 1 form (after Serre's conjecture)

Is it true that in this case, a modular lift $\rho$ will be unique (if it exists)?

1

# Number of modular lifts with prescribed parameters

Let $\bar{\rho} : Gal(\bar{\mathbb{Q}}/ \mathbb{Q} ) \rightarrow GL_2(\bar{\mathbb{F}}_p)$ be an odd, irreducible Galois representation mod p which is unramified outside $S$, where $S$ is a finite set of primes which contains $p$. Fix an integer $k \geq 2$ and a local Galois representation $\rho _p ' : Gal(\bar{\mathbb{Q}} _p/ \mathbb{Q} _p ) \rightarrow GL_2(\bar{\mathbb{Q}} _p)$

Question: Is there a way to compute precisely a number (which is finite) of modular lifts $\rho : Gal(\bar{\mathbb{Q}}/ \mathbb{Q} ) \rightarrow GL_2(\bar{\mathbb{Q}}_p)$ of $\bar{\rho}$, such that

1) $\rho _{|Gal(\bar{\mathbb{Q}} _p/ \mathbb{Q} _p )} \simeq \rho _p '$

2) $\rho$ comes from a modular form of weight $k$.

3) $\rho$ is unramified outside $S$.

I'd like to understand it even in the simplest (?) case, when $S =$ {$p$ }, $k=2$ so that $\bar{\rho}$ actually comes from a reduction of some level 1 form (after Serre's conjecture)

Is it true that in this case, a modular lift $\rho$ will be unique (if it exists)?