Finiteness and bounds for elliptic curves realizing a given galois representation

Question

Let $\rho: \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to \text{GL}_2(\mathbb{Z}_p)$ be a continuous, irreducible Galois representation. Consider the set $\mathcal{L}_\rho$ of all elliptic curves $E/\mathbb{Q}$ such that the Galois representation $\rho_{E,p}: \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to \text{GL}_2(\mathbb{Z}_p)$ on the $p$-adic Tate module of $E$ is isomorphic to $\rho$.

Under what conditions on $\rho$ is the set $\mathcal{L}_\rho$ finite? Can one give an effective upper bound on the cardinality of $\mathcal{L}_\rho$ in terms of the invariants of $\rho$, such as its conductor and the weight of the associated modular form (if it exists)?

Answer 1

The set $\mathcal{L}_{\rho}$ is either empty, or a singleton finite. This follows from Faltings' isogeny theorem, which states that for any two elliptic curves (or, more generally, abelian varieties) $E, E'$ over a number field $K$, we have $$ Hom_{G_K}(T_p E, T_p E') = Hom_K(E, E') \otimes_{\mathbb{Z}}\mathbb{Z}_p. $$ For more information on Faltings' theorem see here.

EDIT. Sorry, I was too hasty (shouldn't answer MO questions before breakfast!); if $T_pE$ and $T_pE'$ are isomorphic, Faltings' theorem only shows that there is an isogeny of prime-to-$p$ degree, not an isomorphism. But the isogeny class of an elliptic curve over a number field is known to be finite.

Possibly the question you meant to ask is to fix a mod $p$ representation $\bar{\rho}$, and ask for all $E$ with $E[p] \cong \bar{\rho}$. This set may be empty, nonempty and finite, or infinite, depending on $p$ and $\bar{\rho}$; it is parametrised by a modular curve, which is a twisted form of the modular curve of level $\Gamma(p)$. Studying these sets plays a crucial role in modularity theorems (and hence, in particular, in the proof of Fermat's Last Theorem).