#### Vol. 16, No. 4, 2022

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Galois representations attached to elliptic curves with complex multiplication

### Álvaro Lozano-Robledo

Vol. 16 (2022), No. 4, 777–837
##### Abstract

We give an explicit classification of the possible $p$-adic Galois representations that are attached to elliptic curves $E$ with CM defined over $ℚ\left(j\left(E\right)\right)$. More precisely, let $K$ be an imaginary quadratic field, and let ${\mathsc{𝒪}}_{K,f}$ be an order in $K$ of conductor $f\ge 1$. Let $E$ be an elliptic curve with CM by ${\mathsc{𝒪}}_{K,f}$, such that $E$ is defined by a model over $ℚ\left(j\left(E\right)\right)$. Let $p\ge 2$ be a prime, let ${G}_{ℚ\left(j\left(E\right)\right)}$ be the absolute Galois group of $ℚ\left(j\left(E\right)\right)$, and let ${\rho }_{E,{p}^{\infty }}:{G}_{ℚ\left(j\left(E\right)\right)}\to \mathrm{GL}\left(2,{ℤ}_{p}\right)$ be the Galois representation associated to the Galois action on the Tate module ${T}_{p}\left(E\right)$. The goal is then to describe, explicitly, the groups of $\mathrm{GL}\left(2,{ℤ}_{p}\right)$ that can occur as images of ${\rho }_{E,{p}^{\infty }}$, up to conjugation, for an arbitrary order ${\mathsc{𝒪}}_{K,f}$.

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