This text is a summary of Jean-Pierre Serre's work on the Galois properties of finite order points of elliptic curves. The goal is to show that the Galois groups associated with these points are as large as possible. The methods used by André Weil are effective when restricting to points of order $ l^n $, where $ l $ is a fixed prime, but become insufficient when varying $ l $. The author focuses on the variation of $ l $. Let $ K $ be a number field, $ \overline{K} $ its algebraic closure, and $ G $ the Galois group of $ \overline{K} $ over $ K $. Let $ E $ be an elliptic curve over $ K $. The Galois group $ G $ acts naturally on the group $ E(\overline{K}) $ of $ \overline{K} $-rational points of $ E $. For each integer $ n \geq 1 $, $ E_n $ is the subgroup of $ E(\overline{K}) $ consisting of points of order dividing $ n $, and $ G $ acts on $ E_n $ via a homomorphism $ \varphi_n: G \to \mathrm{Aut}(E_n) \simeq \mathrm{GL}_2(\mathbb{Z}/n\mathbb{Z}) $. The author considers the subgroup $ E_\infty $ of $ E(\overline{K}) $ consisting of all torsion points, which is the union of all $ E_n $. The automorphism group of $ E_\infty $ is the projective limit of the automorphism groups of $ E_n $, and is isomorphic to $ \mathrm{GL}_2(\widehat{\mathbb{Z}}) $, where $ \widehat{\mathbb{Z}} $ is the profinite completion of $ \mathbb{Z} $. The action of $ G $ on $ E_\infty $ defines a continuous homomorphism $ \varphi_\infty: G \to \mathrm{Aut}(E_\infty) $. The main result is that $ \varphi_\infty(G) $ is a finite index subgroup of $ \mathrm{Aut}(E_\infty) $. This is equivalent to saying that $ \varphi_\infty(G) $ is open in $ \mathrm{Aut}(E_\infty) $, i.e., that it contains all automorphisms of $ E_\infty $ that act trivially on $ E_m $ for some $ m \geq 1 $. The author then considers the $ l $-primary component $ E_{l^\infty} $ of $ E_\infty $, which is the union of $ E_{l^n} $ for $ n \geq 1 $. Its automorphismThis text is a summary of Jean-Pierre Serre's work on the Galois properties of finite order points of elliptic curves. The goal is to show that the Galois groups associated with these points are as large as possible. The methods used by André Weil are effective when restricting to points of order $ l^n $, where $ l $ is a fixed prime, but become insufficient when varying $ l $. The author focuses on the variation of $ l $. Let $ K $ be a number field, $ \overline{K} $ its algebraic closure, and $ G $ the Galois group of $ \overline{K} $ over $ K $. Let $ E $ be an elliptic curve over $ K $. The Galois group $ G $ acts naturally on the group $ E(\overline{K}) $ of $ \overline{K} $-rational points of $ E $. For each integer $ n \geq 1 $, $ E_n $ is the subgroup of $ E(\overline{K}) $ consisting of points of order dividing $ n $, and $ G $ acts on $ E_n $ via a homomorphism $ \varphi_n: G \to \mathrm{Aut}(E_n) \simeq \mathrm{GL}_2(\mathbb{Z}/n\mathbb{Z}) $. The author considers the subgroup $ E_\infty $ of $ E(\overline{K}) $ consisting of all torsion points, which is the union of all $ E_n $. The automorphism group of $ E_\infty $ is the projective limit of the automorphism groups of $ E_n $, and is isomorphic to $ \mathrm{GL}_2(\widehat{\mathbb{Z}}) $, where $ \widehat{\mathbb{Z}} $ is the profinite completion of $ \mathbb{Z} $. The action of $ G $ on $ E_\infty $ defines a continuous homomorphism $ \varphi_\infty: G \to \mathrm{Aut}(E_\infty) $. The main result is that $ \varphi_\infty(G) $ is a finite index subgroup of $ \mathrm{Aut}(E_\infty) $. This is equivalent to saying that $ \varphi_\infty(G) $ is open in $ \mathrm{Aut}(E_\infty) $, i.e., that it contains all automorphisms of $ E_\infty $ that act trivially on $ E_m $ for some $ m \geq 1 $. The author then considers the $ l $-primary component $ E_{l^\infty} $ of $ E_\infty $, which is the union of $ E_{l^n} $ for $ n \geq 1 $. Its automorphism