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For $p$ a prime number and $q$ a non trivial $p$th root of 1, we present themain steps of the construction of a local $q$-deformation of the "Simpsoncorrespondence in characteristic $p$" found by Ogus and Vologodsky in 2005. Theconstruction is based on the Morita-equivalence between a ring of $q$-twisteddifferential operators and its center. We also explain the expected relationsbetween this construction and those recently done by Bhatt and Scholze. For thesake of readability, we limit ourselves to the case of dimension 1.

Sur une $q$-d\'eformation locale de la th\'eorie de Hodge non-ab\'elienne en caract\'eristique positive