It is a theorem of Kim-Tamagawa that the Q -pro-unipotent Kummer map associated to a smooth projective curve Y over a finite extension of Qp is locally constant when = p. The present paper establishes two generalisations of this result. Firstly, we extend the Kim-Tamagawa Theorem to the case that Y is a smooth variety of any dimension. Secondly, we formulate and prove the analogue of the Kim-Tamagawa Theorem in the case = p, again in arbitrary dimension. In the course of proving the latter, we give a proof of an étale-de Rham comparison theorem for pro-unipotent fundamental groupoids using methods of Scholze and Diao-Lan-Liu-Zhu. This extends the comparison theorem proved by Vologodsky for certain truncations of the fundamental groupoids. Contents 1. Introduction 1 2. Local constancy of the Q -pro-unipotent Kummer map 5 3. The unipotent Tannakian formalism 6 4. The unipotent Riemann-Hilbert fundamental groupoid 11 5. Comparison between étale and de Rham fundamental groupoids 17 6. Horizontal de Rham local systems over polydiscs 25 7. Local constancy of the Q p -pro-unipotent Kummer map mod H 1 e 29 References 30
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