Calcul effectif sur les courbes hyperelliptiques à réduction semi-stable

Abstract : In this thesis we study the weight filtration on the De Rham cohomology of an hyperelliptic curve C defined over a finite extension of Qp and with semi-stable reduction. The goal is to provide algorithms computing explicitly, given an equation of C, the basis of the weight filtration’s spaces as well as the matrix of the Poincaré pairing. In the first chapter we introduce tools related to the algebraic De Rham cohomology of the hyperelliptic curve. We build a suitable basis of the De Rham cohomology of C, we establish explicit formulae for the cup-product and the trace, and we give an algorithm computing the matrix of the Poincaré pairing. The second chapter is dedicated to the explicit description of the morphism induced by the inclusion of the tube of a double point on the cohomology spaces. It is the main ingredient that allows us to describe the weight filtration on the De Rham cohomology of C. To achieve that, we use the framework of the Berkovitch analytical geometry. We introduce and then we develop the notion of standard residually singular points and the notion of apparent form of the curve’s equation. In the third and last chapter, we synthesize all the results and we complete the description of the weight filtration. Finally, we give the algorithms that compute the basis of Fil0 and Fil1. For each of our algorithm, we propose a sage implementation and concrete examples on genus one and two curves.
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Yvan Ziegler. Calcul effectif sur les courbes hyperelliptiques à réduction semi-stable. Géométrie algébrique [math.AG]. Université Rennes 1, 2019. Français. ⟨NNT : 2019REN1S023⟩. ⟨tel-02286074⟩

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