Le théorème de concentration et la formule des points fixes de Lefschetz en géométrie d’Arakelov

Abstract : In the nineties of the last century, R. W. Thomason proved a concentrationtheorem for the algebraic equivariant K-theory on the schemes which are endowed withan action of a diagonalisable group scheme G. As usual, such a concentration theoreminduces a fixed point formula of Lefschetz type which can be used to calculate theequivariant Euler-Poincaré characteristic of a coherent G-sheaf on a proper G-schemein terms of a characteristic on the fixed point subscheme. It is the aim of this thesis togeneralize R. W. Thomason’s results to the context of Arakelov geometry. In this work,we consider the arithmetic schemes in the sense of Gillet-Soulé and we first prove anarithmetic analogue of the concentration theorem for the arithmetic schemes endowedwith an action of the diagonalisable group scheme associated to Z/nZ. The proof is acombination of the algebraic concentration theorem and some analytic arguments. Inthe last chapter, we formulate and prove two kinds of arithmetic Lefschetz formulae.These two formulae give a positive answer to two conjectures made by K. Köhler, V.Maillot and D. Rössler.
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Shun Tang. Le théorème de concentration et la formule des points fixes de Lefschetz en géométrie d’Arakelov. Mathématiques générales [math.GM]. Université Paris Sud - Paris XI, 2011. Français. ⟨NNT : 2011PA112015⟩. ⟨tel-00574296⟩

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