Arithmétique et D-modules

Abstract : This text is a survey of my research articles. It consists of two independent parts. In the first we present the results obtained in collaboration with T. Abe and published mainly in the article "Product formula for $p$-adic epsilon factors". Let $X$ be a proper and smooth curve over a finite field of characteristic $p$. This formula describes the constants appearing in the functional equations of $L$-functions for rigid cohomology of $X$, as a product of local invariants (the epsilon factors) at closed points of $X$. It is the counterpart in rigid cohomology of the Deligne-Laumon formula for epsilon factors in $\ell$-adic étale cohomology. We give an introduction to the context and to the main tools intervening in the proof: the stationary phase formula for arithmetic $D$-modules and the $p$-adic microlocal analysis. We end this part with a theorem describing the Frobenius of Fourier--Huyghe transform and an application to Gross-Koblitz formula. In the second part we present a result in collaboration with A. Iovita published in the article "On the continuity of the finite Bloch--Kato cohomology".
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Habilitation à diriger des recherches
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Submitted on : Friday, November 17, 2017 - 3:54:21 PM
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Adriano Marmora. Arithmétique et D-modules. Théorie des nombres [math.NT]. IRMA, Université de Strasbourg, 2017. ⟨tel-01628170v2⟩

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