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Motivic Serre invariants of curves

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Abstract

In this article, we generalize the theory of motivic integration on formal schemes topologically of finite type and the notion of motivic Serre invariant, to a relative point of view. We compute the relative motivic Serre invariant for curves defined over the field of fractions of a complete discrete valuation ring R of equicharacteristic zero. One aim of this study is to understand the behavior of motivic Serre invariants under ramified extension of the ring R. Thanks to our constructions, we obtain, in particular, an expression for the generating power series, whose coefficients are the motivic Serre invariant associated to a curve, computed on a tower of ramified extensions of R. We give an interpretation of this series in terms of the motivic zeta function of Denef and Loeser.

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Authors and Affiliations

  1. Université Lille 1, Laboratoire Painlevé, CNRS - UMR 8524, Cité Scientifique, 59655, Villeneuve d’Ascq Cédex, France

    Johannes Nicaise

  2. Université Bordeaux 1, IMB, Laboratoire A2X, 351 cours de la libération, 33405, Talence cedex, France

    Julien Sebag

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  1. Johannes Nicaise
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  2. Julien Sebag
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Correspondence to Julien Sebag.

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Nicaise, J., Sebag, J. Motivic Serre invariants of curves. manuscripta math. 123, 105–132 (2007). https://doi.org/10.1007/s00229-007-0088-0

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  • DOI: https://doi.org/10.1007/s00229-007-0088-0

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