Paris Diderot Sorbonne Université CNRS

Séminaire des thésards

A Deligne-Riemann-Roch isometry for modular curves

Mathieu DUTOUR - (IMJ-PRG)

mercredi 20 février 2019 à 18:00
Sophie Germain, salle 2015

Bâtiment Sophie Germain, av. de France
métro Bibliothèque F. Mitterrand, Paris 13e

In 1987, Deligne proved a type of Riemann-Roch theorem, which aims to relate geometric and arithmetic properties of compact Riemann surfaces endowed with smooth hermitian metrics.
When trying to apply this result to the case of modular curves, we find that there is a crucial hypothesis that is not satisfied : the Poincaré metric does not behave nicely and has singularities at some points.
The purpose of this talk is to present a method, called analytic surgery, which we can use to avoid these singularities and get a variation of Deligne’s results. Some unexpected applications stem from these considerations, such as explicit values of some derivatives of Selberg zeta functions.

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