Cours de l'IHES 2016-2017
# On the Arakelov theory of arithmetic surfaces (4/4)

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Amphithéâtre Léon Motchane (IHES)
### Amphithéâtre Léon Motchane

#### IHES

Le Bois-Marie
35, route de Chartres
91440 Bures-sur-Yvette

Description

Let *X* be a semi-stable arithmetic surface of genus at least two and *$\omega$* the relative dualizing sheaf of *X*, equipped with the Arakelov metric. Parshin and Moret-Bailly have conjectured an upper bound for the arithmetic self-intersection of $*\omega*$. They proved that a weak form of the *abc* conjecture follows from this inequality. We shall discuss a way of making their conjecture more precise in order that it implies the full *abc* conjecture (a proof of which has been announced by Mochizuki).

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