Séminaire de Mathématique
# Vologodsky and Coleman integration on curves with semi-stable reduction

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Amphitéâtre Léon Motchane (IHES)
### Amphitéâtre Léon Motchane

#### IHES

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

Description

Coleman and Vologodsky integration theories give canonical parallel transports for unipotent differential equations - Coleman on overconvergent spaces with good reduction and Vologodsky on algebraic varieties, both over p-adic fields. In Coleman's theory the transport is via a path invariant under the action of Frobenius while in Vologodsky's theory one adds a condition involving the monodromy operator. While both theories can be formulated in fairly similar terms, the precise relationship between them is a bit unclear.
In this talk, based on joint work in progress with Sarah Zerbes, I will describe some background on the two integration theories and I will describe the simplest non-trivial case - a holomorphic form on a curve with semi-stable reduction, where we can say what the relation should be. Time permitting I'll discuss possible applications to syntomic regulators.

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