# A p-adic day in Exeter

University of Exeter

Harrison Building Har004

## Speakers

- Valentina Di Proietto (University Of Exeter)
- Netan Dogra (Imperial College)
- Johannes Nicaise (Imperial College)

## Programme

Let X be a smooth projective curve over the rationals of genus bigger than one. The Chabauty-Kim method attempts to locate the rational points of X inside its adelic points via a local-global obstruction coming from the unipotent fundamental group. In my talk I will describe some recent results in this area, with a focus on understanding local structure.

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**3:00-3:30** Coffee break

Berthelot's conjecture predicts that under a proper and smooth morphism of varieties in characteristic p, the higher direct images of an F-overconvergent isocrystal are F-overconvergent isocrystals. In a joint work with Fabio Tonini and Lei Zhang we prove that this is true for crystals up to isogeny. As an application we prove a Künneth formula for the crystalline fundamental group.

Let A be an abelian variety over a strictly henselian discretely valued field K. In his 1992 paper "Néron models and tame ramification", Edixhoven has constructed a filtration on the special fiber of the Néron model of A that measures the behaviour of the Néron model with respect to tamely ramified extensions of K. The filtration is indexed by rational numbers in [0,1], and if A is wildly ramified, it is an open problem whether the places where it jumps are always rational. I will explain how an interpretation of the filtration in terms of logarithmic geometry leads to explicit formulas for the jumps in the case where A is a Jacobian, which confirms in particular that they are rational. This is joint work with Dennis Eriksson and Lars Halvard Halle.