Hidden structures on semistable curves.(English)Zbl 1251.11047

Berger, Laurent (ed.) et al., Représentations $$p$$-adiques de groupes $$p$$-adiques III: Méthodes globales et géométriques. Paris: Société Mathématique de France (ISBN 978-2-85629-282-2/pbk). Astérisque 331, 179-254 (2010).
Summary: Let $$V$$ be the ring of integers of a finite extension of $$\mathbb Q_p$$ and let $$X$$ be a proper curve over $$V$$ with semistable special fiber and smooth generic fiber. In this article we explicitly describe the Frobenius and monodromy operators on the log crystalline cohomology of $$X$$ with values in a regular $$\log F$$-isocrystal in terms of $$p$$-adic integration. We have a version for open curves and as an application we prove that two differently defined $$\mathcal L$$-invariants, attached to a split multiplicative at $$p$$ new elliptic eigenform, are equal.
For the entire collection see [Zbl 1192.11002].

MSC:

 11G20 Curves over finite and local fields 11G25 Varieties over finite and local fields 11F11 Holomorphic modular forms of integral weight
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