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arXiv · 1912.02167

Semisimplicity and weight-monodromy for fundamental groups

L. Alexander Betts·Daniel Litt
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Abstract

Let X be a smooth, geometrically connected variety over a p-adic local field. We show that the pro-unipotent fundamental group of X (in both the etale and crystalline settings) satisfies the weight-monodromy conjecture, following Vologodsky. We deduce (in the etale setting) that Frobenii act semisimply on the Lie algebra of the pro-unipotent fundamental group of X, and (in the crystalline setting) that the same is true for a K-linear power of the crystalline Frobenius. We give applications to the representability and geometry of the Selmer varieties appearing in the Chabauty-Kim program, even in cases of bad reduction.

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L. Alexander Betts, Daniel Litt. 2019-12-04. Semisimplicity and weight-monodromy for fundamental groups. https://arxiv.org/abs/1912.02167

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