# $p$-adic representations of the fundamental group of a smooth proper curve over a finite field

This question is very general. Let $C$ be a smooth and proper curve over a finite field ${\bf F}_p$. Are there any general results or conjectures on continuous non abelian representations $$\pi_1(C)\to {\rm GL}_n({\bf Q}_p)$$ or more restrictively on continuous non abelian representations $$\pi_1(C)\to {\rm GL}_n({\bf Z}_p)$$ (for some choice of base-point for $\pi_1$)? Does the Langlands programme over function fields of positive characteristic (eg Lafforgue's work) say anything about this? I know very little about automorphic forms so please forgive my ignorance if there is a vast theory behind this. I stress that I am interested in $p$-adic (not $l$-adic, where $l\not=p$) representations.

• arxiv.org/abs/1310.0528 – nfdc23 Jul 26 '16 at 18:23
• To be more precise: 4.2.2 in that arxiv link relates automorphic representations to certain constructions with isocrystals, and 2.4.14-2.4.19 develops isocrystal versions of fundamental groups and global Weil groups. The framework of isocrystals (and variants) provides the "right" coefficients for $p$-adic cohomology theories in characteristic $p$. A $p$-adic representation of the fundamental group gives rise to a very special type of isocrystal, but that is a topic separate from the automorphic context); see section 2 of arxiv.org/pdf/1511.02884v4.pdf and its references in [Kat73]. – nfdc23 Jul 27 '16 at 13:51
• $@$nfdc23: thank you so much for all these references! I shall have a close look and possibly get back to you, if that is alright. – Damian Rössler Jul 27 '16 at 17:53
• In particular, if you are only interested in $p$-adic representations of etale the fundamental group and not the larger crystalline fundamental group, it looks like nfdc23 says that you should only concern yourself with representations of finite order. Unfortunately, there is not any nice automorphic way to distinguish automorphic representations corresponding to crystalline or $\ell$-adic representations of finite order from from general automorphic representations. – Will Sawin Aug 2 '16 at 18:43
• @Will Savin (somewhat belatedly): thank you for your comments! – Damian Rössler Aug 15 '16 at 14:47