# Arithmetic representations of fundamental groups I

@article{Litt2018ArithmeticRO,
title={Arithmetic representations of fundamental groups I},
author={Daniel Litt},
journal={Inventiones mathematicae},
year={2018},
volume={214},
pages={605-639}
}
• Daniel Litt
• Published 22 August 2017
• Mathematics
• Inventiones mathematicae
Let X be a normal algebraic variety over a finitely generated field k of characteristic zero, and let $$\ell$$ℓ be a prime. Say that a continuous $$\ell$$ℓ-adic representation $$\rho$$ρ of $$\pi _1^{\acute{\mathrm{e}}\text {t}}(X_{\bar{k}})$$π1e´t(Xk¯) is arithmetic if there exists a finite extension $$k'$$k′ of k, and a representation $$\tilde{\rho }$$ρ~ of $$\pi _1^{\acute{\mathrm{e}}\text {t}}(X_{k'})$$π1e´t(Xk′), with $$\rho$$ρ a subquotient of \tilde{\rho }|_{\pi _1(X_{\bar{k…
A note on images of Galois representations (with an application to a result of Litt).
• Mathematics
• 2018
Let $X$ be a variety (possibly non-complete or singular) over a finitely generated field $k$ of characteristic $0$. For a prime number $\ell$, let $\rho_\ell$ be the Galois representation on the
Arithmetic representations of fundamental groups, II: Finiteness
Let $X$ be a smooth curve over a finitely generated field $k$, and let $\ell$ be a prime different from the characteristic of $k$. We analyze the dynamics of the Galois action on the deformation
Level structure, arithmetic representations, and noncommutative Siegel linearization
• Mathematics
Journal für die reine und angewandte Mathematik (Crelles Journal)
• 2022
Abstract Let ℓ{\ell} be a prime, k a finitely generated field of characteristic different from ℓ{\ell}, and X a smooth geometrically connected curve over k. Say a semisimple representation of
Geometrically irreducible $p$-adic local systems are de Rham up to a twist
We prove that any geometrically irreducible Qp-local system on a smooth algebraic variety over a p-adic field K becomes de Rham after a twist by a character of the Galois group of K. In particular,
Mini-Workshop: Arithmetic Geometry and Symmetries around Galois and Fundamental Groups
• Mathematics
Oberwolfach Reports
• 2019
The geometric study of the absolute Galois group of the rational numbers has been a highly active research topic since the first milestones: Hilbert’s Irreducibility Theorem, Noether’s program,
Canonical representations of surface groups
• Mathematics
• 2022
Let Σg,n be an orientable surface of genus g with n punctures. We study actions of the mapping class group Modg,n of Σg,n via Hodge-theoretic and arithmetic techniques. We show that if ρ : π1(Σg,n)→
Semisimplicity and weight-monodromy for fundamental groups
• Mathematics
• 2019
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
Arithmetic subspaces of moduli spaces of rank one local systems
• Mathematics
Cambridge Journal of Mathematics
• 2020
We show that closed subsets of the character variety of a complex variety with negatively weighted homology, which are $p$-adically integral and Galois invariant, are motivic. Final version:
Survey on special subloci of the moduli spaces of local systems on complex varieties
It is a short report for the ICCM2019 Proceedings on recent results obtained with Michael Groechenig and Moritz Kerz concerning special subloci of the Betti moduli space of irreducible complex local

## References

SHOWING 1-10 OF 73 REFERENCES
Gonality of abstract modular curves in positive characteristic
• Mathematics
Compositio Mathematica
• 2016
Let $C$ be a smooth, separated and geometrically connected curve over a finitely generated field $k$ of characteristic $p\geqslant 0$ , $\unicode[STIX]{x1D702}$ the generic point of $C$ and
GALOIS ACTIONS ON FUNDAMENTAL GROUPS OF CURVES AND THE CYCLE $C-C^{-}$
• Mathematics
Journal of the Institute of Mathematics of Jussieu
• 2005
Suppose that $K$ is a subfield of $\mathbb{C}$ for which the $\ell$-adic cyclotomic character has infinite image. Suppose that $C$ is a curve of genus $g\geq3$ defined over $K$, and that $\xi$ is a
P-torsion monodromy representations of elliptic curves over geometric function fields
• Mathematics
• 2014
Given a complex quasiprojective curve $B$ and a non-isotrivial family $\mathcal{E}$ of elliptic curves over $B$, the $p$-torsion $\mathcal{E}[p]$ yields a monodromy representation
Gonality of modular curves in characteristic~$p$
Let $k$ be an algebraically closed field of characteristic $p$. Let $X(p^e;N)$ be the curve parameterizing elliptic curves with full level $N$ structure (where $p \nmid N$) and full level $p^e$ Igusa
Note on the Gonality of Abstract Modular Curves
Let S be a curve over an algebraically closed field k of characteristic $$p \geq 0$$. To any family of representations $$\rho= ({\rho }_{\mathcal{l}}\, :\ {\pi }_{1}(S) \rightarrow {\mbox{ On 3-Nilpotent Obstructions to π1 Sections for \( \mathbb{P}^{1}_\mathbb{Q}$$−{0,1, $$\infty$$}
We study which rational points of the Jacobian of ℙ k 1 − { 0, 1, ∞} can be lifted to sections of geometrically 3-nilpotent quotients of etale π1 over the absolute Galois group. This is equivalent to
The geometric torsion conjecture for abelian varieties with real multiplication
• Mathematics
Journal of Differential Geometry
• 2018
The geometric torsion conjecture asserts that the torsion part of the Mordell--Weil group of a family of abelian varieties over a complex quasiprojective curve is uniformly bounded in terms of the
INTEGRAL AND ADELIC ASPECTS OF THE MUMFORD–TATE CONJECTURE
• Mathematics
Journal of the Institute of Mathematics of Jussieu
• 2018
Let $Y$ be an abelian variety over a subfield $k\subset \mathbb{C}$ that is of finite type over $\mathbb{Q}$. We prove that if the Mumford–Tate conjecture for $Y$ is true, then also some refined
Assuming Vojta’s conjecture, and building on recent work of the authors, we prove that, for a fixed number field $K$ and a positive integer $g$ , there is an integer $m_{0}$ such that for any