# 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} }

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…

## 9 Citations

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

- MathematicsDuke Mathematical Journal
- 2021

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

- MathematicsJournal 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

- Mathematics
- 2020

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

- MathematicsOberwolfach 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

- MathematicsCambridge 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

- Mathematics
- 2019

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

GALOIS ACTIONS ON FUNDAMENTAL GROUPS OF CURVES AND THE CYCLE $C-C^{-}$

- MathematicsJournal 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~$p$

- Mathematics
- 2007

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

- Mathematics
- 2012

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, \(\infty\)}

- Mathematics
- 2012

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

- MathematicsJournal 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

- MathematicsJournal 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…

On ℓ -adic Iterated Integrals V: Linear Independence, Properties of ℓ -adic Polylogarithms, ℓ -adic Sheaves

- Mathematics
- 2012

In a series of papers we have introduced and studied l-adic polylogarithms and l-adic iterated integrals which are analogues of the classical complex polylogarithms and iterated integrals in l-adic…

Level structures on abelian varieties and Vojta’s conjecture

- MathematicsCompositio Mathematica
- 2017

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…

Uniform boundedness of level structures on abelian varieties over complex function fields

- Mathematics
- 2006

Let X = Ω/Γ be a smooth quotient of a bounded symmetric domain Ω by an arithmetic subgroup . We prove the following generalization of Nadel's result: for any non-negative integer g, there exists a…