Are all representations of the geometric étale fundamental group subquotients of representations of the arithmetic étale fundamental group?

Question

Let $X$ be a variety over a field $k$. The étale fundamental group of $X$ fits into the exact sequence: $$1 \to \pi_1^{\text{geom}}(X) \to \pi_1^{\text{arith}}(X) \to \text{Gal}(\overline{k}/k) \to 1,$$ where $\pi_1^{\text{geom}}(X) = \pi_1(X_{\overline{k}})$ is the geometric étale fundamental group, and $\pi_1^{\text{arith}}(X) = \pi_1(X)$ is the arithmetic étale fundamental group.

I am interested in understanding the relationship between the representations of these two groups. Specifically, is every representation of $\pi_1^{\text{geom}}(X)$ a subquotient of a representation of $\pi_1^{\text{arith}}(X)$?

This would be true, for example, by the Tannakian formalism, if the induced map on the associated algebraic envelopes is a closed embedding.

Any insights or references on this topic would be greatly appreciated.

Thank you!

Answer 1

This is very far from true. In fact, a representation extending to the arithmetic fundamental group is such a severe restriction that these satisfy strong finiteness results; see [Litt21].

An easier (and much older) constraint is that if $\rho \colon \pi_1(X_{\bar k}) \to \operatorname{GL}_n(\mathbf Q_\ell)$ extends to a representation of $\pi_1(X)$, then the action of monodromy around any puncture is quasi-unipotent [SGA7$_{\text{I}}$, Exp. I, Prop. 1.1]. You can see quite explicitly why this is true for $X = \mathbf G_{m,\mathbf F_q}$: the (tame) geometric fundamental group is $\hat{\mathbf Z}{}^{(p')}$ where $\operatorname{Gal}(\bar{\mathbf F}_q/\mathbf F_q) \cong \hat{\mathbf Z}$ acts by multiplication by $q$. Then the (tame) arithmetic fundamental group has presentation $\langle F,M\ |\ FMF^{-1} = M^q\rangle$, so $M$ is sent to a matrix that is conjugate to its $q$-th power. Thus the generalised eigenvalues are all $(q-1)$-st roots of unity.

To be very concrete, this implies that a 1-dimensional representation $M \mapsto a \in \mathbf Z_\ell^\times$ can appear as a subquotient of an arithmetic representation if and only if $a$ is a $(q-1)$-st root of unity. (Allowing finite extensions of $k$, it is still necessary that $a$ is a root of unity.) So for instance $a = p$ (which is a unit as $\ell \neq p$) can never appear.

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References.

[Litt21] D. Litt, Arithmetic representations of fundamental groups. II: Finiteness. Duke Math. J. 170.8 (2021), p. 1851-1897. ZBL1520.14041.

[SGA7$_{\text{I}}$] A. Grothendieck, M. Raynaud, D. S. Rim, Séminaire de Géométrie Algébrique Du Bois-Marie 1967–1969. Groupes de monodromie en géométrie algébrique (SGA 7$_{I}$). Lecture Notes in Mathematics 288. Springer-Verlag (1972). ZBL0237.00013.