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Let $C_1,\dots, C_n$ be smooth curves over $\overline{\mathbb F}_p$, not necessarily proper. Let $X$ be a subvariety of $C_1 \times \dots \times C_n$. I'm interested in the natural map:

$$ \pi_1^{ab}(X) \to \pi_1^{ab} (C_1 ) \times \dots \times \pi_1^{ab} (C_n)$$

and whether or not the image has finite index.

One possible explanation for infinite index comes from algebraic groups: Let $G$ be an abelian algebraic group of positive dimension and let $f_i: C_i \to G$ be a tuple of maps, not all constant. Consider the equation in the group law of $G$:

$$f_1(x)f_2(x) \dots f_n(x) = e$$

If all $x \in X$ satisfy this equation, then the map on $\pi_1$s has infinite index: Using the Lang isogeny, one can construct an infinite quotient of $\pi_1(G)$ on which Eckman-Hilton holds, so composition in the fundamental group is the same as composition in the group. If all $x\in X$ satisfy that relation in $G$, then all elements of $\pi_1(X)$ must satisfy the analogous relation in $\pi_1(G)$, forcing $\pi_1(X)$ to have infinite index.

Is the converse true?

That is, if the abelianization of $\pi_1$ does in fact have infinite index, then is it always explained by the presence in the ideal of $X$ of an equation coming from an algebraic group?

This is certainly true if we replace $\pi_1^{ab} (C_i)$ with its quotients corresponding to only those \'{e}tale covers with bounded ramification at the missing points. The bounded-wild-ramification fundamental group of $C$ over $\mathbb F_q$ can be expressed by geometric class field theory as the $\mathbb F_q$-points of the generalized Jacobian of $C$. If the image of $X$ inside the products of the generalized Jacobians of the $C_i$ is contained in a proper algebraic subgroup, then by taking $G$ to be the quotient by that subgroup we obtain a relation of this type. If not, then the map on groups of $\mathbb F_q$-points has finite index independent of $q$, and hence the map on fundamental groups has finite index.

Hence it is true in the proper case, or in characteristic $0$. But does unbounded wild ramification cause a problem?

(I'm actually interested in the index of the map on full $\pi_1$s, but an easy argument lets you reduce to the abelian case.)

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