The following is a question I've been asking people on and off for a few years, mostly out of idle curiosity, though I think it's pretty interesting. Since I've made more or less no progress, I figured I'd ask it here.

Let $C_1, C_2$ be smooth projective curves over $\overline{\mathbb{F}_q}$ of genus at least $2$. Do $C_1$ and $C_2$ necessarily have a finite etale cover in common?

In other words, does there exist a third smooth projective curve $D/\overline{\mathbb{F}_q}$, and finite etale morphisms $\pi_i: D\to C_i$?

Surely the answer is "no," but I haven't been able to prove it.

**Remarks**

- The answer is probably "yes" if one allows only requires that one of the $\pi_i$ be etale, as opposed to both of them. This is a conjecture of Bogomolov and Tschinkel, and is true if one of the $C_i$ is hyperelliptic and the characteristic of $\mathbb{F}_q$ is greater than $5$.
- By virtue of the above, there is no "abelian" obstruction to a positive answer to this question -- for any hyperelliptic curve $C$ over $\overline{\mathbb{F}_q}$ as above,
*every*Abelian variety over $\overline{\mathbb{F}_q}$ appears as an isogeny factor of the Jacobian of some finite etale cover of $C$. - The answer to this question is clearly "no" if $\overline{\mathbb{F}_q}$ is replaced by any uncountable field, by dimension considerations; in characteristic zero, it is also "no" for countable fields, by a result of Mochizuki (Theorem A of this paper).
- A positive answer would be a weak $p$-adic analogue of the Ehrenpreis conjecture; more generally, one can ask if two smooth projective curves of genus $>2$ over $\mathbb{Z}_p^{un}$ have finite etale covers which are isomorphic mod $p^n$. Perhaps a better thing to do would be to ask only that the covers be etale over the generic fiber, and to allow ramified extensions of the base ring. I have no idea whether this weaker version should be true or not; it would imply some kind of $p$-adic uniformization theorem for curves of good reduction. (This is actually the version of the question I am most interested in, but it is clearly harder.)
- This problem admits an anabelian rephrasing; given $C_1, C_2$, smooth, projective, geometrically connected curves over $\mathbb{F}_q$ of genus at least $2$, one can ask if $\pi_1^{et}(C_1), \pi_1^{et}(C_2)$ have finite index subgroups in common.

9more comments