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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

  1. 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 $5$.
  2. 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 $C$.
  3. 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).
  4. 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.)
  5. 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.
Daniel Litt
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  • What happens for modular curves? Do we pick out more correspondences than the Hecke operators in char p? $X(mn)$ covers $X(m)$ and $X(n)$ but the maps are ramified at the cusps. You can apply Hecke operators on both sides to get more correspondences and should be feasible to check whether they are etale. – Felipe Voloch May 09 '17 at 20:35
  • @FelipeVoloch: Good question! Of course modular curves have many etale covers which are not modular (by the failure of the congruence subgroup problem...). I suggest thinking about compact Shimura curves for a source of curves with lots of etale covers in common. In fact one natural subquestion is: is every smooth proper curve over $\overline{\mathbb{F}_q}$ the reduction of a Shimura curve? I think Jie Xia has thought about these sorts of questions... – Daniel Litt May 09 '17 at 20:39
  • Yes, modular curves have lots of other etale covers but, in char zero at least, correspondences between two modular curves have to come from Hecke ops. So that could be a source of a counterexample. – Felipe Voloch May 09 '17 at 20:49
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    @FelipeVoloch: Ah, I understand. I think this is no longer true in positive characteristic, at least if interpreted literally (consider e.g. correspondence coming from the Igusa curve). I don't know if there's a correct statement; Raju Krishnamoorthy considered this question in his (very nice) thesis, but it seems to be open. – Daniel Litt May 09 '17 at 20:55
  • @FelipeVoloch This would rather an example of a more precise conjecture which could be of much appeal to the André-Oort-Zilber-Pink afficionados. – ACL May 09 '17 at 20:56
  • @FelipeVoloch Any Hecke correspondence of compactified modular curves will be ramified at some cusps (9.2, 9.3 of arXiv:1704.00335.) Can you provide a reference for "correspondence between modular curves" come from Hecke ops? – Raju May 09 '17 at 22:05
  • @Raju Not sure what a ref would be but here is an argument. If $Y$ maps to $X(n),X(m)$ then pull back the maps via $X(nm) \to X(m)$ and $X(n)$ to get two curves $Z,W$ say mapping to both $Y$ and $X(mn)$, taking the fiber product of $Z,W$ over $Y$ gives a curve mapping to $X(mn)$ in two ways, so an element of $End(Jac(X(nm)))$. The latter is the Hecke algebra of level $mn$. – Felipe Voloch May 09 '17 at 22:44
  • @FelipeVoloch: Great, so the failure in positive characteristic comes from the fact that here $\text{End}(\text{Jac}(X(nm)))$ is a CM algebra, which is typically not the case in characteristic zero. – Daniel Litt May 09 '17 at 22:46
  • @FelipeVoloch: Sorry for the billion comments, but would you mind saying a bit more about that argument? $\text{Pic}(X(nm)^2)=\mathbb{Z}^2\oplus End(Jac(X(nm)))$, so really the correspondence gives us an element of that group. I see no reason that this element can't be a linear combination of vertical and horizontal divisors -- in other words, I don't see why a linear comb of vertical and horizontal divisors can't be linearly equivalent to a curve whose normalization is etale over both sides; (likewise w/ a linear comb of Hecke correspondences, in fact...) – Daniel Litt May 10 '17 at 00:23
  • I didn't mean to imply I solved your problem. My comment is just that for modular curves we have a more or less explicit description of the correspondences and so there is a hope that one could check they are not etale on both sides. I have made no attempt to check that's actually the case. – Felipe Voloch May 10 '17 at 00:45
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    I have nothing to add except the observation that this is an old folk question that many people are interested in! And one more in the genre of "are statements that are obviously false for an arbitrary complex foo because countability perhaps true for an arbitrary foo over Fpbar where countability is not a problem?" Of course one could say the same about Qbar but apparently Mochizuki has solved the problem in this case? – JSE May 10 '17 at 02:29
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    @JSE: Yes, it's Theorem A of this paper of Mochizuki: http://www.kurims.kyoto-u.ac.jp/~motizuki/Correspondences%20on%20Hyperbolic%20Curves.pdf – Daniel Litt May 10 '17 at 03:51
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    @DanielLitt The paper you linked actually proves a much stronger fact (as you surely have noticed), which is that there are only finitely many $C_2$ of a given genus satisfying this condition for any $C_1$. Could this stronger fact be true for curves over $\overline{\mathbb F}_q$, or is there a disproof? – Will Sawin May 10 '17 at 04:58
  • @WillSawin: As far as I know it could be true; I certainly don't know a counterexample. – Daniel Litt May 10 '17 at 13:49

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