Timeline for Does a curve over a number field have a finite etale cover of given degree
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12 events
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| Oct 15, 2012 at 5:52 | comment | added | Will Sawin | Elementary group theory shows that a 3-fold cover implies the existence of either a point of order $2$ or a subgroup of order $3$ in the Jacobian, a $4$-fold cover implies the existence of either a point of order $2$, a subgroup of order $3$, or a Klein $4$ subgroup, and that no similar nice results exist for covers of order $5$ and above. | |
| Oct 15, 2012 at 5:50 | comment | added | Will Sawin | @Harry: it would be very surprising if a d-cover of X implies the existence of an element of order d in the Jacobian, because the proof uses the fact that index two subgroups are normal, which fails to hold in general. @Rene: Can't one always chose a morphism $X \to J$ that induces an isomorphism on the first homology? Then it is clear that the covering is nontrivial. | |
| Oct 14, 2012 at 19:46 | comment | added | R.P. | You're right... still thinking about an argument for that. | |
| Oct 14, 2012 at 15:25 | comment | added | Dan Petersen | @rene. One needs some kind of argument for why the resulting cover is not trivial. E.g. take $f$ constant. | |
| Oct 14, 2012 at 13:04 | answer | added | Niels | timeline score: 4 | |
| Oct 14, 2012 at 12:38 | comment | added | R.P. | (What I wrote before was mostly wrong -- so I have deleted it.) Elliptic curves always have étale covers corresponding to mult. by $n$. Generically, these are the only ones. As for higher genus curves, these too always admit non-trivial étale coverings: Choose a non-trivial morphism $f:X \rightarrow J = \mathrm{Jac}(X)$ (no need to assume it injective), then base change $[n]:J \rightarrow J$ along $f$ to obtain a map $Y \rightarrow X$, which is étale since it is the base change of an étale map. I imagine it might well be true that, generically, this is all there is, but I have no proof. | |
| Oct 14, 2012 at 11:32 | comment | added | Harry | Fair enough. That solves the problem for $d=2$. Does your remark also generalize to $d>2$ in some way? I have the feeling that the existence of a $d$-cover of $X$ implies the existence of an element of order $d$ in its Jacobian. This means that it should be possible that $X$ has no finite etale covers at all. Is that true? (I think it's true for an elliptic curve with $X(K)$ trivial.) | |
| Oct 14, 2012 at 11:24 | comment | added | R.P. | Unramified double covers of a curve correspond to elements of order 2 in its Jacobian, so MP's observation generalizes to curves whose Jacobian has only the trivial 2-torsion, of which there exist plenty. | |
| Oct 14, 2012 at 11:15 | history | edited | Harry | CC BY-SA 3.0 |
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| Oct 14, 2012 at 11:15 | comment | added | Harry | Right. I forget to mention that $X$ is hyperbolic. But probably, the answer will be "not in general" again. I just can't think of a good reason. | |
| Oct 14, 2012 at 10:57 | comment | added | M P | Not in general: if $X$ is an elliptic curve with trivial two-torsion subgroup over $K$, then there is no non-trivial étale double cover of $X$. | |
| Oct 14, 2012 at 10:26 | history | asked | Harry | CC BY-SA 3.0 |