Let \(X/\mathbf{Q}\) be a smooth projective curve. I would like to be able to say that the motive \(M\) associated to \(X\) “generally” determines \(X\). That is, I would like to say it in a talk without feeling like I’m telling too much of a fib. But is this true? There are two issues. Recall that, by the Torelli Theorem, the Jacobian together with a principle polarization determines \(X/\mathbf{C}\). So there are two things to worry about:
- Knowing \(M\) only recovers the Jacobian up to isogeny, and you can certainly have two different curves with isogenous Jacobians, even isomorphic Jacobians with different polarizations.
- Knowing \(X/\mathbf{C}\) does not determine \(X/\mathbf{Q}\).
To overcome the second issue, it is sufficient and necessary to assume that \(\mathrm{Aut}_{\mathbf{C}}(X)\) is trivial. Let me ignore the first point, since I both assume it generically doesn’t happen but since I can’t even address the second point yet I haven’t thought about it yet.
Perhaps this is obvious to a geometer, but I don’t see why a “random” curve \(X/\mathbf{Q}\) doesn’t have automorphims. My model of a random curve is to take, for example, an embedding of \(M_g\) into projective space and then count points by the ambient height function and see what ratio of points has trivial automorphisms. (Presumably any other counting function like Faltings height or whatever will more or less be the same.) Certainly a generic \(\mathbf{C}\)-point of \(M_g\) has no automorphisms (at least for \(g > 2\)), but since \(M_g\) is of general type for large enough \(g\) I don’t whether one can find enough rational points which are generic!
Probably the most natural way to answer this is to give a positive answer to the following question:
Question: Does \(M_g\) contain a subvariety \(X\) which is unirational over \(\mathbf{Q}\) and has dimension strictly greater than the hyperelliptic locus?
Or, to put it more naturally, can you just explicitly write down enough generic curves which don’t have any automorphisms to see that they dominate any point count?
Trigonal curves. See the comments (especially the one from Jason Starr) to this answer of mine on Mathoverflow:
https://mathoverflow.net/questions/138581/are-most-curves-over-q-pointless/138592#138592
Or 5-gonal curves. See the comments (especially the two from Angelo) to this question of mine on Mathoverflow:
https://mathoverflow.net/questions/187116/what-is-the-expected-dimension-of-the-zariski-closure-of-the-rational-points-on
There’s reason to suspect, at least a little bit, that this stops at 5 for the same reason MB was only able to count S_n fields for n up to 5, i.e. one runs out of nice parameterizations.
In terms of upper bounds, I don’t think anything is known – I don’t think anyone can rule out that there is a unirational subvariety of dimension 3g-4 for g arbitrarily large, for example.
Even if that was known, that wouldn’t answer your question about heights, since a lower-dimensional subvariety can have a larger count of points with bounded height than a higher-dimensional one.
I don’t think there’s much hope to formulate the claim for anything but a specific family -plane curves, n-gonal curves for n from 2 to 5, complete intersections in higher-dimensional space…
(either of you) Willing to stake a guess to the question at hand?
Perhaps more relevantly, if I said (not in this language of course) in a colloquium talk that the motive \(M\) generally determines the curve \(X/\mathbf{Q}\), would you let that pass you by or would it be a distraction…?
It is possible that, ordered by some height (Faltings height?) most curves over 𝐐 are hyperelliptic. I don’t really have a sense of what is true here.
On the other hand, I think the statement that the motive 𝑀 generally determines the curve (up to automorphism) 𝑋/𝐐 is true, even for hyperelliptic curves.
I wrote a paper with Drew Sutherland looking at which L-functions would determine a curve over a finite field. The paper has conjectures with heuristics and numerical evidence. In a subsequent paper (in RNT) with Jeremy Booher, we prove a variant of one of the conjectures.
Maybe I should add that, in the case of a hyperelliptic curve over 𝐐 and its quadratic twist by d, all you want is a prime p for which d is a quadratic non-residue mod p and the number of points on the curve is not p+1. There should be lots of those.
Last comment (I hope). First, I meant number of points mod p above. Second, an example of two curves with the same motive would then be a curve of genus 2 whose Jacobian is the square of a CM elliptic curve and its quadratic twist corresponding to the CM field. This should be extremely rare among all curves.
One can apply here the inexplicably-frequently useful lemma – a representation is isomorphic to its quadratic twist if and only if it’s induced from the corresponding index 2 subgroup. So we’re looking for curves with monodromy contained in some fairly explicit subgroups.
In genus 2, this suggests to look at curves whose Jacobian is the Weil restriction of an elliptic curve over a quadratic extension (or really isogenous to this by a suitable 2-isogeny, putting some conditions on the 2-torsion) which gives a whole two-dimensional family of such curves.
Here is a list of examples of the type of genus 2 curves Will Sawin suggested (these should all have Jacobians that are isogenous to the Weil restriction of an elliptic curve over a quadratic field). You want curves whose Jacobian is not geometrically simple and whose Sato-Tate group is the normalizer of SU(2)xSU(2) inside USp(4) — this will guarantee that the Jacobian is isogenous to the Weil restriction of an elliptic curve over a quadratic extension that is not a Q-curve.
Sorry, I’m confused, what’s the “question at hand”? The bolded question and the last line of your post both have negative answers.
It wouldn’t bother me! But I’m not usually the person who complains about this sort of thing. (Of course if someone does complain you can say “I mean a generic member among those curves you can write down, for example by one equation in two variables” or something like that so the issue is with the people who may be bothered but not enough to complain.)
I meant does a random curve over \(\mathbf{Q}\) of genus \(g\) (counted by height) have trivial automorphism group.
I think under the general principle of “You can’t calculate a proportion if you can’t estimate the denominator,” this is beyond current technology.
Maybe conditional on Manin’s conjecture one can prove that if there are too many curves with automorphisms for some g there must be too many curves without automorphisms for a smaller g and from this deduce that the proportion with automorphisms must be zero for infinitely many g, but I think unconditionally even this kind of reasoning is problematic.
Since Mg is of general type for g sufficiently large, I am not seeing how one applies Manin’s conjecture in order to count genus g curves. I am aware that rational points should be sparse on varieties of general type, but is there a conjecture for the growth of the number of rational points of bounded height on varieties of general type?
There’s a general philosophy related to the Bombieri-Lang and Manin conjectures – I don’t remember exactly which combination of conjectures one could assume to draw this as a conclusion but probably there is one – that most of the rational points on a general type variety should lie on its Fano subvarieties, whichever ones have the most Manin-predicted points, if it has any Fano subvarieties at all.
The point being that all but finitely many rational points should be explained by varieties that have a lot of rational points (Fano, abelian varieties, Calabi-Yaus) but Fano varieties have vastly more points than the others (except Calabi-Yaus which themselves contain Fanos). For example, any rational curve contained in any variety has vastly more rational points than any abelian subvariety, no matter the ample height function you choose.
Thus, Manin-type predictions could come from looking for Fano subvarieties and trying to find the ones that have the most predicted rational points. I don’t remember my idea above but I think it had to do with assuming that if there are many ration points of M_g with automorphisms then there must be a Fano variety parameterizing rational points of M_g with automorphisms and then showing that this maps to a Fano variety parameterizing points of M_{g’} for g'<g and trying to compare the numbers of rational points.
Thank you for explaining that to me. This is an interesting idea, but I am guessing that it is difficult to identify the relevant Fano subvarieties of M_g (?).
How about this example, the genus 3 curves
\(y^2 + (x^4 + x^3 + x^2 + 1)y = x^7 – 8x^5 – 4x^4 + 18x^3 – 3x^2 – 16x + 8\)
and
\(x^3z+x^2yz+x^2z^2+xy^3-xy^2z+y^4-y^3z-yz^3\)
are both generic in the sense that the have no extra endomorphisms, but one is hyperelliptic and the other is not.
While I have not tried to run a Faltings-Serre computation to prove it, I’m morally certain these two genus 3 curves have the same L-function (their Frobenius traces agree out to 2^28 and the conductor is only 8233), hence they have isogenous Jacobians. But these two curves really are different in meaningful sense.
Nice example! There is a related example over \(\mathbf{F}_3\) in this paper as well, but I certainly haven’t seen an example over \(\mathbf{Q}\) before.