Tag Archives: rational points

How many rational distances can there be between N points in the plane?

Terry has a nice post up bout the Erdös-Ulam problem, which was unfamiliar to me.  Here’s the problem:

Let S be a subset of R^2 such that the distance between any two points in S is a rational number.  Can we conclude that S is not topologically dense?

S doesn’t have to be finite; one could have S be the set of rational points on a line, for instance.  But this appears to be almost the only screwy case.  One can ask, more ambitiously:

Is it the case that there exists a curve X of degree <= 2 containing all but 4 points of S?

Terry explains in his post how to show something like this conditional on the Bombieri-Lang conjecture.  The idea:  lay down 4 points in general position.  Then the condition that the 5th point has rational distances from x1,x2,x3, and x4 means that point lifts to a rational point on a certain (Z/2Z)^4-cover Y of P^2 depending on x1,x2,x3,x4.  (It’s the one obtained by adjoining the 4 distances, each of which is a square root of a rational function.)

With some work you can show Y has general type, so under Lang its rational points are supported on a union of curves.  Then you use a result of Solymosi and de Zeeuw to show that each curve can only have finitely many points of S if it’s not a line or a circle.  (Same argument, except that instead of covers of P^2 you have covers of the curve, whose genus goes up and then you use Faltings.)

It already seems hard to turn this approach into a proof.  There are few algebraic surfaces for which we can prove Lang’s conjecture.  But why let that stop us from asking further questions?

Question:  Let S be a set of N points on R^2 such that no M are contained in any line or circle.  What is the maximal number of rational distance among the ~N^2 distances between points of S?

The Erdos-Ulam problem suggests the answer is smaller than N^2.  But surely it’s much smaller, right?  You can get at least NM rational distances just by having S be (N/M) lines, each with M rational points.  Can you do better?

 

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How many points does a random curve over F_q have?

So asks a charming preprint by Achter, Erman, Kedlaya, Wood, and Zureick-Brown.  (2/5 Wisconsin, 1/5 ex-Wisconsin!)  The paper, I’m happy to say, is a result of discussions at an AIM workshop on arithmetic statistics I organized with Alina Bucur and Chantal David earlier this year.

Here’s how they think of this.  By a random curve we might mean a curve drawn uniformly from M_g(F_q).  Let X be the number of points on a random curve.  Then the average number of points on a random curve also has a geometric interpretation: it is

|M_{g,1}(\mathbf{F}_q)|/|M_{g}(\mathbf{F}_q)|

What about

|M_{g,2}(\mathbf{F}_q)|/|M_{g}(\mathbf{F}_q)|?

That’s just the average number of ordered pairs of distinct points on a random curve; the expected value of X(X-1).

If we can compute all these expected values, we have all the moments of X, which should give us a good idea as to its distribution.  Now if life were as easy as possible, the moduli spaces of curves would have no cohomology past degree 0, and by Grothendieck-Lefschetz, the number of points on M_{g,n} would be q^{3g-3+n}.  In that case, we’d have that the expected value of X(X-1)…(X-n) was q^n.  Hey, I know what distribution that is!  It’s Poisson with mean q.

Now M_g does have cohomology past degree 0.  The good news is, thanks to the Madsen-Weiss theorem (née the Mumford conjecture) we know what that cohomology is, at least stably.  Yes, there are a lot of unstable classes, too, but the authors propose that heuristically these shouldn’t contribute anything.  (The point is that the contribution from the unstable range should look like traces of gigantic random unitary matrices, which, I learn from this paper, are bounded with probability 1 — I didn’t know this, actually!)  And you can even make this heuristic into a fact, if you want, by letting q grow pretty quickly relative to g.

So something quite nice happens:  if you apply Grothendieck-Lefschetz (actually, you’d better throw in Kai Behrend’s name, too, because M_g is a Deligne-Mumford stack, not an honest scheme) you find that the moments of X still agree with those of a Poisson distribution!  But the contribution of the tautological cohomology shifts the mean from q to q+1+1/(q-1).

This is cool in many directions!

  • It satisfies one’s feeling that a “random set,” if it carries no extra structure, should have cardinality obeying a Poisson distribution — the “uniform distribution” on the groupoid of sets.  (Though actually that uniform distribution is Poisson(1); I wonder what tweak is necessary to be able to tune the mean?)
  • I once blogged about an interesting result of Bucur and Kedlaya which showed that a random smooth complete intersection curve in P^3 of fixed degree had slightly fewer than q+1 points; in fact, about q+1 – 1/q + o(q^2).  Here the deviation is negative, rather than positive, as the new paper suggests is the case for general curves; what’s going on?
  • I have blogged about the question of average number of points on a random curve before.  I’d be very interested to know whether the new heuristic agrees with the answer to the question proposed at the end of that post; if g is a large random matrix in GSp(Z_ell) with algebraic eigenvalues, and which multiplies the symplectic form by q, and you condition on Tr(g^k) > (-q^k-1) so that the “curve” has nonnegatively many points over each extension of F_q, does this give something like the distribution the five authors predict for Tr(g)?  (Note:  I don’t think this question is exactly well-formed as stated.)

 

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Puzzle: low-height points in general position

I have no direct reason to need the answer to, but have wondered about, the following question.

We say a set of points P_1, \ldots, P_N in \mathbf{A}^2 are in general position if the Hilbert function of any subset S of the points is equal to the Hilbert function of a generic set of |S| points in \mathbf{A}^n.  In other words, there are no curves which contain more of the points than a curve of their degree “ought” to.  No three lie on a line, no six on a conic, etc.

Anyway, here’s a question.  Let H(N) be the minimum, over all N-tuples P_1, \ldots, P_N \in \mathbf{A}^2(\mathbf{Q}) of points in general position, of

\max H(P_i)

where H denotes Weil height.  What are the asymptotics of H(N)?  If you take the N lowest-height points, you will have lots of collinearity, coconicity, etc.  Does the Bombieri-Pila / Heath-Brown method say anything here?

 

 

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Gonality, the Bogomolov property, and Habegger’s theorem on Q(E^tors)

I promised to say a little more about why I think the result of Habegger’s recent paper, ” Small Height and Infinite Non-Abelian Extensions,” is so cool.

First of all:  we say an algebraic extension K of Q has the Bogomolov property if there is no infinite sequence of non-torsion elements x in K^* whose absolute logarithmic height tends to 0.  Equivalently, 0 is isolated in the set of absolute heights in K^*.  Finite extensions of Q evidently have the Bogomolov property (henceforth:  (B)) but for infinite extensions the question is much subtler.  Certainly \bar{\mathbf{Q}} itself doesn’t have (B):  consider the sequence 2^{1/2}, 2^{1/3}, 2^{1/4}, \ldots  On the other hand, the maximal abelian extension of Q is known to have (B) (Amoroso-Dvornicich) , as is any extension which is totally split at some fixed place p (Schinzel for the real prime, Bombieri-Zannier for the other primes.)

Habegger has proved that, when E is an elliptic curve over Q, the field Q(E^tors) obtained by adjoining all torsion points of E has the Bogomolov property.

What does this have to do with gonality, and with my paper with Chris Hall and Emmanuel Kowalski from last year?

Suppose we ask about the Bogomolov property for extensions of a more general field F?  Well, F had better admit a notion of absolute Weil height.  This is certainly OK when F is a global field, like the function field of a curve over a finite field k; but in fact it’s fine for the function field of a complex curve as well.  So let’s take that view; in fact, for simplicity, let’s take F to be C(t).

What does it mean for an algebraic extension F’ of F to have the Bogomolov property?  It means that there is a constant c such that, for every finite subextension L of F and every non-constant function x in L^*, the absolute logarithmic height of x is at least c.

Now L is the function field of some complex algebraic curve C, a finite cover of P^1.  And a non-constant function x in L^* can be thought of as a nonzero principal divisor.  The logarithmic height, in this context, is just the number of zeroes of x — or, if you like, the number of poles of x — or, if you like, the degree of x, thought of as a morphism from C to the projective line.  (Not necessarily the projective line of which C is a cover — a new projective line!)  In the number field context, it was pretty easy to see that the log height of non-torsion elements of L^* was bounded away from 0.  That’s true here, too, even more easily — a non-constant map from C to P^1 has degree at least 1!

There’s one convenient difference between the geometric case and the number field case.  The lowest log height of a non-torsion element of L^* — that is, the least degree of a non-constant map from C to P^1 — already has a name.  It’s called the gonality of C.  For the Bogomolov property, the relevant number isn’t the log height, but the absolute log height, which is to say the gonality divided by [L:F].

So the Bogomolov property for F’ — what we might call the geometric Bogomolov property — says the following.  We think of F’ as a family of finite covers C / P^1.  Then

(GB)  There is a constant c such that the gonality of C is at least c deg(C/P^1), for every cover C in the family.

What kinds of families of covers are geometrically Bogomolov?  As in the number field case, you can certainly find some families that fail the test — for instance, gonality is bounded above in terms of genus, so any family of curves C with growing degree over P^1 but bounded genus will do the trick.

On the other hand, the family of modular curves over X(1) is geometrically Bogomolov; this was proved (independently) by Abramovich and Zograf.  This is a gigantic and elegant generalization of Ogg’s old theorem that only finitely many modular curves are hyperelliptic (i.e. only finitely many have gonality 2.)

At this point we have actually more or less proved the geometric version of Habegger’s theorem!  Here’s the idea.  Take F = C(t) and let E/F be an elliptic curve; then to prove that F(E(torsion)) has (GB), we need to give a lower bound for the curve C_N obtained by adjoining an N-torsion point to F.  (I am slightly punting on the issue of being careful about other fields contained in F(E(torsion)), but I don’t think this matters.)  But C_N admits a dominant map to X_1(N); gonality goes down in dominant maps, so the Abramovich-Zograf bound on the gonality of X_1(N) provides a lower bound for the gonality of C_N, and it turns out that this gives exactly the bound required.

What Chris, Emmanuel and I proved is that (GB) is true in much greater generality — in fact (using recent results of Golsefidy and Varju that slightly postdate our paper) it holds for any extension of C(t) whose Galois group is a perfect Lie group with Z_p or Zhat coefficients and which is ramified at finitely many places; not just the extension obtained by adjoining torsion of an elliptic curve, for instance, but the one you get from the torsion of an abelian variety of arbitrary dimension, or for that matter any other motive with sufficiently interesting Mumford-Tate group.

Question:   Is Habegger’s theorem true in this generality?  For instance, if A/Q is an abelian variety, does Q(A(tors)) have the Bogomolov property?

Question:  Is there any invariant of a number field which plays the role in the arithmetic setting that “spectral gap of the Laplacian” plays for a complex algebraic curve?

A word about Habegger’s proof.  We know that number fields are a lot more like F_q(t) than they are like C(t).  And the analogue of the Abramovich-Zograf bound for modular curves over F_q is known as well, by a theorem of Poonen.  The argument is not at all like that of Abramovich and Zograf, which rests on analysis in the end.  Rather, Poonen observes that modular curves in characteristic p have lots of supersingular points, because the square of Frobenius acts as a scalar on the l-torsion in the supersingular case.  But having a lot of points gives you a lower bound on gonality!  A curve with a degree d map to P^1 has at most d(q+1) points, just because the preimage of each of the q+1 points of P^1(q) has size at most d.  (You just never get too old or too sophisticated to whip out the Pigeonhole Principle at an opportune moment….)

Now I haven’t studied Habegger’s argument in detail yet, but look what you find right in the introduction:

The non-Archimedean estimate is done at places above an auxiliary prime number p where E has good supersingular reduction and where some other technical conditions are met…. In this case we will obtain an explicit height lower bound swiftly using the product formula, cf. Lemma 5.1. The crucial point is that supersingularity forces the square of the Frobenius to act as a scalar on the reduction of E modulo p.

Yup!  There’s no mention of Poonen in the paper, so I think Habegger came to this idea independently.  Very satisfying!  The hard case — for Habegger as for Poonen — has to do with the fields obtained by adjoining p-torsion, where p is the characteristic of the supersingular elliptic curve driving the argument.  It would be very interesting to hear from Poonen and/or Habegger whether the arguments are similar in that case too!

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Bourqui on spaces of rational curves and motivic Batyrev-Manin

David Bourqui just posted a really nice paper, “Asymptotic Behavior of Rational Curves,” notes from a lecture series he gave last summer at the Institut Fourier at Grenoble.  I’ll try to sum up here what it’s about and hopefully entice people to have a look!

Let N_X(B) net of rational points of height at most B on a Fano variety X endowed with an ample line bundle L over a global field K.  By now we are used to the idea (summed up in greatest generality by the Baytrev-Manin conjecture, refined by Peyre) that N_X(B) satisfies an asymptotic regularity — it approaches

c B^a (log B)^{b-1}

for some constants a and b (which are integers) and c (a real number.)  Batyrev and Manin gave predictions for a and b, Peyre pinned down c.  To fix ideas, let’s suppose that the projective embedding of X is given by the anticanonical divisor.  Then a is 1 and b is the Picard rank of X (over K.)  So for instance when X = P^1, you can see immediately that the number of points of height B should be linear in B, and that’s true (remembering that the canonical height is the square of the usual Weil height on P^1.)

Now these conjectures are not exactly right.  There is the problem of accumulating subvarieties, like the lines on a cubic surface, which have way too many rational points; you have to strip these out before you can expect to get down to the expected asymptotic.  And there are more subtle counterexamples, like the one produced by Batyrev and Tschinkel, where the number of rational points is too high by some power of log B.  But the conjecture has been proved for many classes of varieties (toric, homogeneous, very low degree relative to dimension…)

Bourqui’s approach starts from the consideration of these conjectures over the global function field K = k(t), where k = F_q is a finite field.  For simplicity let’s take X to be defined over the constant field k.  Now N_K(B) has two meanings.  You can think of it as a set of K-points of X of bounded height — or you can think it as the number of k-points on the space C_d(X) of degree-d rational curves on X, where q^d = B.  The Batyrev-Manin conjecture, as we phrased it here, is about the first interpretation.  But you can also read it as a statement about the varieties C_d, and it turns out what it says is that

|C_d(k)| / |A^(k)|^d

approaches a limit as d goes to infinity.

Doesn’t this seem a rather astonishing claim at first glance?  These are higher and higher-dimensional varieties over a fixed finite field; the Weil conjectures offer us no useful control.  Why shouldn’t their point-counts fluctuate wildly?

To get some idea of what might be gone, think of a family of varieties that’s easily seen to display this behavior:  the projective spaces P^d.  Evidently,

\lim_{d \rightarrow \infty} (|\mathbf{P}^d(k)| / |\mathbf{A}^1(k)|^d) = (1-1/q)^{-1}.

Why is there a limit?  I can think of two reasons.

First reason:  projective spaces have stable cohomology;  the compactly supported cohomology has one dimension in degrees 2d,2d-2, … 0 and is empty in the odd degrees, and all the cohomology is of Tate type.  Note that (apart from the location of the top cohomology group) this description is independent of d, and it follows that the point count provided by the Lefschetz trace formula is (apart from a prepended power of q) independent of d as well.

Second reason:  Forget about finite fields — the expression

\lim_{d \rightarrow \infty} ([\mathbf{P}^d] / [\mathbf{A}^1]^d) = (1-1/[A^1])^{-1}

remains true motivically (in the suitably completed version of the Grothendieck ring of varieties.)  To be fair, this alone doesn’t quite imply that the point-counting limits hold.  (For instance, the sequence $1 + latex 2^{2^d} [\mathbf{A}^1]^{-d}$ converges motivically to 1, while none of its point-counts converge)  But the motivic convergence is highly suggestive.

My own work in this circle of ideas is mostly concerned with stable cohomology.  What Bourqui is interested in, on the other hand, is whether one has a motivic Batyrev-Manin conjecture; is it the case that

[C_d] / [A^1]^d

approaches a limit in d, and what is this limit?  (This is question 1.11.2 in Bourqui’s paper — to be precise, Bourqui asks for something more precise where one breaks up C_d according to the numerical equivalence class of the rational curve.)  Bourqui proves this is indeed the case when X is a smooth projective toric variety over a field of characteristic 0.  This is by no means a straightforward imitation of the proof of Batyrev-Manin for toric varieties over global fields:  proving motivic identities is hard!

In the case of toric varieties, by the way, both routes to Batyrev-Manin are available; the fact that spaces of rational curves on toric varieties have stable cohomology was proved by Martin Guest.  Guest shows that the cohomology stabilization holds for all smooth projective toric varieties and some of the singular ones as well — the main tool is a diffeomorphism between this space of rational curves and a certain kind of decorated configuration space on the sphere.  I wonder, is Guest’s configuration space implicitly present in Bourqui’s proof?

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JMM, Golsefidy, Silverman, Scanlon

Like Emmanuel, I spent part of last week at the Joint Meetings in New Orleans, thanks to a generous invitation from Alireza Salefi Golsefidy and Alex Lubotzky to speak in their special session on expander graphs.  I was happy that Alireza was willing to violate a slight taboo and speak in his own session, since I got to hear about his work with Varju, which caps off a year of spectacular progress on expansion in quotients of Zariski-dense subgroups of arithmetic groups.  Emmanuel’s Bourbaki talk is your go-to expose.

I think I’m unlike most mathematicians in that I really like these twenty-minute talks.  They’re like little bonbons — you enjoy one and then before you’ve even finished chewing you have the next in hand!  One nice bonbon was provided by Joe Silverman, who talked about his recent work on Lehmer’s conjecture for polynomials satisfying special congruences.  For instance, he shows that a polynomial which is congruent mod m to a multiple of a large cyclotomic polynomial can’t have a root of small height, unless that root is itself a root of unity.  He has a similar result where the implicit G_m is replaced by an elliptic curve, and one gets a lower bound for algebraic points on E which are congruent mod m to a lot of torsion points.  This result, to my eye, has the flavor of the work of Bombieri, Pila, and Heath-Brown on rational points.  Namely, it obeys the slogan:  Low-height rational points repel each other. More precisely — the global condition (low height) is in tension with a bunch of local conditions (p-adic closeness.)  This is the engine that drives the upper bounds in Bombieri-Pila and Heath-Brown:  if you have too many low-height points, there’s just not enough room for them to repel each other modulo every prime!

Anyway, in Silverman’s situation, the points are forced to nestle very close to torsion points — the lowest-height points of all!  So it seems quite natural that their own heights should be bounded away from 0 to some extent.  I wonder whether one can combine Silverman’s argument with an argument of the Bombieri-Pila-Heath-Brown type to get good bounds on the number of counterexamples to Lehmer’s conjecture….?

One piece of candy I didn’t get to try was Tom Scanlon’s Current Events Bulletin talk about the work of Pila and Willkie on problems of Manin-Mumford type.  Happily, he’s made the notes available and I read it on the plane home.  Tom gives a beautifully clear exposition of ideas that are rather alien to most number theorists, but which speak to issues of fundamental importance to us.  In particular, I now understand at last what “o-minimality” is, and how Pila’s work in this area grows naturally out of the Bombieri-Pila method mentioned above.  Highly recommended!

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Lieblich’s counter counterexample example

Max Lieblich gave a great talk at WAGS yesterday about something that looks like a counterexample to the Hasse principle, but secretly isn’t!  All mistakes in this summary are my own.  For a more authoritative take on the material below, see Max’s recent arXiv preprint.

The counterexample Max countered is the central simple algebra A over the field Q(t) obtained as the tensor product of the two generalized quaternion algebras (17,t) and (13,6(t-1)(t-11)).  This algebra has index 4, which is to say (if I understand correctly) that its Brauer class isn’t the cup product of two elements in Q(t)^*.  On the other hand, it turns out that A \otimes \mathbf{Q}_v has index 1 or 2 — that is, it’s either trivial or a cup product — for all places v of Q.

This seems like an example of a situation where there’s no Hasse principle; A fails to be a cup product despite the fact that A is a cup product over every completion of Q.  But the truth, as Max explained, is more complicated.

We know (and if we don’t, we read A Course in Arithmetic to remember) that the quaternion algebra (a,b) is trivial (i.e., has index 1) over a field k precisely when the conic x^2 - ay^2 - bz^2 has a k-rational point.  You might ask whether there’s a similar criterion for the tensor product of two quaternion algebras to have index 2.  It turns out that (at least over Q(t)) there is indeed a variety that does this trick — it’s a coarse moduli space M parametrizing certain twisted vector bundles on — well, not quite P^1/Q, but a certain orbifold version of P^1 with a bunch of stacky points with inertia Z/2Z.  And the assertion that A has index 4 is equivalent to the assertion that M(Q) is empty.

To really talk about what M is would take us too far afield; I just want to record Max’s observation that the definition of M is truly global, in the sense that the scheme M_{\mathbf{Q}_v} is not determined by A \otimes \mathbf{Q}_v.  In particular, the fact that A \otimes \mathbf{Q}_v has index 2 doesn’t imply that M has a \mathbf{Q}_v-point.  And indeed, in the case at hand, M has no points over \mathbf{Q}_{17}.  So there is, after all, a local obstruction to A having index 2; but it’s a local obstruction which, it seems, can’t be seen except in this rather intricate geometric way.

It makes you wonder what should actually be meant by “Hasse principle.”  Suppose, for instance, you had some class C of varieties X/Q, and suppose you had some construction which attached to each X in C a variety Y/Q such that X(Q) is empty if and only if Y(Q) is empty.   Now one way to prove X(Q) empty would be to prove that Y(Q) was empty, which you could in turn prove if you knew that Y(Q_v) was empty for some place v.  Do you consider this a local obstruction to rational points on X?

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