Tag Archives: algebraic geometry

Rational points on solvable curves over Q via non-abelian Chabauty (with Daniel Hast)

New paper up!  With my Ph.D. student Daniel Hast (last seen on the blog here.)

We prove that hyperelliptic curves over Q of genus at least 2 have only finitely many rational points.  Actually, we prove this for a more general class of high-genus curves over Q, including all solvable covers of P^1.

But wait, don’t we already know that, by Faltings?  Of course we do.  So the point of the paper is to show that you can get this finiteness in a different way, via the non-abelian Chabauty method pioneered by Kim.  And I think it seems possible in principle to get Faltings for all curves over Q this way; though I don’t know how to do it.

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Multiple height zeta functions?

Idle speculation ensues.

Let X be a projective variety over a global field K, which is Fano — that is, its anticanonical bundle is ample.  Then we expect, and in lots of cases know, that X has lots of rational points over K.  We can put these points together into a height zeta function

\zeta_X(s) = \sum_{x \in X(K)} H(x)^{-s}

where H(x) is the height of x with respect to the given projective embedding.  The height zeta function organizes information about the distribution of the rational points of X, and which in favorable circumstances (e.g. if X is a homogeneous space) has the handsome analytic properties we have come to expect from something called a zeta function.  (Nice survey by Chambert-Loir.)

What if X is a variety with two (or more) natural ample line bundles, e.g. a variety that sits inside P^m x P^n?  Then there are two natural height functions H_1 and H_2 on X(K), and we can form a “multiple height zeta function”

\zeta_X(s,t) = \sum_{x \in X(K)} H_1(x)^{-s} H_2(x)^{-t}

There is a whole story of “multiple Dirichlet series” which studies functions like

\sum_{m,n} (\frac{m}{n}) m^{-s} n^{-t}

where (\frac{m}{n}) denotes the Legendre symbol.  These often have interesting analytic properties that you wouldn’t see if you fixed one variable and let the other move; for instance, they sometimes have finite groups of functional equations that commingle the s and the t!

So I just wonder:  are there situations where the multiple height zeta function is an “analytically interesting” multiple Dirichlet series?

Here’s a case to consider:  what if X is the subvariety of P^2 x P^2 cut out by the equation

x_0 y_0 + x_1 y_1 + x_2 y_2 = 0?

This has something to do with Eisenstein series on GL_3 but I am a bit confused about what exactly to say.

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Hast and Matei, “Moments of arithmetic functions in short intervals”

Two of my students, Daniel Hast and Vlad Matei, have an awesome new paper, and here I am to tell you about it!

A couple of years ago at AIM I saw Jon Keating talk about this charming paper by him and Ze’ev Rudnick.  Here’s the idea.  Let f be an arithmetic function: in that particular paper, it’s the von Mangoldt function, but you can ask the same question (and they do) for Möbius and many others.

Now we know the von Mangoldt function is 1 on average.  To be more precise: in a suitably long interval ([X,X+X^{1/2 + \epsilon}] is long enough under Riemann) the average of von Mangoldt is always close to 1.  But the average over a short interval can vary.  You can think of the sum of von Mangoldt over  [x,x+H], with H = x^d,  as a function f(x) which has mean 1 but which for d < 1/2 need not be concentrated at 1.  Can we understand how much it varies?  For a start, can we compute its variance as x ranges from 1 to X?This is the subject of a conjecture of Goldston and Montgomery.  Keating and Rudnick don’t prove that conjecture in its original form; rather, they study the problem transposed into the context of the polynomial ring F_q[t].  Here, the analogue of archimedean absolute value is the absolute value

|f| = q^{\deg f}

so an interval of size q^h is the set of f such that deg(f-f_0) < q^h for some polynomial f_0.

So you can take the monic polynomials of degree n, split that up into q^{n-h} intervals of size q^h, and sum f over each interval, and take the variance of all these sums.  Call this V_f(n,h).  What Keating and Rudnick show is that

\lim_{q \rightarrow \infty} q^{-(h+1)} V(n,h) = n - h - 2.

This is not quite the analogue of the Goldston-Montgomery conjecture; that would be the limit as n,h grow with q fixed.  That, for now, seems out of reach.  Keating and Rudnick’s argument goes through the Katz equidistribution theorems (plus some rather hairy integration over groups) and the nature of those equidistribution theorems — like the Weil bounds from which they ultimately derive — is to give you control as q gets large with everything else fixed (or at least growing very slo-o-o-o-o-wly.)  Generally speaking, a large-q result like this reflects knowledge of the top cohomology group, while getting a fixed-q result requires some control of all the cohomology groups, or at least all the cohomology groups in a large range.

Now for Hast and Matei’s paper.  Their observation is that the variance of the von Mangoldt function can actually be studied algebro-geometrically without swinging the Katz hammer.  Namely:  there’s a variety X_{2,n,h} which parametrizes pairs (f_1,f_2) of monic degree-n polynomials whose difference has degree less than h, together with an ordering of the roots of each polynomial.  X_{2,n,h} carries an action of S_n x S_n by permuting the roots.  Write Y_{2,n,h} for the quotient by this action; that’s just the space of pairs of polynomials in the same h-interval.  Now the variance Keating and Rudnick ask about is more or less

\sum_{(f_1, f_2) \in Y_{2,n,h}(\mathbf{F}_q)} \Lambda(f_1) \Lambda(f_2)

where $\Lambda$ is the von Mangoldt function.  But note that $\Lambda(f_i)$ is completely determined by the factorization of $f_i$; this being the case, we can use Grothendieck-Lefschetz to express the sum above in terms of the Frobenius traces on the groups

H^i(X_{2,n,h},\mathbf{Q}_\ell) \otimes_{\mathbf{Q}_\ell[S_n \times S_n]} V_\Lambda

where $V_\Lambda$ is a representation of $S_n \times S_n$ keeping track of the function $\Lambda$.  (This move is pretty standard and is the kind of thing that happens all over the place in my paper with Church and Farb about point-counting and representation stability, in section 2.2 particularly)

When the smoke clears, the behavior of the variance V(n,h) as q gets large is controlled by the top “interesting” cohomology group of X_{2,n,h}.  Now X_{2,n,h} is a complete intersection, so you might think its interesting cohomology is all in the middle.  But no — it’s singular, so you have to be more careful.  Hast and Matei carry out a careful analysis of the singular locus of X_{2,n,h}, and use this to show that the cohomology groups that vanish in a large range.  Outside that range, Weil bounds give an upper bound on the trace of Frobenius.  In the end they get

V(n,h) = O(q^{h+1}).

In other words, they get the order of growth from Keating-Rudnick but not the constant term, and they get it without invoking all the machinery of Katz.  What’s more, their argument has nothing to do with von Mangoldt; it applies to essentially any function of f that only depends on the degrees and multiplicities of the irreducible factors.

What would be really great is to understand that top cohomology group H as an S_n x S_n – representation.  That’s what you’d need in order to get that n-h-2 from Keating-Rudnick; you could just compute it as the inner product of H with V_\Lambda.  You want the variance of a different arithmetic function, you pair H with a different representation.  H has all the answers.  But neither they nor I could see how to compute H.

Then came Brad Rodgers.  Two months ago, he posted a preprint which gets the constant term for the variance of any arithmetic function in short intervals.  His argument, like Keating-Rudnick, goes through Katz equidistribution.  This is the same information we would have gotten from knowing H.  And it turns out that Hast and Matei can actually provably recover H from Rodgers’ result; the point is that the power of q Rodgers get can only arise from H, because all the other cohomology groups of high enough weight are the ones Hast and Matei already showed are zero.

So in the end they find

H = \oplus_\lambda V_\lambda \boxtimes V_\lambda

where \lambda ranges over all partitions of n whose top row has length at most n-h-2.

I don’t think I’ve ever seen this kind of representation come up before — is it familiar to anyone?

Anyway:  what I like so much about this new development is that it runs contrary to the main current in this subject, in which you prove theorems in topology or algebraic geometry and use them to solve counting problems in arithmetic statistics over function fields.  Here, the arrow goes the other way; from Rodgers’s counting theorem, they get a computation of a cohomology group which I can’t see any way to get at by algebraic geometry.  That’s cool!  The other example I know of the arrow going this direction is this beautiful paper of Browning and Vishe, in which they use the circle method over function fields to prove the irreducibility of spaces of rational curves on low-degree hypersurfaces.  I should blog about that paper too!  But this is already getting long….

 

 

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New bounds on curve tangencies and orthogonalities (with Solymosi and Zahl)

New paper up on the arXiv, with Jozsef Solymosi and Josh Zahl.  Suppose you have n plane curves of bounded degree.  There ought to be about n^2 intersections between them.  But there are intersections and there are intersections!  Generically, an intersection between two curves is a node.  But maybe the curves are mutually tangent at a point — that’s a more intense kind of singularity called a tacnode.  You might think, well, OK, a tacnode is just some singularity of bounded multiplicity, so maybe there could still be a constant multiple of n^2 mutual tangencies.

No!  In fact, we show there are O(n^{3/2}).  (Megyesi and Szabo had previously given an upper bound of the form n^{2-delta} in the case where the curves are all conics.)

Is n^{3/2} best possible?  Good question.  The best known lower bound is given by a configuration of n circles with about n^{4/3} mutual tangencies.

Here’s the main idea.  If a curve C starts life in A^2, you can lift it to a curve C’ in A^3 by sending each point (x,y) to (x,y,z) where z is the slope of C at (x,y); of course, if multiple branches of the curve go through (x,y), you are going to have multiple points in C’ over (x,y).  So C’ is isomorphic to C at the smooth points of C, but something’s happening at the singularities of C; basically, you’ve blown up!  And when you blow up a tacnode, you get a regular node — the two branches of C through (x,y) have the same slope there, so they remain in contact even in C’.

Now you have a bunch of bounded degree curves in A^3 which have an unexpectedly large amount of intersection; at this point you’re right in the mainstream of incidence geometry, where incidences between points and curves in 3-space are exactly the kind of thing people are now pretty good at bounding.  And bound them we do.

Interesting to let one’s mind wander over this stuff.  Say you have n curves of bounded degree.  So yes, there are roughly n^2 intersection points — generically, these will be distinct nodes, but you can ask how non-generic can the intersection be?  You have a partition of const*n^2 coming from the multiplicity of intersection points, and you can ask what that partition is allowed to look like.  For instance, how much of the “mass” can come  from points where the multiplicity of intersection is at least r?  Things like that.

 

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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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Squares and Motzkins

Greg Smith gave an awesome colloquium here last week about his paper with Blekherman and Velasco on sums of squares.

Here’s how it goes.  You can ask:  if a homogeneous degree-d polynomial in n variables over R takes only non-negative values, is it necessarily a sum of squares?  Hilbert showed in 1888 that the answer is yes only when d=2 (the case of quadratic forms), n=2 (the case of binary forms) or (n,d) = (3,4) (the case of ternary quartics.)  Beyond that, there are polynomials that take non-negative values but are not sums of squares, like the Motzkin polynomial

X^4 Y^2 + X^2 Y^4 - 3X^2 Y^2 Z^2 + Z^6.

So Greg points out that you can formulate this question for an arbitrary real projective variety X/R.  We say a global section f of O(2) on X is nonnegative if it takes nonnegative values on X(R); this is well-defined because 2 is even, so dilating a vector x leaves the sign of f(x) alone.

So we can ask:  is every nonnegative f a sum of squares of global sections of O(1)?  And Blekherman, Smith, and Velasco find there’s an unexpectedly clean criterion:  the answer is yes if and only if X is a variety of minimal degree, i.e. its degree is one more than its codimension.  So e.g. X could be P^n, which is the (n+1,2) case of Hilbert.  Or it could be a rational normal scroll, which is the (2,d) case.  But there’s one other nice case:  P^2 in its Veronese embedding in P^5, where it’s degree 4 and codimension 3.  The sections of O(2) are then just the plane quartics, and you get back Hilbert’s third case.  But now it doesn’t look like a weird outlier; it’s an inevitable consequence of a theorem both simpler and more general.  Not every day you get to out-Hilbert Hilbert.

Idle question follows:

One easy way to get nonnegative homogenous forms is by adding up squares, which all arise as pullback by polynomial maps of the ur-nonnegative form x^2.

But we know, by Hilbert, that this isn’t enough to capture all nonnegative forms; for instance, it misses the Motzkin polynomial.

So what if you throw that in?  That is, we say a Motzkin is a degree-6d form

expressible as

 

P^4 Q^2 + P^2 Q^4 - 3P^2 Q^2 R^2 + R^6

for degree-d forms P,Q,R.  A Motzkin is obviously nonnegative.

It is possible that every nonnegative form of degree 6d is a sum of squares and Motzkins?  What if instead of just Motzkins we allow ourselves every nonnegative sextic?  Or every nonnegative homogeneous degree-d form in n variables for n and d less than 1,000,000?  Is it possible that the condition of nonnegativity is in this respect “finitely generated?”

 

 

 

 

 

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Boyer: curves with real multiplication over subcyclotomic fields

A long time ago, inspired by a paper of Mestre constructing genus 2 curves whose Jacobians had real multiplication by Q(sqrt(5)), I wrote a paper showing the existence of continuous families of curves X whose Jacobians had real multiplication by various abelian extensions of Q.  I constructed these curves as branched covers with prescribed ramification, which is to say I had no real way of presenting them explicitly at all.  I just saw a nice preprint by Ivan Boyer, a recent Ph.D. student of Mestre, which takes all the curves I construct and computes explicit equations for them!  I wouldn’t have thought this was doable (in particular, I never thought about whether the families in my construction were rational.) For instance, for any value of the parameter s, the genus 3 curve

2v + u^3 + (u+1)^2 + s((u^2 + v)^2 - v(u+v)(2u^2 - uv + 2v))

has real multiplication by the real subfield of \mathbf{Q}(\zeta_7).  Cool!

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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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Gross, Hacking, Keel on the geometry of cluster algebras

I have expressed amazement before about the Laurent phenomenon for cluster algebras, a theorem of Fomin and Zelevinsky which I learned about from Lauren Williams.  The paper “Birational Geometry of Cluster Algebras,”  just posted by Mark Gross, Paul Hacking, and Sean Keel, seems extremely interesting on this point.  They interpret the cluster transformations — which to an outsider look somewhat arbitrary — as elementary transforms (i.e. blow up a codim-2 thing and then blow down one of the exceptional loci thus created) on P^1-bundles on toric varieties.  And apparently the Laurent phenomenon is plainly visible from this point of view.  Very cool!

Experts are highly encouraged to weigh in below.

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Shende and Tsimerman on equidistribution in Bun_2(P^1)

Very nice paper just posted by Vivek Shende and Jacob Tsimerman.  Take a sequence {C_i} of hyperelliptic curves of larger and larger genus.  Then for each i, you can look at the pushforward of a random line bundle drawn uniformly from Pic(C) / [pullbacks from P^1] to P^1, which is a rank-2 vector bundle.  This gives you a measure \mu_i on Bun_2(P^1), the space of rank-2 vector bundles, and Shende and Tsimerman prove, just as you might hope, that this sequence of measures converges to the natural measure.

I think (but I didn’t think this through carefully) that this corresponds to saying that if you look at a sequence of quadratic imaginary fields with increasing discriminant, and for each field you write down all the ideal classes, thought of as unimodular lattices in R^2 up to homothety, then the corresponding sequence of (finitely supported) measures on the space of lattices converges to the natural one.

Equidistribution comes down to counting, and the method here is to express the relevant counting problem as a problem of counting points on a variety (in this case a Brill-Noether locus inside Pic(C_i)), which by Grothendieck-Lefschetz you can do if you can control the cohomology (with its Frobenius action.)  The high-degree part of the cohomology they can describe explicitly, and fortunately they are able to exert enough control over the low-degree Betti numbers to show that the contribution of this stuff is negligible.

In my experience, it’s often the case that showing that the contribution of the low-degree stuff, which “should be small” but which you don’t actually have a handle on, is often the bottleneck!  And indeed, for the second problem they discuss (where you have a sequence of hyperelliptic curves and a single line bundle on each one) it is exactly this point that stops them, for the moment, from having the theorem they want.

Error terms are annoying.  (At least when you can’t prove they’re smaller than the main term.)

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