Tag Archives: curves

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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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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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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Mochizuki on ABC

[Update:  Lots of traffic coming in from Hacker News, much of it presumably from outside the usual pro number theory crowd that reads this blog.  If you’re not already familiar with the ABC conjecture, I recommend Barry Mazur’s beautiful expository paper “Questions about Number.”]

[Re-update:  Minhyong Kim’s discussion on Math Overflow is the most well-informed public discussion of Mochizuki’s strategy.  (Of course, it is still very sketchy indeed, as Minhyong hastens to emphasize.)   Both Kim’s writeup and discussions I’ve had with others suggest that the best place to start may be Mochizuki’s 2000 paper “A Survey of the Hodge-Arakelov Theory of Elliptic Curves I.”]

Shin Mochizuki has released his long-rumored proof of the ABC conjecture, in a paper called “Inter-universal Teichmuller theory IV:  log-volume computations and set-theoretic foundations.”

I just saw this an hour ago and so I have very little to say, beyond what I wrote on Google+ when rumors of this started circulating earlier this summer:

I hope it’s true:  my sense is that there’s a lot of very beautiful, very hard math going on in Shin’s work which almost no one in the community has really engaged with, and the resolution of a major conjecture would obviously create such engagement very quickly.

Well, now the time has come.  I have not even begun to understand Shin’s approach to the conjecture.  But it’s clear that it involves ideas which are completely outside the mainstream of the subject.  Looking at it, you feel a bit like you might be reading a paper from the future, or from outer space.

Let me highlight one point which is clearly important, which I draw from pp.3–6 of the linked paper.

WARNING LABEL:  Of course my attempt to paraphrase is based on the barest of acquaintance with a very small section of the work and is placed here just to get people to look at Mochizuki’s paper — I may have it all wrong!

Mochizuki argues that it is too limiting to think about “the category of schemes over Spec Z,” as we are accustomed to do.  He makes the inarguable point that when X is a kind of thing, it can happen that the category of Xes, qua category, may not tell us very much about what Xes are like — for instance, if there is only one X and it has only one automorphism. Mochizuki argues that the category of schemes over a base is — if not quite this uninformative — insufficiently rich to handle certain problems in Diophantine geometry.  He wants us instead to think about what he calls the “species” of schemes over Spec Z, where a scheme in this sense is not an abstract object in a category, but something cut out by a formula.  In some sense this view is more classical than the conventional one, in which we tend to feel good about ourselves if we can “remove coordinates” and think about objects and arrows without implicitly applying a forgetful functor and referring to the object as a space with a Zariski topology or — ptui! — a set of points.

But Mochizuki’s point of view is not actually classical at all — because the point he wants to make is that formulas can be intepreted in any model of set theory, and each interpretation gives you a different category.  What is “inter-universal” about inter-universal Teichmuller theory is that it is important to keep track of all these categories, or at least many different ones.  What he is doing, he says, is simply outside the theory of schemes over Spec Z, even though it has consequences within that theory — just as (this part is my gloss) the theory of schemes itself is outside the classical theory of varieties, but provides us information about varieties that the classical theory could not have generated internally.

It’s tremendously exciting.  I very much look forward to commentary from people with a deeper knowledge than mine of Mochizuki’s past and present work.

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The conformal modulus of a mapping class

So I learned about this interesting invariant from a colloquium by Burglind Jöricke.

(Warning — this post concerns math I don’t know well and is all questions, no answers.)

Suppose you have a holomorphic map from C^* to M_g,n, the moduli space of curves.  Then you get a map on fundamental groups from \pi_1(\mathbf{C}^*) (otherwise known as Z) to \pi_1(\mathcal{M}_{g,n}) (otherwise known as the mapping class group) — in other words, you get a mapping class.

But not just any mapping class;  this one, which we’ll call u, is the monodromy of a holomorphic family of marked curves around a degenerate point.  So, for example, the image of u on homology has to be potentially unipotent.  I’m not sure (but I presume others know) which mapping classes u can arise in this way; does some power of u have to be a product of commuting Dehn twists, or is that too much to ask?

In any event, there are lots of mapping classes which you are not going to see.  Let m be your favorite one.  Now you can still represent m by a smooth loop in M_g,n.  And you can deform this loop to be a real-analytic function

f: \{z: |z| = 1\} \rightarrow \mathcal{M}_{g,n}

Finally — while you can’t extend f to all of C^*, you can extend it to some annulus with outer radius R and inner radius r.

Definition:  The conformal modulus of a mapping class x is the supremum, over all such f and all annuli, of (1/2 pi) log(R/r).

So you can think of this as some kind of measurement of “how complicated of a path do you have to draw on M_{g,n} in order to represent x?”  The modulus is infinite exactly when the mapping class is represented by a holomorphic degeneration.  In particular, I imagine that a pseudo-Anosov mapping class must have finite conformal modulus.  That is:  positive entropy (aka dilatation) implies finite conformal modulus.   Which leads Jöricke to ask:  what is the relation more generally between conformal modulus and (log of) dilatation?  When (g,n) = (0,3) she has shown that the two are inverse to each other.  In this case, the group is more or less PSL_2(Z) so it’s not so surprising that any two measures of complexity are tightly bound together.

Actually, I should be honest and say that Jöricke raised this only for g = 0, so maybe there’s some reason it’s a bad idea to go beyond braids; but the question still seems to me to make sense.  For that matter, one could even ask the same question with M_g replaced by A_g, right?  What is the conformal modulus of a symplectic matrix which is not potentially unipotent?  Is it always tightly related to the size of the largest eigenvalue?

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What I learned from Zhiwei Yun about Hilbert schemes

One knows, of course, that Hilbert schemes of smooth curves and smooth surfaces are nice, and Hilbert schemes of varieties of dimension greater than two are terrifying.

Zhiwei Yun was here giving a talk about his work with Davesh Maulik on Hilbert schemes of curves with planar singularities, and he made a point I’d never appreciated; it’s not the dimension of the variety, but the dimension of its tangent space that really measures the terrifyingness of the  Hilbert space.  Singular curves C with planar singularities are not so bad — you still have a nice Hilbert scheme with an Abel-Jacobi map to the compactified Jacobian.  But let C be the union of the coordinate axes in A^3 and all bets are off.  Hideous extra high-dimensional components aplenty.  If I had time to write a longer blog post today I would think about what the punctual Hilbert scheme at the origin looks like.  But maybe one of you guys will just tell me.

Update:  Jesse Kass explains that I am wrong about C; its Hilbert scheme has a non-smoothable component, but it doesn’t have any components whose dimension is too large.

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Anabelian puzzle 5: isogenies between Jacobians and metabelian fundamental groups

Another question that came up at the Newton Insitute:  can two different curves X,Y over F_q have the same geometrically metabelian pro-l fundamental group?

I would think not, and here’s why.  First of all, the actions of Frob_q on H_1(X,Z_ell) and on H_1(Y,Z_ell) agree.  This already implies that Jac(X) and Jac(Y) are isogenous.  Can this actually happen in large genus?  Yes:  a recent arXiv preprint by Ben Smith gives lots of explicit examples of pairs of hyperelliptic curves with isogenous Jacobians.  From Smith’s paper I learned about the recent construction by J. F. Mestre of pairs of hyperelliptic curves in every genus with isogenous Jacobians.

In other words, the geometrically abelian fundamental group need not distinguish X from Y.

But the fact that the geometrically metabelian pro-l fundamental groups agree implies the following much stronger fact.  Let X_n be the maximal abelian cover of X/F_qbar whose Galois group has exponent l^n, and define Y_n similarly.  Then X_n and Y_n have isogenous Jacobians for all n.  I would think this would be impossible if X and Y were not isomorphic; but I don’t have the slightest idea for a proof.

Baby version of this question:  do there exist non-isomorphic curves X and Y of large genus (say, for the moment, over C) whose Jacobians are isogenous, and such that each Prym of X is isogenous to a Prym of Y?

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Anabelian puzzle 2: birational sections in the least non-abelian case

(Note:  this is being posted from an airport shortly before boarding, so this is less edited than usual.  I have some more concrete remarks on the puzzle below but will save them for later.)

Let X/K be a variety over a number field; then we have an exact sequence of etale fundamental groups

\pi_1(X/\bar{K}) \rightarrow \pi_1(X/K) \rightarrow G_K

and every point of X(K) induces a section from G_K back to \pi_1(X/K).  Grothendieck’s section conjecture asserts that, at least for certain classes of varieties X including projective smooth curves of genus greater than 1, the group-theoretic sections above are in fact in bijection with X(K).

A point P of X(K) is also a point of U(K), where U is any open subscheme of X containing P.  This gives you a section G_K \rightarrow \pi_1(U/K) for any such U; the limit of all these fundamental groups is the Galois group of the maximal extension of the function field K(X) unramified at P.

In fact, you can associate to P \in X(K) a section from G_K to the whole absolute Galois group of K(X) (or, better, a “bouquet” of related sections.)  The choice of P determines a decomposition group in G_{K(X)}, isomorphic to a semidirect product of its inertia group by a copy of G_K; now a section of this semidirect product is a section from G_K to G_{K(X)}.

So a weaker version of the section conjecture, the birational section conjecture, asserts that all of these sections from G_K to G_{K(X)} come from X(K).  Koenigsmann proved a few years ago that the birational section conjecture holds for K = Q_p.  A recent paper of Florian Pop proves something much stronger; that the birational section conjecture holds over finite extensions of Q_p even when G_{K(X)} is replaced with a really puny quotient.  Namely:  let K’ be the maximal elementary 2-abelian extension of Kbar(X), and let K” be the maximal elementary 2-abelian extension of K”.  Then you have a group Gamma described by an exact sequence

1 \rightarrow G(K''/\bar{K}(X)) \rightarrow \Gamma \rightarrow G_K

and what Pop proves is that this group Gamma, which you might call the “geometrically metabelian mod-2 fundamental group of X,” “remembers” enough about the curve X that already the sections from G_K to Gamma are all given by points of X.  Pop calls this a “minimalist birational section conjecture.”

One then wonders:  just how minimal can one get?  Abelianizing Gal(Kbar(X)) is too brutal; the resulting “geometrically abelian” fundamental group has lots of “extra” sections coming from points of Jac(X).  (Note:  just noticed this paper of Esnault and Wittenberg about exactly this, haven’t read it yet.)

Now here’s the puzzle — suppose we let K’ be the maximal elementary 2-abelian extension of Kbar(X) (i.e. the compositum of all quadratic extensions) and K” be the maximal elementary abelian 2-extension of K’ such that Gal(K”/Kbar(X)) has nilpotence class 2.  Then again you have

1 \rightarrow G(K''/\bar{K}(X)) \rightarrow \Gamma' \rightarrow G_K

where \Gamma' could be called the “geometrically 2-nilpotent mod 2 fundamental group.”

So what are the sections from G_K to \Gamma'?  Is this fundamental group so minimalist that there are tons of extra sections, or is it the maximally minimalist context where a birational section conjecture could hold?

Update:  A bit jetlagged, but let me at least add one concrete question to this post.  I claim that a birational section as above would give you a function

f: U(\mathbf{Q}) \rightarrow \mathbf{Q}^*/(\mathbf{Q}^*)^2

for some open subscheme U of P^1, with the property that the Hilbert symbol

(f(a)/(a-b), f(b)/(b-a))

always vanishes.  Can you think of any such functions besides f(a) = a + constant, or f(a) = constant?

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“Le Groupe Fondamental de la Droite Projective Moins Trois Points” is now online

The three papers that influenced me the most at the beginning of my mathematical career were “Rational Isogenies of Prime Degree,” by my advisor, Barry Mazur; Serre’s “Sur les représentations modulaires de degré 2 de \text {Gal}({\overline {\Bbb Q}}/{\Bbb Q});” and Deligne’s 200-page monograph on the fundamental group of the projective line minus three points.  The year after I got my Ph.D. I used to carry around a battered Xerox of this paper wherever I went, together with a notebook in which I recorded my confusions, questions, and insights about what I was reading.  This was the paper where I learned what a motive was, or at least some of the things a motive should be; where I first encountered the idea of a Tannakian category; where I first learned the definition of a Hodge structure, and what was meant by “periods.” Most importantly, I learned Deligne’s philosophy about the fundamental group:  that the grand questions proposed by Grothendieck in the “Esquisse d’un Programme” regarding the action of Gal(Q) on the etale fundamental group \pi := \pi_1^{et}(\mathbf{P}^1/\overline{\mathbf{Q}} - 0,1,\infty) were simply beyond our current reach, but that the nilpotent completion of \pi — which seems like only a tiny, tentative step into the non-abelian world! — nonetheless contains a huge amount of arithmetic information.  My favorite contemporary manifestation of this philosophy is Minhyong Kim’s remarkable work on non-abelian Chabauty.

Anyway:  Deligne’s article appears in the MSRI volume Galois Groups over Q, which is long out of print; I bought a copy at MSRI in 1999 and I don’t know anyone who’s gotten their hands on one since.  Kirsten Wickelgren, a young master of the nilpotent fundamental group, asked me the obvious-in-retrospect question of whether it was possible to get Deligne’s article back in print.  I talked to MSRI about this and it turns out that, since Springer owns the copyright, the book can’t be reprinted; but Deligne himself is allowed to make a scan of the article available on his personal web page.  Deligne graciously agreed:  and now, here it is, a publicly available .pdf scan of “Le Groupe Fondamental de la Droite Projective Moins Trois Points.”

Enjoy!

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The entropy of Frobenius

Since Thurston, we know that among the diffeomorphisms of surfaces the most interesting ones are the pseudo-Anosov diffeomorphisms; these preserve two transverse folations on the surface, stretching one and contracting the other by the same factor.  The factor, usually denoted \lambda, is called the dilatation of the diffeomorphism and its logarithm is called the entropy. It turns out that \lambda, which is evidently a real number greater than 1, is in fact an algebraic integer, the largest eigenvalue of a matrix that in some sense keeps combinatorial track of the action of the diffeomorphism on the surface.  You might think of it as a kind of measure of the “complexity” of the diffeomorphism.  A recent preprint by my colleague Jean-Luc Thiffeault says much about how to compute these dilatations in practice, and especially how to hunt for diffeomorphisms whose dilatation is as small as possible.

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