Tag Archives: number theory

“On l-torsion in class groups of number fields” (with L. Pierce, M.M. Wood)

New paper up with Lillian Pierce and Melanie Matchett Wood!

Here’s the deal.  We know a number field K of discriminant D_K has class group of size bounded above by roughly D_K^{1/2}.  On the other hand, if we fix a prime l, the l-torsion in the class group ought to be a lot smaller.  Conjectures of Cohen-Lenstra type predict that the average size of the l-torsion in the class group of D_K, as K ranges over a “reasonable family” of algebraic number fields, should be constant.  Very seldom do we actually know anything like this; we just have sporadic special cases, like the Davenport-Heilbronn theorem, which tells us that the 3-torsion in the class group of a random quadratic field is indeed constant on average.

But even though we don’t know what’s true on average, why shouldn’t we go ahead and speculate on what’s true universally?  It’s too much to ask that Cl(K)[l] literally be bounded as K varies (at least if you believe even the most modest version of Cohen-Lenstra, which predicts that any value of dim Cl(D_K)[l] appears for a positive proportion of quadratic fields K) but people do think it’s small:

Conjecture:  |Cl(K)[l]| < D_K^ε.

Even beating the trivial bound

|Cl(K)[l]| < |Cl(K)| < D_K^{1/2 + ε}

is not easy.  Lillian was the first to do it for 3-torsion in quadratic fields.  Later, Helfgott-Venkatesh and Venkatesh and I sharpened those bounds.  I hear from Frank Thorne that he, Bhargava, Shankar, Tsimerman and Zhao have a nontrivial bound on 2-torsion for the class group of number fields of any degree.

In the new paper with Pierce and Wood, we prove nontrivial bounds for the average size of the l-torsion in the class group of K, where l is any integer, and K is a random number field of degree at most 5.  These bounds match the conditional bounds Akshay and I get on GRH.  The point, briefly, is this.  To make our argument work, Akshay and I needed GRH in order to guarantee the existence of a lot of small rational primes which split in K.  (In a few cases, like 3-torsion of quadratic fields, we used a “Scholz reflection trick” to get around this necessity.)  At the time, there was no way to guarantee small split primes unconditionally, even on average.  But thanks to the developments of the last decade, we now know a lot more about how to count number fields of small degree, even if we want to do something delicate like keep track of local conditions.  So, for instance, not only can one count quartic fields of discriminant < X, we can count fields which have specified decomposition at any specified finite set of rational primes.  This turns out to be enough — as long as you are super-careful with error terms! — to  allow us to show, unconditionally, that most number fields of discriminant < D have enough small split primes to make the bound on l-torsion go.  Hopefully, the care we took here to get counts with explicit error terms for number fields subject to local conditions will be useful for other applications too.


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Bourgain, Gamburd, Sarnak on Markoff triples

Such a great colloquium last week by Peter Sarnak, this year’s Hilldale Lecturer, on his paper with Bourgain and Gamburd.  My only complaint is that he promised to talk about the mapping class group and then barely did!  So I thought I’d jot down what their work has to do with mapping class groups and spaces of branched covers.

Let E be a genus 1 Riemann surface — that is, a torus — and O a point of E.  Then pi_1(E-O) is just a free group on two generators, whose commutator is (the conjugacy class of) a little loop around the puncture.  If G is a group, a G-cover of E branched only at O is thus a map from pi_1(E-O) to G, which is to say a pair (a,b) of elements of G.  Well, such a pair considered up to conjugacy, since we didn’t specify a basepoint for our pi_1.  And actually, we might as well just think about the surjective maps, which is to say the connected G-covers.

Let’s focus on the case G = SL_2(Z).  And more specifically on those maps where the puncture class is sent to a matrix of trace -2.  Here’s an example:  we can take

a_0 = \left[ \begin{array}{cc} 1 & 1 \\ 1 & 2 \end{array} \right]

b_0 = \left[ \begin{array}{cc} 2 & 1 \\ 1 & 1 \end{array} \right]

You can check that in this case the puncture class has trace -2; that is, it is the negative of a unipotent matrix.  Actually, I gotta be honest, these matrices don’t generate SL_2(Z); they generate a finite-index subgroup H of SL_2(Z), its commutator.

Write S for the set of all conjugacy classes of pairs (a,b) of matrices which generate H and have commutator with trace -2.  It turns out that this set is the set of integral points of an affine surface called the Markoff surface:  namely, if we take x = Tr(a)/3, y = Tr(b)/3, and z = Tr(ab)/3, then the three traces obey the relation

x^2 + y^2 + z^2 = 3xyz

and indeed every solution to this equation corresponds to an element of S.

So the integral points on the Markoff surface are acted on by an infinite discrete group.  Which if you just look at the equation seems like kind of a ridiculous miracle.  But in the setting of H-covers is very natural.  Because there’s a natural group acting on S: namely, the mapping class group Γ of type (1,1).  This group’s whole purpose in life is to act on the fundamental group of a once-punctured torus!  (For readers unfamiliar with mapping class groups, I highly recommend Benson Farb and Dan Margalit’s wonderful textbook.)   So you start with a surjection from pi_1(E-O) to H, you compose with the action of  Γ, and you get a new homomorphism.  The action of  Γ on pi_1(E-O) is only outer, but that’s OK, because we’re only keeping track of conjugacy classes of homomorphisms from pi_1(E-O) to H.

So Γ acts on S; and now the lovely theorem is that this action is transitive.

I don’t want to make this mapping class group business sound more abstract than it is.  Γ isn’t a mystery group; it acts on H_1(E-O), a free abelian group of rank 2, which gives a map from Γ to SL_2(Z), which turns out to be an isomorphism.  What’s more, the action of Γ on pairs (a,b) is completely explicit; the standard unipotent generators of SL_2(Z) map to the moves

(a,b) -> (ab,b)

(a,b) -> (a,ab)

(Sanity check:  each of these transformations preserves the conjugacy class of the commutator of a and b.)

Sarnak, being a number theorist, is interested in strong approximation: are the integral solutions of the Markoff equation dense in the adelic solutions?   In particular, if I have a solution to the Markoff equation over F_p — which is to say, a pair (a,b) in SL_2(F_p) with the right commutator — can I lift it to a solution over Z?

Suppose I have a pair (a,b) which lifts to a pair (a,b).  We know (a,b) = g(a_0,b_0) for some g in Γ.  Thus (a,b) = g(a_0,b_0).  In other words, if strong approximation is true, Γ acts transitively on the set S_p of Markoff solutions mod p.  And this is precisely what Bourgain, Gamburd, and Sarnak conjecture.  (In fact, they conjecture more:  that the Cayley-Schreier graph of this action is an expander, which is kind of a quantitative version of an action being transitive.)  One reason to believe this:  if we replace F_p with C, we replace S with the SL_2(C) character variety of pi_1(E-O), and Goldman showed long ago that the action of mapping class groups on complex character varieties of fundamental groups was ergodic; it mixes everything around very nicely.

Again, I emphasize that this is on its face a question of pure combinatorial group theory.  You want to know if you can get from any pair of elements in SL_2(F_p) with negative-unipotent commutator to any other via the two moves above.  You can set this up on your computer and check that it holds for lots and lots of p (they did.)  But it’s not clear how to prove this transitivity for all p!

They’re not quite there yet.  But what they can prove is that the action of Γ on S_p has a very big orbit, and has no very small orbits.

Now that G is the finite group SL_2(F_p), we’re in my favorite situation, that of Hurwitz spaces.  The mapping class group Γ is best seen as the fundamental group of the moduli stack M_{1,1} of elliptic curves.  So an action of Γ on the finite set S_p is just a cover H_p of M_{1,1}.  It is nothing but the Hurwitz space parametrizing maps (f: X -> E) where E is an elliptic curve and f an SL_2(F_p)-cover branched only at the origin.  What Bourgain, Gamburd, and Sarnak conjecture is that H_p is connected.

If you like, this is a moduli space of curves with nonabelian level structure as in deJong and Pikaart.  Or, if you prefer (and if H_p is actually connected) it is a noncongruence modular curve corresponding to the stabilizer of an element of S_p in Γ = SL_2(Z).  This stabilizer is in general going to be a noncongruence subgroup, except it is a congruence subgroup in the more general sense of Thurston.

This seems like an interesting family of algebraic curves!  What, if anything, can we say about it?

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Very happy to see that the L-functions and Modular Forms Database is now live!

When I was a kid we looked up our elliptic curves in Cremona’s tables, on paper.  Then William Stein created the Modular Forms Database (you can still go there but it doesn’t really work) and suddenly you could look at the q-expansions of cusp forms in whatever weight and level you wanted, up to the limits of what William had computed.

The LMFDB is a sort of massively souped up version of Cremona and Stein, put together by a team of dozens and dozens of number theorists, including too many friends of mine to name individually.  And it’s a lot more than what the title suggests:  the incredibly useful Jones-Roberts database of local fields is built in; there’s a database of genus 2 curves over Q with small conductor; it even has Maass forms!  I’ve been playing with it all night.  It’s like an adventure playground for number theorists.

One of my first trips through Stein’s database came when I was a postdoc and was thinking about Ljunggren’s equation y^2 + 1 = 2x^4.  This equation has a large solution, (13,239), which has to do with the classical identity

\pi/4 = 4\arctan(1/5) - \arctan(1/239).

It turns out, as I explain in an old survey paper, that the existence of such a large solution is “explained” by the presence of a certain weight-2 cuspform in level 1024 whose mod-5 Galois representation is reducible.

With the LMFDB, you can easily wander around looking for more such examples!  For instance, you can very easily ask the database for non-CM elliptic curves whose mod-7 Galois representation is nonsurjective.  Among those, you can find this handsome curve of conductor 1296, which has very large height relative to its conductor.  Applying the usual Frey curve trick you can turn this curve into the Diophantine oddity

3*48383^2 – (1915)^3 = 2^13.

Huh — I wonder whether people ever thought about this Diophantine problem, when can the difference between a cube and three times a square be a power of 2?  Of course they did!  I just Googled

48383 Diophantine

and found this paper of Stanley Rabinowitz from 1978, which finds all solutions to that problem, of which this one is the largest.

Now whether you can massage this into an arctan identity, that I don’t know!



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Pila on a “modular Fermat equation”

I like this paper by Pila that just went up on the arXiv, which shows the way that you can get Diophantine consequences from the rapid progress being made in theorems of Andre-Oort type.  (I also want to blog about Tsimerman + Zhang + Yuan on “average Colmez” and Andre-Oort, maybe later!)

Pila shows that if N and M are sufficiently large primes, you can’t have elliptic curves E_1/Q and E_2/Q such that E_1 has an N-isogenous curve E_1 -> E’_1, E_2 has an M-isogenous curve E_2 -> E’_2, and j(E’_1) + j(E’_2) = 1.  (It seems to me the proof uses little about this particular algebraic relation and would work just as well for any f(j(E’_1),j(E’_2)) whose vanishing didn’t cut out a modular curve in X(1) x X(1).)  (This is “Fermat-like” in that it asserts finiteness of rational points on a natural countable family of high-genus curves; a more precise analogy is explained in the paper.)

How this works, loosely:  suppose you have such an (E_1, E_2).  A theorem of Kühne guarantees that E_1 and E_2 are not both CM (I didn’t know this!) It follows (WLOG assume N > M) that the N-isogenies of E_1 are defined over a field of degree at least N^a for some small a (Pila uses more precise bounds coming from a recent paper of Najman.)  So the Galois conjugates of (E’_1, E’_2) give you a whole bunch of algebraic points (E”_1, E”_2) with j(E”_1) + j(E”_2) = 1.

So what?  Rational curves have lots of low-height algebraic points.  But here’s the thing.  These isogenous choices of (E’_1, E’_2) aren’t just any algebraic points on X(1) x X(1); they represent pairs of elliptic curves drawn from a {\em fixed pair of isogeny classes}.  Let H be the hyperbolic plane as usual, and write (z,w) for a point on H x H corresponding to (E’_1, E’_2).  Then the other choices (E”_1, E”_2) correspond to points (gz,hw) with g,h in GL(Q).  GL(Q), not GL(R)!  That’s what we get from working in a fixed isogeny class.  And these points satisfy

j(gz) + j(hw) = 1.

To sum up:  you have a whole bunch of rational points (g,h) on GL_2 x GL_2.  These points are pretty low height (for this Pila gestures at a paper of his with Habegger.)  And they lie on the surface j(gz) + j(hw) = 1.  But this surface is a totally non-algebraic thing, because remember, j is a transcendental function on H!  So (Pila’s version of) the Ax-Lindemann theorem (correction from comments:  the Pila-Wilkie theorem) generates a contradiction; a transcendental curve can’t have too many low-height rational points.

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Bounded rank was probable in 1950

Somehow I wrote that last post about bounded ranks without knowing about this paper by Mark Watkins and many other authors, which studies in great detail the variation in ranks in quadratic twists of the congruent number curve.  I’ll no doubt have more to say about this later, but I just wanted to remark on a footnote; they say they learned from Fernando Rodriguez-Villegas that Neron wrote in 1950:

On ignore s’il existe pour toutes les cubiques rationnelles, appartenant a un corps donné une borne absolute du rang. L’existence de cette borne est cependant considérée comme probable.

So when I said the conventional wisdom is shifting from “unbounded rank” towards “bounded rank,” I didn’t tell the whole story — maybe the conventional wisdom started at “bounded rank” and is now shifting back!

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Are ranks bounded?

Important update, 23 Jul:  I missed one very important thing about Bjorn’s talk:  it was about joint work with a bunch of other people, including one of my own former Ph.D. students, whom I left out of the original post!  Serious apologies.  I have modified the post to include everyone and make it clear that Bjorn was talking about a multiperson project.  There are also some inaccuracies in my second-hand description of the mathematics, which I will probably deal with by writing a new post later rather than fixing this one.

I was only able to get to two days of the arithmetic statistics workshop in Montreal, but it was really enjoyable!  And a pleasure to see that so many strong students are interested in working on this family of problems.

I arrived to late to hear Bjorn Poonen’s talk, where he made kind of a splash talking about joint work by Derek Garton, Jennifer Park, John Voight, Melanie Matchett Wood, and himself, offering some heuristic evidence that the Mordell-Weil ranks of elliptic curves over Q are bounded above.  I remember Andrew Granville suggesting eight or nine years ago that this might be the case.  At the time, it was an idea so far from conventional wisdom that it came across as a bit cheeky!  (Or maybe that’s just because Andrew often comes across as a bit cheeky…)

Why did we think there were elliptic curves of arbitrarily large rank over Q?  I suppose because we knew of no reason there shouldn’t be.  Is that a good reason?  It might be instructive to compare with the question of bounds for rational points on genus 2 curves.  We know by Faltings that |X(Q)| is finite for any genus 2 curve X, just as we know by Mordell-Weil that the rank of E(Q) is finite for any elliptic curve E.  But is there some absolute upper bound for |X(Q)|?  When I was in grad school, Lucia Caporaso, Joe Harris, and Barry Mazur proved a remarkable theorem:  that if Lang’s conjecture were true, there was some constant B such that |X(Q)| was at most B for every genus 2 curve X.  (And the same for any value of 2…)

Did this make people feel like |X(Q)| was uniformly bounded?  No!  That was considered ridiculous!  The Caporaso-Harris-Mazur theorem was thought of as evidence against Lang’s conjecture.  The three authors went around Harvard telling all the grad students about the theorem, saying — you guys are smart, go construct sequences of genus 2 curves with growing numbers of points, and boom, you’ve disproved Lang’s conjecture!

But none of us could.

Continue reading

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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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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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Elliptic curves with isomorphic cyclic 13-subgroups?

I liked this MathOverflow question, which asks:  are there two non-isogenous elliptic curves over Q, each one of which has a rational cyclic 13-isogeny, and such that the kernels of the two isogenies are isomorphic as Galois modules?

This is precisely to look for rational points on the modular surface S parametrizing pairs (E,E’,C,C’,φ), where E and E’ are elliptic curves, C and C’ are cyclic 13-subgroups, and φ is an isomorphism between C and C’.

S is a quotient of X_1(13) x X_1(13) by the diagonal in the natural (Z/13Z)^* x (Z/13Z)^* action.

Is S general type, rational, what?



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