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 itself doesn’t have (B):  consider the sequence   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!

Poonen-Rains, Selmer groups, random maximal isotropics, random orthogonal matrices

At the AIM workshop on Cohen-Lenstra heuristics last week I got to hear Bjorn Poonen give a terrific talk about his recent work with Eric Rains about the distribution of mod p Selmer groups in a quadratic twist family of elliptic curves.

Executive summary:  if E is an elliptic curve, say in Weierstrass form y^2 = f(x), and d is a squarefree integer, then we can study the mod p Selmer group Sel_d(E) of the quadratic twist dy^2 = f(x), which sits inside the Galois cohomology H^1(G_Q, E_d[p]).  This is a finite-dimensional vector space over F_p.  And by analogy with the Cohen-Lenstra heuristics for class groups, we can ask whether these groups obey a probability distribution as d varies — that is, does

Pr(dim Sel_d(E) = r | d in [-B, … B])

approach a limit P_r as B goes to infinity, and if so, what is it?

The Poonen-Rains heuristic is based on the following charming observation.  The product of the local cohomology groups H_1(G_v, E[p]) is an infinite-dimensional F_p-vector space endowed with a bilinear form coming from cup product.  In here you have two subspaces:  the image of global cohomology, and the image of local Mordell-Weil.  Each one of these, it turns out, is maximal isotropic — and their intersection is exactly the Selmer group.  So the Selmer group can be seen as the intersection of two maximal isotropic subspaces in a very large quadratic space.

Heuristically, one might think of these two subspaces as being randomly selected among maximal isotropic subspaces.  This suggests a question:  if P_{r,N} is the probability that the intersection of two random maximal isotropics in F_p^{2N} has dimension r, does P_{r,N} approach a limit as N goes to infinity?  It does — and the Poonen-Rains heuristic then asks that the probability that dim Sel_d(E)  = r approaches the same limit.  This conjecture agrees with theorems of Heath-Brown, Swinnerton-Dyer, and Kane in the case p=2, and with results of Bhargava and Shankar when p <= 5 (Bhargava and Shankar work with a family of elliptic curves of bounded height, not a quadratic twist family, but it is not crazy to expect the behavior of Selmer to be the same.)  And in combination with Delaunay’s heuristics for variation of Sha, it recovers Goldfeld’s conjecture that elliptic curves are half rank 0 and half rank 1.

Johan de Jong wrote about a similar question, concentrating on the function field case, in his paper “Counting elliptic surfaces over finite fields.”  (This is the first place I know where the conjecture “Sel_p should have size 1+p on average” is formulated.)  He, too, models the Selmer group by a “random linear algebra” construction.  Let g be a random orthogonal matrix over F_p; then de Jong’s model for the Selmer group is coker(g-1).  This is a natural guess in the function field case:  if E is an elliptic curve over a curve C / F_q, then the Selmer group of E is a subquotient of the etale H^2 of an elliptic surface S over F_q; thus it is closely related to the coinvariants of Frobenius acting on the H^2 of S/F_qbar.  This H^2 carries a symmetric intersection pairing, so Frobenius (after scaling by q) is an orthogonal matrix, which we want to think of as “random.”  (As first observed by Friedman and Washington, the Cohen-Lenstra heurstics can be obtained in similar fashion, but the relevant cohomology is H^1 of a curve instead of H^2 of a surface; so the relevant pairing is alternating and the relevant statistics are those of symplectic rather than orthogonal matrices.)

But this presents a question:  why do these apparently different linear algebra constructions yield the same prediction for the distribution of Selmer ranks?

Here’s one answer, though I suspect there’s a slicker one.

A nice way to describe the distributions that arise in problems of Cohen-Lenstra type is by computing their moments.  But the usual moments (e.g. “expected kth power of dimension of Selmer” or “kth power of order of Selmer” tend not to behave so well.  Better is to compute “expected number of injections from F_p^k into Selmer,” which has a cleaner answer in every case I know.  If the size of the Selmer group is X, this number is

(X-1)(X-p)….(X-p^{k-1}).

Evidently, if you know these “moments” for all k, you can compute the usual moments E(X^k) (which are indeed computed explicitly in Poonen-Rains) and vice versa.

Now:  let A be the random variable (valued in abelian groups!)  “intersection of two random maximal isotropics in a 2N-dimensional quadratic space V” and B be “coker(g-1) where g is a random orthogonal N x N matrix.”

The expected number of injections from F_p^k to B is just the number of injections from F_p^k to F_p^N which are fixed by g.  By Burnside’s lemma, this is the number of orbits of the orthogonal group on Inj(F_p^k, F_p^N).  But by Witt’s Theorem, the orbit of an injection f: F_p^k -> F_p^N is precisely determined by the restriction of the orthogonal form to F_p^k; the number of symmetric bilinear forms on F_p^k is p^((1/2)k(k+1)) and so this is the expected value to be computed.

What about the expected number of injections from F_p^k to A?  We can compute this as follows.  There are about p^{Nk} injections from F_p^k to V.  Of these, about p^{2Nk – (1/2)k(k+1)} have isotropic image.  Call the image W;  we need to know how often W lies in the intersection of the two maximal isotropics V_1 and V_2.  The probability that W lies in V_1 is easily seen to be about p^{-Nk + (1/2)k(k+1)}, and the probability that W lies in V_2 is the same; these two events are independent, so the probability that W lies in A is about p^{-2NK + (1/2)k(k+1)}.  Summing over all isotropic injections gives an expected number of p^{(1/2)k(k+1)} injections from F_p^k to A.  Same answer!

(Note:  in the above paragraph, “about” always means “this is the limit as N gets large with k fixed.”)

What’s the advantage of having two different “random matrix” formulations of the heuristic?  The value of the “maximal isotropic intersection” model is clear — as Poonen and Rains show, the Selmer really is an intersection of maximal isotropic subspaces in a quadratic space.  One value of the “orthogonal cokernel” model is that it’s clear what it says about the Selmer group mod p^k.

Question: What does the orthogonal cokernel model predict about the mod-4 Selmer group of a random elliptic curve?  Does this agree with the theorem of Bhargava and Shankar, which gives the first moment of Sel_4 in a family of elliptic curves ordered by height?

Diophantineness: Mazur-Rubin and Kollar

Last year I blogged about an argument of Bjorn Poonen, which shows that Hilbert’s tenth problem has a negative solution over the ring of integers O_K of a number field K whenever there exists an elliptic curve E/Q such that E(Q) and E(K) both have rank 1.  That is:  there’s no algorithm that tells you whether a given polynomial equation over O_K is solvable.  The idea is that under these circumstances one can construct a Diophantine model for Z inside O_K; one already knows (by Matijasevic, Robinson, etc.) that no algorithm can determine whether a polynomial equation over Z has a solution, and the same property is now inherited by the ring O_K.

The necessary fact about existence of low-rank elliptic curves over number fields (actually, not quite the fact Poonen asked for but something weaker that suffices) has now been proven, subject to a hypothesis on the finiteness of Sha, by Mazur and Rubin: see Theorems 1.11 and 1.12.  So, if you believe Sha to be finite, you believe that Hilbert’s tenth problem has a negative answer for the ring of integers of every number field.

The result of Mazur and Rubin is actually much more substantial than the corollary I mention here, giving for instance quite strong lower bounds on the number of twists of an elliptic curve E with specified 2-Selmer rank.  But I haven’t studied the argument sufficiently to say anything serious about what’s inside.

I recently returned from the “Spaces of curves and their interaction with Diophantine problems” conference at Columbia, where Janos Kollar discussed the question:  Which subsets S of C(t) are Diophantine?  That is, which have the property that they can be written as the set of s in C(t) such that for some polynomial f in k+1 variables with coefficients in C(t).  Kollar explained how to prove that the polynomial ring C[t] is not Diophantine in C(t).  The idea is to show that any “sufficiently large” Diophantine subset S of C(t) contains functions whose denominators are essentially arbitrary; more precisely (but not completely precisely!) if X in Sym^d P^1 is the locus of degree-d denominators of elements of S, the Zariski closure of X needs to be — well, it doesn’t have to contain all degree-d polynomials, but it has to contain a set of the form as F,G range over polynomials of degrees s,t with s+rt = d.  In particular, it’s not possible for the denominator to be identically 1, as would be the case if S were C[t].  In fact, this argument shows that no finitely generated C-subalgebra of C(t) is Diophantine over C[t].

Open question:  is the localization of C(t) at t Diophantine over C(t)?

Update: When I first posted this I didn’t notice that Kollar’s result is already out, in the new journal Algebra and Number Theory, so you can go to the source for more details.  ANT, by the way, is a free electronically distributed journal with a terrific editorial board, and I highly recommend submitting there.

Are there elliptic curves with small rank?

Usually when people muse about distribution of ranks of elliptic curves, they’re wondering how large the rank of an elliptic curve can be. But as Tim Dokchitser pointed out to me, there are questions in the opposite direction which are equally natural, and equally mysterious. For instance: we do not know that, for every number field K, there exists an elliptic curve E/K whose Mordell-Weil rank is less than 100. Isn’t that strange?

Along similar lines, Bjorn Poonen asks: is it the case that, for every number field K, there exists an elliptic curve E/Q such that E(Q) and E(K) have the same positive rank? Again, we don’t have a clue. See Bjorn’s expository article “Undecidability in Number Theory” (.pdf link) to find out what this has to do with Hilbert’s 10th problem.