Tag Archives: elliptic curves


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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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.

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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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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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“Cremona Tables” for elliptic curves over Q(sqrt(5)), by William Stein and his REU

William’s REU at the University of Washington has constructed a table, in the “Cremona format,” of all elliptic curves over Q(sqrt(5)) with conductor of norm up to 1831.  More complete data in computer-readable form here.

The authors of this table, besides William, are Ben Leveque, Ashwath Rabindranath, Ariah Klages-Mundt, Paul Sharaba, Andrew Ohana, Joanna Gaski, Aly Deines, and Jon Bober.

What are good questions to ask this dataset?

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Poonen-Rains and lines on a quadric surface

One feature of the Poonen-Rains heuristics that might seem strange at first is that the dimension of the Selmer group isn’t 0 almost all  the time.  This is by contrast with the Cohen-Lenstra heuristic, where the p-torsion in the class group is indeed trivial about 1-1/p of the time.  Instead, the Poonen-Rains heuristics predict that the p-Selmer rank is 0 about half the time and 1 about half the time, with about 1/p’s worth of measure devoted to ranks 2 or higher.  Of course, given that we expect a random elliptic curve to have Mordell-Weil rank 1 half the time, it would be bad news for their heuristic if it predicted a lower frequency of positive Selmer rank!

But why is the intersection of two maximal isotropic subspaces 1 half the time and 0 half the time?  You can get a nice picture of what’s going on by thinking about the case of  a quadratic form Q in 4 variables.  The vanishing of the quadratic form cuts out a quadric surface in P^3.  A maximal isotropic subspace is a 2-dimensional space on which Q vanishes — in other words, a line on the quadric.  The intersection of two maximal isotropics is o-dimensional if the corresponding lines are disjoint, 1-dimensional if the lines intersect at a point, and 2-dimensional when the lines coincide.  So what’s the probability that two random lines on the quadric intersect?  The key point is that there are two families of lines.  If L1 and L2 come from different families, they intersect; if they come from the same family, they’re disjoint (except in the unlikely event they coincide.)  So there you go — the intersection of the maximal isotropics is split 50-50 between 0-dimensional and 1-dimensional.  More generally, the variety of maximal isotropic subspaces in an even-dimensional orthogonal space has two components, and this explains the leading term of Poonen-Rains.

It would be interesting to understand how to describe the “two types of maximal isotropics” in the infinite-dimensional F_p-vector space considered by Poonen-Rains, and to understand why the two maximal isotropics supplied by a given elliptic curve lie in the same family if and only if the L-function of E has even functional equation, which should lead one to expect that Sel_p(E) has even rank  (or even, thanks to recent progress on the parity conjecture by Nekovar, Kim, los Dokchitsers, etc., implies that Sel_p(E) has even rank, subject to finiteness of Sha.)





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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


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?

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Hain-Matsumoto, “Galois actions on fundamental groups of curves…”

I recently had occasion to spend some time with Richard Hain and Makoto Matsumoto’s 2005 paper “Galois actions on fundamental groups and the cycle C – C^-,” which I’d always meant to delve into.  It’s really beautiful!  I cannot say I’ve really delved — maybe something more like scratched — but I wanted to share some very interesting things I learned.

Serre proved long ago that the image of the l-adic Galois representation on an elliptic curve E/Q is open in GL_2(Z_l), so long as E doesn’t have CM.  This is a geometric condition on E, which is to say it only depends on the basechange of E to an algebraic closure of Q, or even to C.

What’s the analogue for higher genus curves X?  You might start by asking about the image of the Galois representation G_Q -> GSp_2g(Z_l) attached to the Tate module of the Jacobian of X.  This image lands in GSp_{2g}(Z_l).  Just as with elliptic curves, any extra endomorphisms of Jac(X) may force the image to be much smaller than GSp_{2g}(Z_l).  But the question of whether the image of rho must be open in GSp_2g(Z_l) whenever no “obvious” geometric obstruction forbids it is difficult, and still not completely understood.  (I believe it’s still unknown when g is a multiple of 4…?)  One thing we do know in general, though, is that when X is the generic curve of genus g (that is, the universal curve over the function field Q(M_g) of M_g) the resulting representation

\rho^{univ}: G_{Q(M_g)} \rightarrow GSp_{2g}(\mathbf{Z}_\ell)

is surjective.

Hain and Matsumoto generalize in a different direction.  When X is a curve of genus greater than 1 over a field K, the Galois group of K acts on more than just the Tate modules (or l-adic H_1) of X; it acts on the whole pro-l geometric fundamental group of X, which we denote pi.  So we get a morphism

\rho_{X/K}: G_K \rightarrow Aut(\pi)

What does it mean to ask this representation to have “big image”?

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Local points on quadratic twists of X_0(N)

A new paper on the subject by my Ph.D. student, Ekin Ozman, is up on the arXiv today.

The twists in question are isomorphic to X_0(N) over a quadratic field K = Q(sqrt(d)), but not over Q; the twist is via the Atkin-Lehner involution w_N, which is to say that the rational points on such a twist are in bijection with the points P of X_0(N)(K) satisfying

P^\sigma = w_N P

where \sigma is the generator of Gal(K/Q).  Call this twist X^d(N).

Why do we care about these twists?  For one thing, X^d(N) parametrizes certain classes of Q-curves, elliptic curves over Qbar which are isogenous to all their Galois conjugates.  These guys turn out to be more flexible than elliptic curves over Q for constructing “Frey curves” attached to Diophantine equations, but have all the same handsome modularity properties as elliptic curves over Q, allowing the usual Mazur-Frey-Serre-Ribet-Wiles style argument to go through.

More generally, though, the X^d(N) form a very natural class of modular curves whose arithmetic hasn’t been much thought about.  For instance:  is there an analogue of Mazur’s theorem?  That is, do these curves have rational points?  Here one immediately encounters a divergence from the untwisted case — X^d(N) might not even have local points!  The cusps, which provide Q_p-points on X_0(N) for every N and p, are not rational points on X^d(N); without them, there’s no reason for X_0(N)(Q_p) to be nonempty.  In a 2004 survey paper about Q-curves, I asked (Problem A in the linked paper) for which d and N the curve X^d(N) had local points everywhere; this certainly has to be settled before any investigation of the global points can start.  Pete Clark (see the appendix to this paper), Jordi Quer, and Josep Gonzalez got partial results on the problem; now Ekin has almost entirely solved it, getting an exact criterion for the existence of local points whenever K and Q(sqrt(-N)) have no common primes of ramification.

I was never able to see a simple way to do this — and it turns out that’s because the answer, as Ekin works it out, is actually pretty complicated!  So I won’t state her theorem here; I’ll just say that it’s competely explicit and it allows you to compute just about whatever you want.  For example:  the number of squarefree d < X such that X^d(17) has local points everywhere is on order of X / (\log X)^{5/8}.  (She could compute the constant, too, if for some terrifying reason you needed it…)

In the end, a lot of the X^d(N) have local points everywhere; and because they are all covers of the quotient X_0(N)/w_N, which has finitely many rational points once N is big enough, many of them don’t have any global rational points.  In other words, you have a healthy population of curves violating the Hasse Principle.  (This observation is due to Clark.)  Are these failures of the Hasse principle always due, as some people expect, to the Brauer-Manin obstruction?  In the last section of the paper,  Ekin works out one case in full — namely, X^{17}(23) — and shows that it is indeed so.

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