Nicholas Triantafillou, Computing isolated points on modular curves

Rachel Pries: all right thanks everyone for coming today to this series of talks which is an update about the ICERM workshop this summer and the first talk today will be by Nicholas Triantafillou who is speaking about computing isolated points on modular curves. Nicholas, is it all right if we put this talk on youtube afterwards? Nicholas Triantafillou: Sounds good. Rachel Pries: Well take it away. Nicholas Triantafillou: Alright. Thank you very much for the invitation, Rachel and Drew. It’s a real honor to get to kick off this series of VaNTAGe. I think I’ve been watching VaNTAGe since back before COVID hit and I’ve been enjoying it an awful lot this whole time.

00:00 - Everything that I’m talking about today is joint work with Kaan Bilgiin, Atharva Korde, Michelle Manes, Travis Morrison, Matias Giusti, Soumya Sankar, Bianca Viray, and David Zureick-Brown. Let’s get started. We’ve got a couple words here in the title that we need to talk about a little bit. First we’ll motivate studying modular curves for anyone who isn’t as familiar with those and then we’ll jump into talking about isolated points a bit later. To start with talking about modular curves, I want to remind you of two famous theorems about the structure of torsion on elliptic curves. First there’s Mazur’s torsion theorem from 1978 which proved that the the Q-points of the torsion or the torsion of the rational points on an elliptic curve can only be one of these 15 possibilities.

00:06 - Maybe slightly less well-known but also also quite well known is the isogeny theorem which says that the the possible cyclic subgroups of the elliptic curve which can be defined over Q range over this somewhat larger set, but still a finite list of options. Here, when we say that the cyclic subgroup of order n is defined over Q, we don’t mean that the individual points are defined over Q – we can see from the torsion theorem that that often won’t be the case. – but rather that the elements of the cyclic subgroup are Galois conjugate, that is, they’re closed under the action of the Galois group. But how do you really think about this? Well, as an arithmetic geometer, I like to think about it geometrically and I think that’s the popular viewpoint, which is to say that there is an object which parameterizes elliptic curves equipped with a choice of a subgroup isomorphic to Zmod nZ (a cyclic sub-group of order n) up to isomorphism. Alright. What is this object that parametrizes this? Well, it’s the modular curve X-naught of n.

And okay, 01:31 - it’s not not quite a perfect match up. There are some cusps, some extra points which correspond to degenerations of these elliptic curves, but roughly speaking this is the object. And another way of saying Mazur’s isogeny theorem would be that the set of rational points on X-naught of n is contained in the set of cusps unless n is in this finite set of numbers which are all at most 163. And so we understand quite well the the rational points on these modular curves. Another nice thing about this structure is where these elliptic curves exist we have parameterizations.

So, 02:38 - if this modular curve is genus zero then we have a P^1 worth of elliptic curves with a cyclic subgroup of order n defined over Q. If the modular curve is an elliptic curve of positive rank then we could put the structure of a finitely-generated abelian group on these elliptic curves with a cyclic subgroup of order n. We can parametrize them by these Q-points on an elliptic curve. And the remaining case is if we have a higher genus curve or an elliptic curve of rank zero we’re left with a finite set. All of this is to say that we understand rational points on these modular curves X_0 of n, which corresponds to the cyclic subgroup of order n and X_1 of n which corresponds to a torsion point of order n.

We understand 03:38 - these quite well, at least when it comes to the Q-points. So what about over number fields? What happens if we change from Q and allow points over higher degree fields? There’s a lot, quite a lot, that’s known about the torsion subgroup. In 96, Merel proved that if you fix the degree (or you bound the degree) of the number field then there are only finitely many possibilities for the torsion subgroup of the K-points on the elliptic curve. So as you range over all elliptic curves and over all number fields of degree at most d you still have only finitely many possibilities for the torsion subgroup. And that of course asks a very natural question, which is, ``Which subgroups appear?” This has been completely classified in two cases.

04:51 - In the degree two case, quadratic fields, this was classified in the 1980s by Kamienny, Kenku, and Momose. In 2004, Jeon, Kim, and Schweizer answered the question of ``Which torsion subgroups appear infinitely often over cubic fields?” So we’re thinking of things that are maybe parameterized by a P^1 or the rational points on a positive rank elliptic curve. And just very recently Etropolski, Morrow, Zureick-Brown, Derickx and van Hoeij finished the classification of cubic points on the X_1 modular curves. So, they finished the classification of the torsion subgroups of elliptic curves over cubic fields. And so, they found, well they didn’t find, but there is one extra possibility that doesn’t appear infinitely often – this very mysterious point on X_1 of 21 that Najman found in 2014.

06:08 - And this point is very interesting! It doesn’t seem to appear for a geometric reason. And there are some other interesting points, I think, as well that don’t necessarily fit into parametrized families. And those sorts of points will be the the topic of the rest of our talk for today. Now, we’re going to leave the the full torsion subgroup behind and go back to X-naught of n parameterizing these cyclic subgroups. But first, we’ll take a detour to define the other mysterious word from the title of our talk, the word isolated – isolated points.

I think the 07:13 - definition first appears in a fairly recent paper of Bourdon, Ejder, Liu Odumodu, and Viray. As a starting point, we’ll take X to be a nice curve, so smooth proper and geometrically integral. Just think of something nice, an algebraic- geometric curve. And for convenience let’s equip it with a rational point. We’ll see why in a moment. And now this is true in general, not just for curves, but the degree d points on X are basically going to correspond to rational points on the dth symmetric power of X.

(So this is 07:59 - the product of X with itself d times quotiented out by the symmetric group of order d.) And how you should think of this is that a degree d point on X corresponds to the set of all of its conjugates. Now, the symmetric power also has some other points. Say if you had a rational point you could take d copies of that rational point and it would also live there. So you need to be a little bit careful, we can’t write an equals sign, but the point is that we can understand degree d points on x by understanding rational points on a higher dimensional variety.

And this 08:40 - is one of my favorite things in the world. You know, when I’m not working on this project I like to replace K-points on on curves with the restriction of scalars and try to compute them with Chabauty’s method: things like this. Or even to do similar things with the symmetric power. Alright. So what are we going to do now that we’ve parameterized our points as the rational points on some variety? Now we’ll take advantage of the fact that we have a curve equipped with a rational point. You might be familiar that we can map the curve into its Jacobian, an abelian variety defined over Q which parametrizes degree zero divisors on X.

09:28 - We can map the curve into its jacobian by taking a point P to the class of P minus this fixed P-naught. But more generally, we can map we can get a map from the dth symmetric power to the Jacobian that takes a d-tuple of points or a set of d points to the sum of the points minus d times our fixed point on the curve. This map is no longer an embedding, so it’s not quite as nice as the map of the curve to its Jacobian but it’s still quite nice. The fibers of this map are isomorphic to P^n for some n a positive integer. And in fact, it’s not so hard to compute these using linear systems, at least in theory.

10:29 - When X gets complicated and d gets large it can be more difficult in practice. The other nice fact is that the image of this variety, the image of this map, is a sub-variety of an abelian variety. So, by Faltings’s theorem, we know that the rational points on that image are contained in (first of all) a finite union of translates of abelian subvarieties with positive rank and then there are finitely many additional points that lie in the image but don’t lie on a translate of an abelian subvariety of positive rank that’s contained in the image. I’ve got a little picture of this, how you might imagine this would look if you take– maybe this is a surface so maybe it’s d equals two. We see in the picture that there are these fibers in green which contract down to points so we’ve got some projective spaces there, another one over here, contracting down to this green point on the right.

12:02 - And then we have abelian varieties, looks like maybe they’re elliptic curves in this case which i always like to draw as straight lines because you can imagine straightening them out using a logarithm or something like that, and then we have these finitely many points scattered around the picture which are the gold that we’re going to be hunting for. [Rachel Pries:] Nicholas, there’s a question in the chat about whether the d in the union is the same as the d in Sym^d, from Andrew Obus. [Nicholas Triantafillou] Oh, that’s a great question Andrew. No, it’s not. Maybe let’s just say e, give it a different letter. Thank you very much for that Andrew. There’s no reason why the number of translates of abelian varieties should be equal to the dimension.

12:58 - It can definitely be a totally different number. Thank you. Alright, so let’s just summarize what we’ve said. Basically we’ve said the degree d points on X are equal to P^1-parametrized points, so these are the points where the fibers are positive dimensional. They might be parameterized by a higher-dimensional projective space, but you could break that up into P^1s. We’ve got the abelian variety parametrized points, these pink points, and then we’ve got the isolated points which are a finite set.

13:44 - And that’s something that we can really get our hands on and compute. So the goal of this project that we started at ICERM and that is still ongoing is to compute and study the isolated points in all degrees, or as many degrees as we can, at least on X-naught of n, where the rank of the Jacobian is equal to zero. And the nice thing about that, well one thing, is that if the Jacobian has rank zero it certainly can’t have any positive rank abelian sub-varieties so there aren’t any abelian variety parameterized points. And the other nice thing will be that the Jacobian will be torsion so we have a finite set of fibers to look over. As we go through this we’d like to ask lots of questions like: Are the isolated points all cusps? That would be kind of boring.

14:51 - Are they CM points? That is, do they correspond to elliptic curves with extra endomorphisms? We might expect that those curves would be defined over low degree number fields. Maybe this is a good way to find them. Or might they be Q-curves, that is curves that are isogenous to (or that have maps to) all of their Galois conjugates. So again, it might be an interesting place to look for examples. It would be really neat if these had more Q-curves among them than you would expect for random curves or something like that. But in general, these are pretty mysterious points.

15:38 - Now I’d like to roughly outline our strategy, taking advantage of the fact that we’ve restricted to rank zero Jacobians. Our first goal would be to compute the torsion subgroups and I think this is an area that we have a lot of room for improvement. In the second step, once you’ve computed the torsion points you can iterate over points in your Jacobian and really compute these Riemann-Roch spaces, so just computing all of the effective divisors which are linearly equivalent to this given divisor. And the nice thing is that we can determine if a point is isolated just by this computation. If the dimension of the Riemann-Roch space is zero there are no points – it’s not a point in the image of the symmetric power.

If 16:41 - the dimension is 1 we have an isolated point. And if the dimension is at least 2, it’s P^1-parametrized. So, again we are taking advantage of the fact that there are no abelian variety parameterized points in this setup. I’ll mention really quickly that there are very neat conjectures about the torsion subgroup of the rational points on the Jacobian of these modular curves. So the conjecture is that this subgroup is always equal to the lower bound of the subgroup generated by the cusps.

17:29 - And you might try to prove that that’s the case by computing the the reduction modulo p. We know the prime-to-p torsion injects into the the p-torsion on the Jacobian, but in general this upper bound is not tight. It’s sort of a generalized Ax conjecture to prove this in general although i think it is known in the case that n is a prime. Alright, so we’ve got a few minutes left so I’d like to talk about our current progress and the progress of some others before us. I should say before I mention this that the X-naught curves are generally a little more difficult to deal with to do classification than the torsion subgroups just because their their genus stays lower for longer.

They’re not as 18:33 - complicated geometrically for a long time, which means there are many many more cases that you have to address. But nevertheless there has been some progress studying these modular curves with rank zero Jacobians. Bruin and Najman found all of the quadratic points on hyperelliptic X-naught of n so that’s when X-naught of n is equipped with a two-to-one map to the projective line. And in that case you expect there to be lots of quadratic points. You get a P^1-parameterized family from this hyperelliptic map.

And then there can 19:15 - be some other points here and there. And building on this, Ozman and Siksek (also quite recently) computed all the quadratic points on the non-hyperelliptic X-naught of n in genus three, four, and five. These will be some isolated points. And so we’ve pushed beyond these computations. I want to just tell you how you can read this table. I’ve sorted the modular curves with a rank zero Jacobian by their genus so you see a list of n’s in white for each genus up to six and we’ve got a check mark if we’ve computed the isolated points in all degrees.

20:04 - You only have to look up to the degree equal to the genus because beyond that all points will be P^1-parametrized by structure results on Jacobians, or how you construct Jacobians. We also have some entries in the table with inequalities indicating that we’ve computed points up to that degree. In a couple cases we’ve only recomputed the quadratic points or computed the cubic points isolated points and we have a few especially in genus 5 that are labeled by torsion which means that, if it’s green we’ve completed the computation conditionally on the cuspidal subgroup being equal to the full Q-rational torsion subgroup. In genus 2, these aren’t new results because Bruin and Najman computed the quadratic points on all the hyperelliptic modular curves but once we get to genus 3 we have new results in degree 3. In genus 4 we have new results in degree 3 and degree 4 and we even have a couple of the cases in genus 5 where we have new results in degree 3, 4, and 5.

21:40 - It turns out that this is the only modular curve X-naught of n of genus six which has a rank zero Jacobian. It is hyperelliptic and that actually makes it a little bit easier to carry out these computations. We’re able to compute the isolated points all the way up to degree 6. And so you know, we have a lot more information behind beyond just the number of points. We can really nail down minimal polynomials of j-invariants and all sorts of other information that you might want to know about these curves.

22:18 - Just as we close, I want to say what’s coming up next with this project and I think one thing that we’d really like to do is continue these computations and look more for the patterns. Look and really analyze. Are we getting a lot of Q-curves? I know we’ve done a little bit of that we got some help from Drew, looking that over. Are we getting a lot of CM-curves, that sort of thing? But there’s a big technical piece too, which will be computing the torsion subgroups of these modular curves or their Jacobians and improving our Riemann-Roch computations. When the degree is less than the genus you could try to use a Mordell-Weil sieve and use information at different primes to cut out bits of the computation. When degree is equal to genus you expect every point to be either isolated or P^1-parametrized, so Mordell-Weil sieving isn’t going to get you as much, but maybe we can sieve sort of using information from lower degree.

23:27 - And something that’s near and dear to my heart is to extend beyond the rank zero case - try to get to the rank one or even higher rank case, where you might be able to use p-adic methods like Chabauty’s method to cut out a locus, to really cut out this abelian variety- parametrized locus, to cut out the P^1-parameterized and sporadic locus as an additional finite set of points on the abelian variety using sort of p-adic analytic equations and solving those equations. And it would be really cool if we could eventually prove something to the effect that the the set of degree d sporadic j invariants on the X-naught of n is bounded. There are some results like this, conditional on some hypotheses I believe, but some results like this for X_1 of n. I’d love to hear any questions that you have about this project and about what we’ve got going on. Rachel Pries: Thank you Nicholas. That’s wonderful. .