Lori Watson, Odd degree isolated points on X_1(N) with rational j-invariant

Jun 15, 2021 13:55 · 7074 words · 34 minute read

RACHEL: All right, everyone. Thank you for coming back for the second talk of the VaNTAGe seminar today.

00:07 - This is our second talk in the series about modular curves and Galois representations.

00:13 - And for this talk we’re happy to have Lori Watson speaking about our degree isolated points on X1 of N with rational j-invariant.

00:22 - And Lori, is it all right if we video this talk? LORI WATASON: Yes.

00:26 - RACHEL: Oh, great. Well, please go ahead.

00:27 - LORI WATASON: Thank you. So first, thank you to Drew and Rachel for organizing this seminar, and thank you of course for the invitation to speak and thank you to everyone who’s taking the time to give me a chance to share some of my recent work with you.

00:43 - And this talk is based on joint work with Abbey Bourdon, David R. Gill and Jeremy Rouse.

00:50 - And for the first part of the talk, my goal is to just explain what isolated points are because the notion of an isolated point applies to curves outside of X1 of N and then midway through the talk will switch our focus to the modular curves X1 of N in particular.

01:06 - So the question of when a curve can have infinitely many rational points, was of course answered in Faltings’ celebrated theorem.

01:23 - And Theorem says that if you have a number field K, then any curve of genus at least 2, has only finitely many K-rational points.

01:33 - And so in particular, if you’re given a curve over Q, then we know that C of Q can be infinite only if the genus is either 0 or 1, and we know that both of those cases happen.

01:47 - So for a G in 0 curve like P1, if it has even 1 rational point, then it will have infinitely many.

01:53 - For a G it is one curve if it has a rational point, then unselect the curve and to say that it has infinitely many Q rational points, just means that you’re dealing with a positive rank elliptic curve and those are known to exist over to you.

02:06 - And that behavior is unchanged if you consider an arbitrary number field instead of just Q.

02:14 - And I’m going to talk a lot about degrees of points.

02:17 - And so I’ll start here with a definition. By the degree of a closed point, I just mean the degree of the extension that you get when you would join that point.

02:26 - So as an example, if I have the curve C defined here by y squared equals x to the fifth plus x squared plus 1, then I have a closed point given by the Galois orbit of the 0. 1 square root of 3 and 1 minus square root of 3.

02:43 - And I consider this a degree to point over the rational numbers, because in order to realize this point, we have to be working over the quadratic extension Q or join the square root of 3.

02:54 - So in order to see this point, we have to make a quadratic extension.

02:57 - So we’ll consider this a degree to point over Q.

03:03 - And so in the language of degrees of points, what Falting’s Theorem tells us, is that for any curve of genus at least 2 and any number field K, the set of degrees 1 points always going to be finite.

03:13 - But that can change when you allow the degree to be something greater than 1.

03:22 - So we’ll return to that last curve C, again it’s defined by y squared equals x to the 5th plus squared plus 1.

03:30 - This is a curve of genus too, so we only have finitely many rational points finally we need one points.

03:37 - And if we fix a rational number a and allow a to serve as the x-coordinate of this point, then we’ll get a point over any feel that contains the square root of a squared plus– a to the 5th plus a squared plus 1.

03:51 - And I’ll denote such a field by K sub a. Now by faulting stem for any given case of a I’ll still only have finitely many Ka rational points.

04:02 - But if we allow a to run through the rational numbers, then for most values of a, a to the 5th plus a squared plus 1 won’t be a square.

04:11 - So we’ll get an honest to goodness degree to extension of the rational numbers.

04:15 - And in that way, we can obtain many degrees too close points on this curve C. So again, it has genus 1.

04:22 - It can only have finitely many degree 1 points over Q.

04:26 - But as soon as I allow myself to consider a degree two points, then we have infinitely many degrees two points on this curve C.

04:36 - And what’s really making all of this work is that we have a degree to morphism from the curve to P1.

04:42 - So in other words, what we have is a hyperelliptic a curve.

04:46 - And in this case, our morphism is defined by just sending a point with coordinates xy to x.

04:52 - And are infinitely many degrees two points on our curve C, are really coming from the infinitely many degree 1 points on the genus 0 curve P1.

05:02 - And there was nothing special about 2 in this case.

05:05 - If we have a degree dimorphism from C to P1, then there will be infinitely many degrees 1 points on P1 that give rise to infinitely many degrees d points of C.

05:16 - And this is a consequence of Hilbert’s irreducibility theorem.

05:19 - As long as I have a dominant degree d, that from the curve C to P1, we can expect it to be infinitely many degrees d points.

05:27 - So that’s one way that we can hope to find infinitely many degree d points.

05:32 - But Gina 0 curves, aren’t the only curves that can have infinitely many rational points.

05:37 - So for example, maybe I don’t quite have a degree d map to be P1, but maybe instead, we have a degree d map to annalytic curve like in the second example.

05:46 - So for this example, I’m going to let C, it’s again a hyper elliptic curve, but this time it’s defined by y squared equals x to the 9 plus e cubed plus 1.

05:56 - And just like before, it admits a degree to morph into P1, so we would still expect to see infinitely many degree two points on this curve.

06:04 - But this curve also admits a degree 3 map to the elliptic curve, y squared equals x cubed plus x plus 1.

06:11 - And in this case, you send a point on C with coordinates x, y to point on your elliptic curve with coordinates x cubed, y.

06:19 - This tells us that we can expect there to be two big points whenever we have some rational point a, b on the elliptic curve.

06:27 - And since I’ve sort of rigged the game so that I have an elliptic curve with positive rank, there will be infinitely many such cubic points.

06:34 - OK, so we have a second example where we’re getting infinitely many degree d points parameterized by P1 or positive rank at curve.

06:45 - And you might wonder, are those the only ways in which these things can happen.

06:50 - And not quite. To give a sense of the more general way that you can get infinitely many degree d points, I’ll point you to these examples of Debarre and Fahlaoui, they provided examples of curves that did admit infinitely many degree d points despite not having any maps of degree less than or equal to d to P1 or to an elliptic curve.

07:12 - And their construction instead involved the deep symmetric product which we saw in Ekin’s talk a little while ago.

07:20 - So for this, just for the sake of simplicity, I’m going to assume that I’m dealing with a curb over a number field K, and I’m assuming that it has at least one k rational point.

07:29 - A closed point of degree d on this curve C, gives rise to a K rational point on the dth symmetric product.

07:36 - And there’s a natural map from dth symmetric product to the Jacobian of the curve.

07:40 - So if this map isn’t objective, then we can conclude that there’s a dominant morphism from the curve C to P1 of degree D, and as before, it will have infinitely many degree d points.

07:53 - Well, what if this map actually is injective, is it still possible that you can get degree d points from this map.

08:00 - And the answer is yes, under the right conditions.

08:04 - So if the degree d map from– if the natural map from the dth symmetric product to the Jacobian is not injective, then by Falting’s Theorem implies that there are finitely make K-rational of abelian subvarieties of the Jacobian, and finitely many key rational points such that the image of the dth symmetric product is equal to the union of is translates of these abelian subvarieties.

08:29 - And so in this situation if you hope to have infinitely many degree d points, it’s necessary that one of these A sub is will have positive degree. rank.

08:37 - And that really does capture the true story about when you can have infinitely many degree d closed points.

08:46 - So if you have a number of fields K, and you have a curve C, in order to have infinitely many degree d closed points, one of two things like we’ve seen has to happen.

08:56 - Either the curve C admits a dominant map of degree d to P1, or it’s more like the define our y examples where the set of degree d points of C inject into the set of k rational points of some translate of a positive rank abelian subvariety of the Jacobian of the curve.

09:12 - But even when you have infinitely many degree d points, that doesn’t mean that those two constructions really tell you the whole story of degree d points.

09:25 - So as a third example, I have this curve here, it’s another hyperelliptic curve.

09:29 - You can tell that I like those. This is a hyperelliptic curve defined by y squared equals this 8 degree polynomial that I’m calling f of x.

09:36 - Again, it’s hyperelliptic. We still expect it to have infinitely many quadratic points, and most of these are going to be of the form a, plus or minus the square root of f of a where a is rational number.

09:48 - But there is at least one point that does not arise in this fashion.

09:53 - You can check that the point i plus or minus 4i is a quadratic point of this curve, it’s certainly quadratic, because we have to be working over the field that contains the square root of negative 1 in order to realize this point.

10:06 - But it’s definitely not like these other examples, because for example the x-coordinate isn’t rational.

10:12 - And so it’s not coming from the sort of natural map where you send a point xy on the curve, to x on P1.

10:18 - But you might be wondering, well, is it possible that there’s another degree to morph into P1, and the answer is no, because for hyperelliptic curves of genus at least 2, there is a unique degree to map to P1.

10:30 - So this isn’t explained by the dominant map that I have to P1, but maybe it’s explained by something having to do with the Jacobian.

10:37 - The answer is still no. It’s not part of an infinite family of quadratic points of– and abelian subvariety of the Jacobian, and we know this because the Jacobian of this curve has rank 0 over Q. So we have here a point that even though it’s lying in a degree where there are infinitely many other points of that same degree, it’s somehow not explained by the construction from what you can get infinitely many degrees d points.

11:03 - And this leads us to the definition of an isolated point.

11:07 - So if we have a curve C to find over a number field K, and we have a closed point of degree d, we say that that point is isolated if it doesn’t belong to an infinite family of degree d points parameterized, either by P1 or by a translate of a positive rank abelian subvariety of the curve’s Jacobian.

11:27 - And this term isolated points was first defined in a paper of Bourdon, Ejder, Liu, Odumodu, and Viray, and their focus was on modular curves, but I should note that isolated points even though they weren’t given that that term, they were identified earlier than that.

11:48 - These are not new kinds of points, it’s just that the term is relatively new.

11:51 - And one of the things that they showed is that in order to have infinitely many degree d points on a curve C, there has to be at least one degree d point that’s not isolated.

12:03 - But there’s one type of point that’s guaranteed to be isolated.

12:09 - So as a reminder in order to be, in order to be isolated, we have to say that it’s not part of an infinite family of points parameterized either by P1 or something having to do with an abelian subvariety of the Jacobian.

12:23 - But if you already know, for example, that there just are not infinitely many points in that degree at all, or even stronger that there are only finite remaining points of degree less than or equal to d than any point and degree d is going to be isolated, and we call such a point sporadic.

12:39 - So definition, if C is a curve to find over a number field K, then we say that a closed point x is sporadic if there are only finitely many other close points with degree less than or equal to the degree of x.

12:53 - And some of the first isolated points on modular curves that people found were actually sporadic points.

13:02 - OK, and so now let’s start talking about modular curves just as a reminder.

13:06 - If I fix a positive integer N, and I refer to the modular curve X1 of N then I’m talking about an algebraic curve that can be defined over the rational numbers.

13:16 - And the k rational non-cuspidal points on such a curve correspond up to isomorphism to a pair of elliptical and a distinguished point of order N defined over the field K.

13:27 - And because I’m also going to talk in at points about X0 of N, let me remind you what the non-cuspidal K-rational points on x not even correspond to.

13:35 - The non-cuspidal K-rational points there correspond up to isomorphic to a pair of elliptic curve and a cyclic subgroup of order N or equivalently and elliptic curve E and a k-rational cyclic isogeny So sporadic points.

13:58 - Again, some of the first sporadic points that were identified by some of the first isolated points on modular curves that were identified were in fact sporadic points.

14:06 - And these points were actually not terribly surprising.

14:09 - For example, you might expect to see more points to give rise naturally to sporadic points, and this is because if you have a CM of elliptic curve with a CM by an imaginary quadratic field K, then you can get isogenies and very low degree as long as you’re working over that imaginary quadratic field K.

14:28 - And as you might imagine once you get those isogenies in low degree, it’s maybe not too unreasonable to expect that you’ll also get torsion points in relatively low degree.

14:37 - And so, work of Clark, Cook, and Stankewicz shows that you’ll CM isolated points on X1 of L for sufficiently large primes L. Sorry, I said isolated, they really mean sporadic.

14:50 - So something even stronger. So we’re all sufficiently large primes L, you can find CM sporadic points.

14:56 - And I believe Drew actually gave an argument that extended this to compositing and more recently work of Clark, Janelle, Polich, and Seyah also extends this kind of result to composite in for arbitrarily large N.

15:14 - OK, so some sporadic points are really not terribly surprising, but there were other examples of sporadic points that were a bit more surprising.

15:24 - So for example Mark Van Hoeij found a point of degree 6 on X1 of 37, and for those of you who are a little bit familiar with carnality, the personality of that curve is 18.

15:34 - And so to have a point and degree 6 it’s so far below the carnality that it has to be sporadic.

15:42 - And here Philip Nyman found a point of degree 3 on the curve X1 of 21, and that example is especially interesting because the point that he found on X1 of 21 it corresponds to an elliptic curve that actually has rational j-invariant, and that attains a point of order 21 over the maximum real cyclic, sorry, the maximal real subfield of the day/night cycle atomic field.

16:10 - OK, so I’m not saying that necessarily the elliptic curve has rational coefficients, although in this case, I believe that this is a very special example because I think actually you can choose a model that has rational coefficients.

16:25 - But in general, even if the j-invariant is rational, that’s not necessarily saying that the elliptic curve that realizes your point of order N, is also going to have rational coefficients.

16:37 - OK, so for a fixed curve C, there are only finitely many isolated points of any given degree d, and once the degree is large enough, the no point of degree d can be isolated.

16:52 - And if you put those two facts together, that means that for any given curve there could only be finitely many isolated points at all.

17:00 - And so in particular, if you fix a positive integer N, there can only be finitely many isolated points on X1 of N.

17:09 - But as we saw in that result from Clark, Cook, and Stankewicz, the set of isolated points if you allow yourself to range over all positive integers N is actually an infinite set.

17:20 - And there are even CM elliptic curves with rational j-invariant that give rise to isolated points of arbitrarily large degree.

17:28 - And so even if you can say well, let’s only consider the isolated points corresponding to elliptic curve with rational j-invariant, we’re still dealing with an infinite set.

17:38 - I said that we actually believed to be finite though, is the set of j-invariants themselves.

17:45 - In other words, it’s expected that only finitely many rational j-invariants can correspond to isolated points on X1 of N, and this is one of the main theorems, and that paper by put on Ejder, Liu, Odumodu, and Viray.

17:59 - If you let I denote the set of all isolated points of modular curves on X1 of N as you range through all positive integers N, and if you assume there’s uniformity conjecture.

18:10 - Then the set of rational j-invariants that give rise to isolated points on modular curve, is finite.

18:17 - So what this result is saying is that if you allow, if you assume there’s uniformity conjecture, then even though we know there are infinitely many isolated points on modular curves that correspond to elliptic curves with rational j-invariant only finitely many gene variants are actually accounting for that infinite set of isolated points.

18:37 - And as a brief reminder, Sarah’s uniformity conjecture.

18:43 - It was originally a question posed by Sarah, but I think formally conjectured by Sutherland and zwirner.

18:49 - And it says, that there exists a constant M, such that for all non-CM elliptic curves E/Q, and for all primes of all sufficiently large primes, the mod p Galois representation is surjective, and expectation is that you should be able to take M to be 37.

19:06 - And so if that result is true, if that conjecture is true, then we’ll know that even though we have infinitely many isolated points corresponding to elliptic curves of rational j-invariant, only finitely many j in-variants are actually giving right to those examples.

19:21 - What Abby, David, Jeremy and I were interested in, was trying to find some sort of unconditional result along the same lines.

19:32 - And what we were able to show is that if you just add in 1 not small but one assumption, namely that the degree of the isolated point is odd, then you actually can get an unconditional result.

19:47 - So at the expense of putting this restriction on the degree of your isolated point, you get something that no longer relies on Serre’s uniformity conjection.

19:57 - And so what we show is that if you let i sub odd denote the set of isolated points of odd degree on all modular curves where you again range through all positive integers.

20:07 - Then the set of rational j-invariants that correspond to odd degree isolated points, is one of only possibly 5 j-invariants.

20:15 - On the left-hand side, we had two non-CM j-invariants and on the right hand side, we have 3 CM J-invariants.

20:22 - And we know that the two non-CM j-invariants actually occur.

20:26 - And if you’re paying close attention to 9 months example, this CM j-invariant here, the first one in the list on the left-hand side, sorry non-CM j-invariant, this is the one corresponding to fill up Nyman’s example of a sporadic point of degree 3 on X1 of 21.

20:41 - OK, and where do we get all of this mileage just by assuming that the degree is odd? Well, really the key thing is that it allows us to exploit this connection that we have in between an odd degree point on X1 of N and rational cyclic isogenies.

21:02 - What we show is that if you have a point of our degree on X1 of N, then as long as you stay away from 1 exceptional j-invariant, you can say something about primes p that have to divide N, and you can also say– so for example, the prime p has to be one that appears in this list here.

21:20 - In Moreover, what you can say is that the j-invariant of that point is going to be equal to the j-invariant of some rational elliptic curve that has a cyclic P isogeny over Q.

21:32 - And even more, we can tell you something about what the shape of them looks like.

21:36 - So for example, you couldn’t take all of these primes and multiply them together and get some horrendously large thing, instead N has to look like 2 to a times b to the b, sorry, p of the b times q to the c, and we even have some control over what a is.

21:51 - So this puts lots of restrictions on the kinds of points that can arise on modular odd degree points that can arise on modular curves.

22:04 - OK, in the paper we end up having to treat CM and non-CM points mostly separately, but there’s one result that gets used again and again throughout the paper.

22:15 - And it says that if you have a finite map of curves, and you have some isolated point on a curve C, then you can push that isolated point on C down to an isolated point on the curve D, as long as you have the maximum possible degree growth on residue fields.

22:32 - So if the degree growth from x to f of excess as large as possible, then an isolated point gets pushed down to an isolated point.

22:42 - And one of the results that allows us to exploit that, is a result of Greenberg, and Greenberg, Rubin, Silverberg, and Stoll.

22:50 - But the goal of– so with that, that last result in mind, it gives us one approach for identifying isolated points on X1 of N.

23:01 - So as you can imagine trying to do computations on X1 of N can be really hard, especially as it gets large.

23:06 - But what that then allows us to do is in some cases, look at the natural map from X1 of N to X1 of m for some m that divides N.

23:15 - And then your hope is that you can push things down to some level where you can better understand what’s going on.

23:20 - In this result of Greenberg and Greenberg, Rubin, Silverberg, and Stoll, is about images of p-adic Galois representations for elliptic curves that have cyclic p-isogenies.

23:30 - And what the result does, is it gives us a chance to determine some nice values of m for which that necessary condition on the growth of the residue field is going to hold.

23:42 - And what that result is really about is, what it tells us is that the image of a p-adic Galois representation is somehow as large as possible.

23:50 - And by that, I mean, as large as the cyclic p-isogeny will allow for.

23:54 - And an example of how this gets used, in order to show that you have no non-CM isolated points of our degree on X1 of 2 times 7 to the B, with a rational j-invariant.

24:09 - What we do is we show that any such odd degree isolated point would actually have to map down to a non-custodial of isolated point on X1 of 14.

24:19 - But X1 of 14, has no non-cuspidal Q rational points.

24:23 - And so if we’re dealing with the point that has our degree, the degree has to be at least 3, but X1 of 14 is an elliptic curve.

24:30 - So it has genus 1, which means that nothing in degree 3 can actually be isolated.

24:35 - We use lots of techniques in this paper for trying to address different N.

24:43 - And in general the is approach is to just again try to push an isolated point on X1 of N down to some isolated point on another curve.

24:52 - Often the hope is that you can push it all the way down to something that has such low genus that may be no isolated points can exist.

24:58 - For example, if you take an isolated or potentially isolated point, and you push it all the way down to genus 0 curve, then you’re done because you don’t have isolated points.

25:08 - Or barring that, maybe you can push it down to some lower genus curve where the genus is at least small enough so that we can perform the sorts of computations that will allow us to answer the question.

25:19 - But as you can imagine, primes like 2 and 3, sometimes throw a wrench into things, and so that nice technique of just push things down doesn’t always work as easily as it did in the X1 of 2 times 7 to b example.

25:36 - For example, to deal with curves, for X1 of 2, to the a times 3 to the b, what we first had to show was that an isolated point of odd degree corresponding to a non similar to curb with rational j-invariant on one of these curves, would actually have to map down to an isolated point on either X1 of 54 or X1 of 162.

25:58 - But you’re still dealing with powers in 2 and 3, so it’s still a little bit difficult.

26:02 - And so in order to show that these curves had no non-CM isolated points of our degree, we had to use things like a 3 at a classification that’s in progress work of Rouse, Sutherland, and Zureick-Brown.

26:14 - But we also at some point, especially when dealing with X1 of 54, we had to really deal with something known as entanglement.

26:21 - So we had to explicitly find all of the rational points on a genus for curve that was really characterizing the possible entanglement of little elliptic curves to find over the rational numbers.

26:35 - So for a product of a power of 2 and a power of 3, again, as you can imagine, things get a little bit complicated.

26:44 - And that captures a lot of the kinds of things that we had to do for non-CM j-invariants.

26:50 - And for CM j-invariants, well, as a reminder, if an elliptic curve over Q has CM by an order O, then the discriminant of that order can be one of only 13 integers.

27:01 - So we we will only have a finite list of things that we have to check, and we show that there are no isolated points of our degree with rational j-invariant corresponding to an elliptic curve and by order of 10 of those discriminant.

27:15 - But for the remaining three discriminant negative 43, negative 67, negative 163, those actually correspond to the 3 j-invariants that appeared in the main theorem.

27:26 - And so you might be wondering, OK, what’s the obstruction there, well, in order to complete the classification of our degree isolated points, we have to determine whether or not there are points of degree 2133 and 81 on one X1 of 43, X1 of 67, and X1 of 163 respectively, that are isolated.

27:48 - And the difficulty lies in the fact that the Jacobins of each of these curves have positive rank.

27:54 - And so, it’s really a computational thing that’s standing in the way of a complete classification of the odd degree isolated points that correspond to rational j-invariants.

28:04 - And so to end, I just want to talk a little bit about some of the remaining problems and questions that naturally arise.

28:14 - So that results are word on Ejder, Liu, Odumodo, and Virays says that, once we know there’s uniformity conjecture or if we know that it’s true, then we’ll get an unconditional result that applies to even an art degree.

28:26 - But until we have such a result, is there any other way of getting some unconditional results for isolated points of even degree that’s 1 natural question? Another question.

28:39 - So, CM j-invariants, as I said, there are natural class of sporadic points on.

28:45 - They give you a natural class of sporadic points on modular curves.

28:49 - And actually, for many CM j-invariants, you can find a point on a modular curve that’s in such low degree, that if you allow yourself to go up the tower and keep lifting to points on higher and higher modular curves, they’re all occurring in such low degree that they still have to be sporadic points on those modular curves.

29:09 - But that’s something that seems so far to be special to CM them.

29:12 - So one question is, is there a non CM j-invariant that will also give rise to similar behavior.

29:17 - Is there some non CM j-invariant for which you can always provably show that if you go up and up and up a tower, that same j-invariant is always giving rise to isolate or is often giving rise to isolated points on X1 event for larger and larger N.

29:35 - And finally, what’s the proportion of non-CM to CM isolated j-invariants.

29:40 - So in our list we have two non CM j-invariants that definitely occur, and 3 CM invariants that might occur, or what’s the proportion overall.

29:50 - So we know that you can have both non-CM and CM isolated j-invariants.

29:55 - Like if you ask me, I would say it’s probably the case that most of them are coming from CM points, but what’s the proportion overall.

30:03 - And there’s one other question that I’ve often been asked.

30:06 - And so I’ll just acknowledge it here, a few times people have asked me what about X0 of N? And that’s a great question.

30:13 - And Zonia Menendez, a student at Wesleyan, and is currently working on X0 of N. So it’s in progress, but it’s not my question, so that’s why it’s not included in this list.

30:24 - And with that, I’ll stop. Thank you.

30:25 - RACHEL: Thank you Lori. So this would be a great time for questions.

30:40 - Edjer asked the question. EDJER: Yes.

31:02 - RACHEL: So one of the examples that you gave, you were looking at the modular curves X1 of 2 times 7 to the b, I think? EDJER: Yes.

31:11 - EDJER: And then you said that you lower the isolated points on that material curve to X1 of 14.

31:18 - So how do you get rid of the power b, do you show that the curve has some 7 isogeny? LORI WATASON: Yeah, it’s really leveraging.

31:27 - That result of Greenberg, Rubin, Silverberg and Stoll to say that for high enough powers, yeah, you sort of have to have a certain isogeny that’s telling you that you would have the maximal residual field growth that you would have to have in order to push things down as far as you can push them down.

31:44 - Thank you. EDJER: Thank you.

31:49 - CLARK: I’m kind of curious about these CM gender invariance that you don’t know whether they’re isolator or not.

32:03 - So you know that you have points of these degrees, but then the problem as you said is that, in order to figure out whether they’re isolated you need to figure out whether they lie in this family of rational points parameterized by a positive rank abelian subvarieties.

32:25 - So it’s kind of, I don’t know, it’s kind of ironic because you’re trying– it sounds like a question about CM points, but the answer has to do with all these other points.

32:34 - Is there any– do you have any intuition or any feelings about what the answer would be? LORI WATASON: Honestly, no.

32:48 - If you handed me a random CM point and said, what to you like, what are your odds on the degree? I’m going to say it’s probably even.

32:55 - But that’s a slightly different question than what you’re asking.

32:58 - And I would be surprised if none of them gave rise to isolated gene variants.

33:04 - I think it would be interesting and strange if the only odd degree isolated j-invariants were all from non-CM elliptic curves.

33:15 - But as to which of them might actually give you an isolated odd degree, sorry an odd degree isolated point.

33:25 - I don’t know that I would put money down on any one of them.

33:29 - CLARK: I mean, to me I think it’s a very interesting question, because as I understand it, is the philosophy behind these isolated versus non-isolated points, is we’re trying to kind of qualitatively understand infinite families of points on curves, and we’re trying to understand why do we have this point, this low degree.

33:53 - And because it’s a CM point, we already have a complete, we have a different kind of understanding.

34:00 - AUDIENCE: [INAUDIBLE] is that [SINGING] CARK: OK, in the CM, we have like a totally CM explanation for why we have this point of a law degree, I mean, we know all lots of stuff about degrees of CM points, but then, because you have this definition of isolated in order to figure out whether it’s isolators or not, you need to figure out whether it occurs in this other a geometric family of points and it’s just like so do you do you expect that the CM point occurs in this interesting non-trivial geometric family or not.

34:47 - You sort of, you don’t need it to occur in that geometric family in order to know that you have the point, but maybe it does anyway.

34:54 - So if I find that to be a kind of interesting and somewhat ironic question.

34:59 - LORI WATASON: Yeah, and that’s really, I think, where my– that to me, that feels like the sort of natural question to ask.

35:05 - Next is to understand like really where are these points coming from, because like you said, for CM points there they’re mostly for the CM reasons.

35:13 - And so these other points sort of how do you explain them? Why are they there? That’s really the question that I’ve been trying to think more about.

35:19 - So it’s like if it’s not for the same reason, then is it something having to do with the geometry of the curve itself like how do you actually explain the existence of those points.

35:28 - And they are so weird because it’s like well, they’re not explained by sort of any of the natural maps that you would expect them to be explained by.

35:34 - So what is the reason? Like, for what reason are they there? So I agree.

35:39 - It’s an interesting question and that’s, I think that’s where my interests are leading me right now, is trying to understand the why of these points as much as their existence.

35:48 - And then also the fact that, I mean, I look back at the paper, which I guess I’ve seen before, certainly, but haven’t read in as much detail as I should.

35:55 - I mean, you rule out the other 10 CM j-invariants by some other argument.

35:59 - So it’s also kind of interesting that a lot of the CM cases get ruled out sort of no problem, and then and then there’s just this extra phenomenon.

36:08 - Anyway, it’s interesting. JEREMY ROUSE: The fact I could make a comment that the discriminant minus 27 CM point was not actually so easy.

36:20 - That was, we know that that point is not isolated as a consequence of a very long rebound computation which David guilded.

36:27 - CLARK: OK, thanks. RACHEL: So I had a question about the odd degree like, does it seem feasible to handle maybe next degrees that are like 2 times an odd number? ? Or do you think something really different will have to be used to approach those? LORI WATASON: So, when we first completed this one, I had sort of a hope that I expressed to Abby like, maybe you can sort of twist your way to or 2 times an odd thing.

37:05 - But I didn’t quite see a way to make it work because so much of what we get out of assuming our degree really does come from being able to exploit rational cyclic isogenies.

37:17 - And so without having sort of as complete a classification over quadratic fields, I can imagine that it would be a little bit harder.

37:24 - And so we’ve been trying to look for different ways to approach this problem.

37:28 - And I believe Abby and Phillip Nyman are working on a related are working on an approach that feels somewhat similar, but still different enough that I don’t think that the work that we did here quite generalizes to even degree and the way one might hope.

37:45 - Thank you. Yeah. ANDREW SUTHERLAND: A question, maybe this is early to ask when there’s still so much to do for rational isolated points, but I’m curious if you thought at all about what if I wanted to ask about isolated chain variance over quadratic fields.

38:08 - LORI WATASON: I haven’t touched that one yet, mostly because it’s like it’s sticking in my craw a little bit that we haven’t been able to say anything about even degree for rational j-invariants.

38:24 - And so I haven’t given much thought of it, but that does feel like the natural maybe not next step, but next step after the next step.

38:32 - Yeah. Thanks. BARINDER BANWAIT: And so hi, Lori, I just have one question.

38:43 - I’m wondering about these isolated points that you found in the tape.

38:49 - Do any of them correspond to sporadic points in that? LORI WATASON: , Yes the degree– so this first CM j-invariant, this degree 3 point is a sporadic point originally found by Philip Nayman, and its degree 3 over Q.

39:07 - BARINDER BANWAIT: OK. But the others are definitely not.

39:12 - LORI WATASON: Yeah, these, the CM j-invariants are definitely not because yeah, those aren’t and Jeremy, remind me, I think this one is also, it’s isolated, but in a degree weather infinitely any other points of that same degree.

39:26 - I don’t believe this one is isolated. JEREMY ROUSE: Yeah, that’s correct.

39:30 - LORI WATASON: Yeah. Isolated with an [? explanic. ?] JEREMY ROUSE: OK, thank you.

39:34 - LORI WATASON: Thank you. RACHEL: Wonderful.

39:44 - All right, let’s think Lori again. Thank you. .