Jeremy Booher, Can you hear the shape of a curve

00:02 - It was very nice to be able to do something with people I knew back in the U. S. when I’m so far away. So this is with Vishal Arul, Steven Gruen, Everett Howe, Wanlin Li, Vlad Matei, Rachel Pries, and Caleb Springer, and before I explain the title, I just want to start with a warm-up, something very concrete with two elliptic curves.

00:24 - So I have these two elliptic curves, and I’ve counted the number of points on them modulo p for various small primes, and my question is how often do they have the same number of points mod p? So this happens sometimes, but not always.

00:40 - So it happens a couple of times for small p.

00:43 - And the question you might ask is how often does this happen? Well, barring other information, you could say, well the Weil bounds give you a certain amount of information on the range of possible points mod p, and so you’d expect because they’re around some constant times the square root of p points, the chance that they randomly are the same is around a constant over square root of p.

01:10 - Now, it turns out this is not true, there’s something special, these elliptic curves mod p I’m getting are coming from reducing these special elliptic occurs with CM, so it’s a little different, but this is your first sort of guess.

01:27 - Here’s another example. I kept the same first curve but now I change the third one to something a little bit different. And now you’ve noticed I start getting the same point counts for every prime.

01:40 - What’s going on? Well your first guess probably is that I just did some change of variable and I’m looking at the same elliptic curve, but that’s not true. I can compute the j-invariants, the j-invariant for the first one is 1728 and for the second one it is this rather large number.

02:00 - Okay, they have different j-invariants, so they’re not isomorphic.

02:07 - Of course they’re still closely related, and it turns out the relationship is that E_1 and E_3 are 2-isogenous over the rationals.

02:24 - So an isogeny is a non-constant homomorphism between elliptic curves, and it implies this strong relationship between the number of points mod p, which I’ll review in a minute, but my first question is how did I know they’re 2-isogenous? Well this is a role for the classical modular polynomials.

02:46 - If I plug in the j-invariants for the two elliptic curves into this modular polynomial I get zero, and that’s exactly saying there’s a cyclic 2-isogeny in this case between the elliptic curves.

03:02 - Okay so why is this property of being isogenous telling us something about the point counts? Well the nicest way to think about this is by introducing the zeta function.

03:16 - So for a nice scheme like a curve over F_q I can define a zeta function either by an Euler product, this looks a lot like the Riemann zeta function if you were taking X to be Spec Z which you can’t do in my definition but if you imagine that you’re looking at the primes, which are the points of Spec Z and then you’re having a normal euler product and instead you do this for a curve, you can make a short argument and rewrite it as sort of a generating function that involves the point counts on your curve over various extensions of your finite field.

03:53 - So for an elliptic curve we can actually compute this zeta function directly.

03:58 - The zeta function for an elliptic curve E, well it’s a rationaL-function, the numerator is 1 - aT + qT^2 where a is this number which depends on the point count, it’s often called the trace of Frobenius.

04:21 - In fact the numerator of this function is sort of the reversed characteristic polynomial of the Frobenius on the l-adic Tate module when you pick l to be a prime which is relatively prime to q.

04:43 - So what you’ll notice from this zeta function is that it includes information about the point count mod q, and also includes information about the action of Frabenius on the Tate module.

04:56 - So it sort of shows the first three of these things are equivalent for two elliptic curves. You can have they have the same zeta function, have the Tate modules be isomorphic as Galois modules, or have the same number of points mod q.

05:12 - Note that at first it looks like the zeta function should depend on the points on your elliptic curve over extensions also but for elliptic curves the explicit formula shows no it just depends on the number of points mod q.

05:25 - Anyway it’s a not so easy theorem of Tate that these three conditions also imply that E and E’ are isogenous.

05:37 - And in particular, that’s why the existence of an isogeny between the curves E1 and E3 I was looking at earlier, that was an isogeny over Q, so it gives you an isogeny over Fp for most primes p, that’s why that had this implication about the point counts mod p would be the same.

05:59 - All right, so you can generalize this to higher dimensional curves, higher genus curves, and what it turns out is that the zeta functions of two curves being the same implies that the Jacobians are isogenous.

06:15 - So for elliptic curves the Jacobian is just an elliptic curve, but for higher genus curves you actually need to use with this higher dimensional abelian variety.

06:29 - Okay, so now I’m ready to actually explain the title which was “Can you hear the shape of a curve?” and this goes back to a very famous question “Can you hear the shape of a drum?” So maybe a hexagonal drum, and by this it means if you strike it you know it produces certain sounds and you can encode the frequencies of those sounds as the spectrum of a differential operator, in this case a Laplacian, the question was if I just tell you the eigenvalues of this operator on this domain can you determine the domain? And the answer is no, but you sort of know a lot but not everything.

07:16 - And so our question today is an analogous question for curves, and the question is can you determine a curve by giving some sort of spectral information, in particular using the zeta function, and the zeta function really just encodes the spectrum of the Frobenius, because one way to write the zeta function down is some sort of characteristic polynomial or reverse characteristic polynomial for the Frobenius acting on some module.

07:58 - So as we saw, the answer to this question is no, you can’t hear the shape of a curve, because having the same zeta function, having the same spectral information, only determines a curve up to isogeny, or rather an isogeny for its Jacobian.

08:20 - Okay, so that’s sort of a disappointing answer, no we can’t hear the shape of a curve.

08:25 - But we can ask a sort of more broad question: can additional information help us do this? In particular can additional information about covers of our curve help? So for example, suppose I have some nice cover, constructed in some natural way of a curve C, called C-tilde, and suppose I were to tell you that the zeta function for C-tilde and the zeta function for C-tilde’ of an analogous cover for the other curve, suppose those were the same also, is that enough extra information to determine that C and C’ are in fact isomorphic? And you can imagine using more than one cover, and you can also imagine just sort of studying the more general question how much information does like a single nicely chosen cover give you? And that’s really the question I want to focus on, and what that’s what I mean by can you hear the shape of a curve.

09:21 - Can you determine the curve of the isomorphism given zeta functions maybe of the curve, maybe some covers and then there’s a bunch of variants you can also look at. You can for example think about which covers are you supposed to use, cna you be explicit about them? what if you wanted to use the L-functions or various characters or the Galois groups of Galois covers, those are sort of smaller pieces of information than the Zeta function.

does that help? Can you get by with less? What if you wanted to work with curves in a family so instead of trying to deal with all curves maybe it’d be easier if we pick our favorite family of covers and look at what occurs from there.

10:00 - And then also how effective and explicit can you be? Can you write down an explicit list of zeta functions or L-functions to use, can you pick ones which are actually very fast to compute? Okay so I’m going to focus on a couple of versions of this, mostly one in an ICERM project, but I’ll also briefly mention some work with Felipe Voloch.

10:32 - And the first question to ask is, so what covers are we going to use? Well we can get covers from geometric class field theory.

10:43 - So if I have a curve I can embed it into its Jacobian by sending a point to the divisor of the point minus some fixed degree one divisor. I get a map into the Jacobian, and if I have a finite etale cover of the Jacobian, I can pull it back to get a cover of C.

11:09 - And that’s a great way to produce covers of C, and in fact this is basically the only way to do it, that’s what class field theory says.

11:19 - So what covers of C do I want to use? We’re going to focus on the ones that come from taking X-tilde to just be the Jacobian of C, and there are a couple of simple choices you can make. One thing you could try is multiplication by two, it’s a great isogeny, and then the Galois group of this geometrically is the 2-torsion in the Jacobian, just because that’s the fiber above the identity element of the Jacobian, and this is the kind of the cover that the ICERM project focused on.

11:53 - You can also look at pulling back by the Frobenius minus the identity.

12:00 - The fiber of identity there is the rational points of the Jacobian, the Fq points, and these are the kind of covers that my work with Felipe Voloch focused on, these are sort of analogs of a Hilbert class field.

12:15 - So i’ll talk about these at the end if I have any time, but I’m going to focus on the ICERM project instead.

12:21 - The rough difference between these two is that the multiplication-by-two map has a much lower degree, it gives you a cover a much lower degree, and sort of includes less information.

12:38 - All right, so let’s now actually start where we started back in June. So we were looking at an explicit family of hyperelliptic curves.

12:50 - these curves have genus two, and there’s a family where the parameter is s, and you can vary s, and you get these different hyperlliptic curves, and what’s special is this is a family where there’s an automorphism of order four. So in particular the dihedral group of order 8 is inside the automorphism group of these curves, because you also have the hyperelliptic involution.

13:18 - Okay, and so what we’re going to do is we’re going to form the cover, which is a degree 16 cover, given by pulling back the multiplication-by-2 map.

13:30 - So the Jacobian of Z is supposed to be dimension two, because the genus is two, and in fact it’s just the square of some elliptic curve up to isogeny.

13:44 - And the genus of the cover is actually 17, which you can compute using the Riemann-Hurwitz formula, and then the 17-dimensional Jacobian is actually isogenous to E^2 times a product of 15 other elliptic curves.

14:10 - Now maybe this is not quite not so surprising because we have all these symmetries, we sort of expect the Jacobian to break up into a bunch of pieces, and so my question then is how much information does the zeta function give us about this family. So as sort of a first question, let’s try to count the number of pairs of elliptic curves in this family where the Jacobians are isogenous.

14:36 - Okay, disregarding the cover just work directly with the curves.

14:41 - And our sort of expectation is, well there are roughly q^2 members in the family, and sort of their Jacobians are just given by some elliptic curve and the probability that two elliptic curves are isogenous mod q is roughly a constant over the square root of q, so we’re expecting c times q to the three halves.

15:08 - And if you compute and check this is what you’re seeing.

15:12 - And in fact we can prove this for this particular family up to log factors.

15:17 - So seems pretty good, in particular the data function tells you something, but there are lots of collisions.

15:26 - So now we’re going to try adding some more information. So I’m going to say that Z1 and Z2 are doubly isogenous if their Jacobians are the same.

15:40 - up to isogeny and likewise the Jacobians of these covers are isogenous.

15:52 - Okay so how often does this happen? That’s the question we focused on.

16:02 - So sort of zeroth guess, which is a very bad one, would be to guess that two members of the family are doubly isogenous if these 16 elliptic curves showing up are isogenous sort of just by chance.

16:13 - So you could imagine that you know, maybe the first one is isogenous to the the first one for the other curve or the first one is isogenous to the second one for the other curve and so you can sort of imagine different permutations of how to match them up, but ultimately you just need these sort of 16 coincidences to get doubly isogenous curves. This is just completely wrong, and the reason is that there are actually only six different elliptic curves in the Jacobian of Z-tilde that depend on the parameter.

16:46 - And again this is not so unexpected because we had this automorphism group of the dihedral group acting on this Jacobian, so the elliptic curves get shuffled around and the same ones show up multiple times.

17:03 - So given that expectation, we actually only expect a constant times q squared times one over the square root of q to the sixth power, the six because we need six coincidences and the q squared because there are roughly g squared pairs of curves in this family.

17:20 - And so that says we’re expecting c/q elliptic occurs in this family to be doubly isogenous.

17:30 - Okay, that’s not so common, but maybe it will happen sometimes, and experimentally we can actually look and see how often it happens. So because we’re expecting it to happen rarely we need to look at a bunch of primes all at once. So we’ve been looking at 1024 primes closest to 2^n for various values of n to sort of amplify the number of times this happens so we can actually see it pretty easily, and then here’s what happens.

We have a fairly good number of doubly isogenous pairs in our family and they get rarer as n increases which is what we’d expect because that’s increasing q, and in fact when we increase n by one we’re expecting q to roughly double.

18:16 - so we’d expect to have half the number of double isogenous pairs.

18:21 - And if you look we see sort of halving behavior, but we have to increase n by two.

18:30 - So it really looks like the number of double isogenous pairs is a constant over the square root of q, instead of a constant over q.

18:40 - So what our project does is it tries to explain this discrepancy, what was wrong with our heuristic.

18:52 - Okay, so what’s going on? What’s going on is that there are a bunch of unexpected coincidences in the Jacobians of these covers.

19:03 - So remember, you know these covers are made up of six varying elliptic curves, that you know in terms of some parameters s1 and s2 that describes the member of our family, and what happens is that when s1 and s2 satisfy certain polynomial relationships you actually only need three coincidences to be doubly isogenous, instead of the six we were imagining.

19:39 - Okay, so then what does that predict then about how many pairs of doubly isogenous curves we should get? Well if you have a fixed s1 they’re a finite number of s2 which would satisfy this relationship, so there are roughly q sort of cube pairs which satisfy this relation, and then we need three coincidences, so we need a constant over the square root of q cubed, and then that that predicts that we’re going to be seeing a constant over the square root of q doubly isogenous pairs.

And that’s exactly what we were seeing. So this is a better heuristic, and it’s just explaining some of this discrepancy.

20:34 - Now the next question is like where are these coincidences coming from? Well, what it turns out happens is if you start looking for isogenies between the various elliptic curves showing up, you do that by by writing down a modular polynomial, plugging in the j-invariants which are some function of the perimeter s, and then setting it equal to zero.

20:58 - And then that gives you a relation for one particular isogeny between some of the elliptic curves showing up.

21:04 - What turns out to happen is that these keep repeating. For many different choices of elliptic curves showing up and different choices of n you sometimes get the same relationship.

21:16 - And in particular sometimes you’ll get three of the elliptic curves to be isogenous just if there’s one relation, and then that means you only need fewer relationships, fewer coincidences, to end up being doubly isogenous.

21:33 - You can also express these relationships in terms of Prym varieties if you want to sort of have a geometric explanation.

21:41 - So if you look at the degree 16 cover Z-tilde of Z, you can also look at intermediate covers and look at their Prym varieties which tell you chunks of the Jacobian of Z-tilde, and these relationships say when s1 and s2 satisfy some relationship, then you can express it as different Prym varieties that are automatically isogenous, which means chunks of the Jacobian or Z-tilde are isogenous, and then you just have to have some sub-piece of that to actually be isogenous just by chance.

22:14 - So finally let’s go back to our data, we found lots of examples of pairs that were doubly isogenous, and then we can take out all the ones that are isogenous because they were in these special families. And once what’s left over is less common, but is a little more random, but it’s very much in line with our original heuristic c/q.

22:42 - Roughly every time you increase n by one you’re going down by a factor of two, except it’s not nearly as clean anymore because the numbers are smaller.

22:51 - but this is saying that yeah our original heuristic about sort of elliptic curves behaving more or less randomly mod q outside these special families that’s a reasonable guess.

23:05 - All right so in the last couple of minutes I want to just briefly talk about a sort of analogous question that i’ve been working on with Felipe Voloch, and we’re using Hilbert class field covers which are the covers of a curve you get by pulling back the Frobenius minus the identity, and as I said the Galois group is the rational points of the Jacobian, which under the analog between function fields and number fields is just a class group, so these are really similar to the Hilbert class fields you see in class field theory for number fields.

23:42 - And the question is to what extent is C determined by the zeta function, maybe the data function of the Hilbert class field, or what we actually do is we look at L-functions, because they’re easier to work with and they sort of give us pieces of the zeta function.

24:16 - And we look at various Artin L-functions for the characters of the Galois group of this extension.

24:21 - Okay, so I don’t have enough time to talk about this in any detail, but let me just sort of say that the answer is the zeta functions, or rather the pieces of the zeta functions, the L-functions for the Hilbert class field, and the Hilbert class fields when you extend the ground field to all the extensions of Fq, together they’re enough to actually determine the curve up to isomorphism when the curve is genus at least two.

24:52 - All right so I’m going to stop here and I’m happy to answer questions.

24:59 - Rachel Pries: Thanks Jeremy, that was a Wonderful talk. Are there any questions? Could you just move your slide up a little bit so we can see the Mochizuki fact? Booher: Okay, yeah so the sort of other inspiration for why you might expect this is that for hyperbolic curves, so for example productive curves or genus at least two the etale fundamental group, which tells you about all the etale covers, determines your curve.

And if you’re given information about the Hilbert class fields for your curve and also the scalar extensions of your curve, that’s giving you all the differen etale extensions, after you’re sort of representing your ground field again.

25:56 - Pries: Okay I think there’s a question from Sam wondering about whether anything special happens at singular members of the family for the isogeny factors of Jacobians of the curves? Booher: I do not know off the top of my head what happens.

26:19 - Certainly when we’ve been analyzing things I think we just throw out any singular members of the family immediately and then don’t think about them.

26:28 - Pries: They’re only one or two points where the curves have automorphisms.

26:40 - Could you say a little bit more about the setup for your theorem? So is C initially defined over Fq? Booher: Yes, so this is a curve over Fq and then the setup is let’s take the Hilbert class field for this curve and for this curve extended to Fq^n, and then to actually compare the L-functions you need to have a group isomorphism of the Fq-bar points of the Jacobian, so you can start comparing characters, and then once you have that you can ask do all the characters match up, do all the L-functions for the characters match up? And then the answer is when they do then your curves are actually isomorphic over Fq, up to maybe some Frobenius twist.

27:32 - Pries: Would it be a very hard question to figure out what extension of Fq you need in order to be sure? Booher: I don’t know how to do that.

27:43 - So I do know Felipe and I have an example written down where you can’t just do this over Fq you actually do need at least one or two extensions, and in the kind of proof we use iit’s very not clear whether you you really need all n or just some n.

28:03 - It ends up using this result that is proven by Zilberg using model theory and it’s opaque as to what information is really necessary when you do that.

28:15 - Pries: Cool. Andrew Sutherland: Actually can I ask, on that topic, Jeremy, so do you have an effective algorithm here? Booher: So it’s not like, so like somehow the problem with using all of these sort of very nice covers that come from the sort of Hilbert class field is that the cover is rather big, it’s you know depending on the points of this Jacobian so it’s going to be like…

28:52 - Sutherland: I wasn’t asking so much about practical as effective, I mean if you don’t know how big the extension of Fq is that’s what’s worrying me, how do you know when to stop? Booher: Okay, so yeah I do not think it can be made into an algorithm, this particular theorem, but other things we’ve been thinking about are more effective but still not like computationally effective.

29:23 - Pries: Well thanks and there’s just one question from Vladimir.

29:26 - i don’t know if you can see it Vladimir would you like to ask your question yourself or should I read it? Vladimir Dokchitser: Oh sure, yes, I was just thinking, because you’re looking at covers of curves, is there any direct way to compare their Galois representations? After all, so what I have in mind is you know you you know the relation between the genera, because Riemann-Hurwitz tells you that, now from point of view of Galois representations that means you know their dimensions, how to compare them right because that’s the dimmension of the Tate module.

30:08 - So is there any result that actually tells you how to compare the Galois reps beyond the dimension? I don’t know, I’m really curious if there is.

30:26 - Booher: I don’t know of anything that’s very direct.

30:29 - I mean there’s there are various other pieces of information that are sort of encoded in like, like here’s an example.

30:44 - So for example Joe Kramer-Miller has recently been investigating if you have various kinds of covers of curves, what can you say about the Newton polygon of the curve which is which is encoding the Frobenius action on the Tate module, of a cover in terms of the ramification of the cover and the the newton polygon in the base curve and you can say something, although it’s not it’s not like a unique answer, like the Riemann-Hurwitz formula, and that’s sort of saying something along those lines but it’s not even phrased directly in terms of like describing the Galois action directly.

I think that’s sort of a very subtle and interesting question.

31:29 - I guess I’m not sure if anyone else knows a clean way to answer it.

31:32 - Pries: I think there’s a very interesting paper I think it’s by Borne, about extending the Riemann Hurwitz formula to more of a Galois representation context, but I would have to take a minute to look it up.

31:51 - Okay, Jeremy thank you so much let’s thank Jeremy again. .