Bianca Viray: Unlikely ramification in residue fields of points on curves (NTWS 291)

Thank you Philip for the uh introduction and thank you to the organizers for inviting me. And thanks everybody for for coming. Uh I see many familiar names, so it's great to see you virtually. Um yeah, let me just echo what Philip said about questions. I really love getting questions. That's one of my favorite parts of giving talks, so please uh please feel free to ask at any time, and I will try to keep an eye on the chat and uh and the raised hands as well, so that I can Never able to hear it. Never. >> [clears throat] >> Um so today the I'm going to speak about unlikely ramification and residue fields on curves. All of this is based on joint work with Isabel Bot, who's at Brown University. And uh my work in this was supported by the National Science Foundation and the my institution, the Department of Math at University Washington, which is on the ancestral unceded lands and waters of the Duwamish, Suquamish, Tulalip, and Muckleshoot nations. Okay, so before I get into any of the words in the title, well, maybe okay, the first five words in the title. Let Let me start with one of my favorite theorems. And maybe favorite theorem of many people in the audience. So, the Mordell conjecture or uh Faltings' curve theorem, so proved by Faltings in 1983. And is maybe familiar to to many of you. So, it says that if you have a nice curve, so smooth projective and geometrically integral over some number field, then if the genus is at least two, you can only have finitely many rational points. Just points defined over K. So, this is a very very strong example I'm I'm going of geometry controlling arithmetic. So, the genus of the curve, that's a geometric invariant that we can read off after passing to the complex numbers, forgetting all of the arithmetic structure. But that that invariant is strong enough to control the arithmetic properties. And of course I wrote this over K, but if I take any finite extension L over K, that L is still a number field. So if when I base change C to L, that still satisfies the hypothesis. So it tells you that over any finite extension, we still only have finitely many rational points. So by making a finite extension, I can never break uh break this uh infinite barrier. But of course if I take the union over all of them, then that is the points over the algebraic closure, and there we have infinitely many points because whenever I fix any number of the coordinates, solving the last one any any number of the coordinates in K bar, solving the last one, I still stay in K bar. And so this is this is infinite. So the union is infinite, uh but each each piece that we're taking the union over is finite. And so the vague motivating question I want to take to start is sort of how these K bar points distribute over over the L L points. So this is a little imprecise at the moment, but hopefully gives you um a sense. And uh so let me try to make this a little bit more precise. Okay, so everything just moved up. Um if we're trying to understand the set of K bar points, and we're just working over K, then what Galois theory tells us is well we can't really understand an individual K point K bar point on the nose, what we can understand is its a orbit because we can't distinguish between different points in the same Galois orbit using only information over K. So, when I think of the K bar points in a Galois orbit, I can also describe that just scheme theoretically as talking about closed points on the curve. So, a closed point on on the curve X, you can also think of it a Galois orbit of K bar points. And then given some Galois orbit of a K bar point, I can ask um what is a minimal field of definition of some point in the orbit. The exact field as a subfield of K bar will depend on which choice of point I've taken, but if I if I quotient by isomorphism, then then this will be well-defined. Okay, so if I do this instead thinking in the language of closed points, then what I'm doing is taking the residue field of the closed point. Okay, so throughout the talk, I'm going to use the language of closed points and residue fields, but if you prefer, you can just replace all of that in your head with Galois orbit and minimal field of definition up to isomorphism. Uh Okay. So, the question I want to ask is how these residue fields compare to the set of number fields. So, the set of if you fix a curve and you range over all closed points, you get a bunch of residue fields. That's some subset of the set of number fields. What uh what subset do we get? Okay, so just to dial in a little bit more, so really this map uh you know, like any map, you can you can factor to the image and then include the image. And so, Faltings' theorem is about the vertical map in this picture and it Faltings theorem phrased in another way tells us that if the genus is at least two, then this vertical map is finite to one. So in some sense, it's taking all of the closed points, the infinitely many closed points and spreading them out sort of as much as it can because there's no fibers with infinitely many points. So this infinite set is being spread out. Okay, what are some other uh observations? So this this image of the vertical map is infinite because the the top is infinite. Okay, even on a genus zero or one curve, the image is infinite. So that's always true. And then if you have some curve at some point on your curve of a fixed degree and it's large compared to the genus, then you know you're actually going to get infinitely many points of that same degree, so infinitely many residue fields of that degree. Well, okay. By Faltings, then infinitely many residue fields of that same degree. And in fact, you get a Zariski open of some large projective space corresponding to these degree D points. So um So I'm not explaining that this is basically Riemann-Roch and Hilbert irreducibility, but you do have this very large set uh of of degree D points. So in some sense, you can think of this as saying that points of large degree on any curve are very common. But a curve, if your curve is genus at least two, it's I mean, it still has complicated geometry. So by this philosophy of geometry controlling arithmetic, even though the points of large degree are common, there should be some sense in which they're less common for higher genus curves than they are for a genus zero or one curves. Like in some big big sense of that word. So this is the more precise question of this talk that I'd like to explore. Is that is can you show that the image of this map is constrained when the genus is at least two? So that somehow the set of residue fields of a genus at least two curve cannot sort of fill out all of number fields. There should be should be constrained in some way. So this is the the question that Isabel Voigt and I set out to explore and we were particularly focused on curves whose Jacobian has rank zero. And we focused on rank zero because we thought that the structure of complete linear systems would give us extra leverage to be able to study points of arbitrarily high degree. And so uh let me say up front we don't do not have a complete answer to this question. I cannot characterize all residue fields of a particular a genus two curve. You know, I think that would be a very difficult um question but we do have some results some results that I find exciting in this direction. I think it's we're we're showing that there's there's a lot of interesting um arithmetic to explore. Okay. Maybe I should pause to see there any questions. Okay. So Okay, as you can tell from um how I set this up This is sort of a different take on the geometry controlling arithmetic than is than is has been the object of most study. So, the the one result that I know in this direction is by Mazur and Rubin in 2018. So, they studied something that they called L-diophantine stability. Uh I think they were motivated by applications to Hilbert's 10th problem. And so, among many many results in their paper, one thing that they proved is if you have a curve of genus one whose all of whose endomorphism geometric endomorphisms are defined over your ground field, then there's a positive density set of primes L such that for every N, if I take the residue fields that are cyclic of degree L to the N, so, just the cyclic ones, then that will exclude infinitely many cyclic extensions. So, as you go up, so this L to the N as you go up, so for every L in this positive density set, you can go up and you'll see that you're you're not getting everything in this image, and particularly you have to miss infinitely many cyclic extensions. Okay? So, uh that's what they show. So, uh in joint work with Isabelle, what we show is that if we take a curve, so the hypotheses are a little bit different. So, we want Jacobian the K points of the Jacobian to be finite, as I said, and then also no rational points on the curve. Then what we can show is that for all sufficiently large primes L, so excluding finitely many primes, um that if you take uh the cyclic extensions of degree L. So, we know by Mazur and Rubin that this is going to exclude infinitely many cyclic extensions. Uh but what we show is that the ones you obtain, that that is actually a finite set. Just we so that we just do this for prime degree cyclic extensions. So, there's a stronger result when you look at prime power prime Yeah, prime power, not prime. Um but this is what we get for prime degree extensions. So, it's it's giving showing that there's actually quite a strong bias for a curve of genus at least two without rational points should be biased away from cyclic residue fields. Okay. So, how How do we obtain this theorem? Uh and particularly where does the prime hypothesis come from instead of prime powers? So, we actually obtain our result as a corollary of a of a different result. So, it's not really the Galois behavior that we intrinsically control, it's something else. But just before um I explain the theorem, I just need to give some context about the general structure of these residue fields. So, let me just look at that. So, when you have a residue field, there is a discrete invariant related to that, the degree of the residue field. So, you know, we're really concerned about the top arrow, but of course the bottom uh the bottom map is going to constrain what can happen. And if you're looking at a set of integers, well, we know one of the constraints that can happen, something is divisibility. So, you could look at the subgroup generated by the degrees of the residue fields. The GCD that that ideal will be generated by a number that we call the index. So, these residue fields always sit uh inside of this subgroup just by definition. Some The index can be one. That That certainly happens a lot. So, some sometimes this is not a constraint, but the index can also be greater than one. For instance, if you take a curve over Q that has no real points, then its index will always be even. So, and the degrees, you can show by Riemann-Roch and Hilbert irreducibility, it's actually cofinite in this subgroup multiples of the index. So, every sufficiently large multiple of the index is the degree of some field extension. So, this is the only constraint. So, the theorems are only interesting when we look in residue fields that are divisible by the index. So, in the theorem statement and in subsequent things, you will see multiples of the index everywhere. That's not an If you do it outside of that case, it's just not interesting. There's nothing to say because there's no points of of degree divisible by the index. If you don't think about the index that much, you can just take You can just think of that I'm writing one in a very complicated way and respect to that case, and that's totally fine. Okay. So, now that we we've set the context, so what what do we show? So, we take a curve over a number field whose Jacobian is finite, and then I have some other condition which I've grayed out for now cuz I don't want you to focus on it too much, but I promise we will come back to it. So, some geometric condition. Then, you have some positive integer M depending on the curve C so that for all D co-prime to this M if I take the points of degree that multiple of the index D times the index and now I'm going to not put a Galois theoretic condition on the residue field. What I want to do is assume that you have a totally ramified residue field totally ramified at some prime. But now instead of keeping track of the residue fields, what I want to do is keep track of the prime where you have this total ramification. Okay, so we range over all points of this fixed degree um and we take the one we take the places where that residue field can be totally ramified. And then what we show is that this set is finite. So the slogan of how I think about this is that total ramification in this multiple this degree D times the index is unlikely on the curve. So total ramification can only occur at finitely many places. So these residue fields that you're getting, they're biased away from having total ramification. And then once you have this theorem, well how do we get the corollary? Then we apply the wonders of class field theory. So a cyclic extension if it's any prime that's ram cyclic extension of prime degree if you have a prime that's ramified, it will be totally ramified. So now the theorem tells you, okay, well that prime can only be in this finite set. So that tells you any cyclic extension will be contained in the ray class field of some fixed modulus. And so that's a number field that's a finite extension, we get finitely many possible cyclic fields of prime degree. Okay, so that and that's why we have to stick to prime degree case because if you have cyclic extensions of composite degree, then you can have ramified primes that are not totally ramified and uh we can't appeal to class field theory. Yeah. Okay. So uh my plan for the next large chunk of the talk is to talk through different aspects of this theorem. So first I'm going to explain where this M comes from and uh what what what values of M you get in practice. Then as I promised, we're going to come back to this assumption. Uh I'm going to tell you where that comes in and then the lastly I'll talk about how this Jacobian uh being rank zero gives us the leverage to prove a result like this. Uh so just before starting with these, I'm going to um act like a computer scientist and index with zero to begin with and I just want to return to this question about this image being constrained. Okay, so um of course if you look at number fields of high degree there's a lot we don't know about what kind of what the distribution of number fields is and and what you know, how many number fields there are. So if we don't know the big set on the right like it's hard to to quantitatively describe how much you're missing. But what we can do is is throw some of the heuristics at the problem. So there are there are juristics due to Manjul Bhargava about um sort of the proportion of of number fields with specific behaviors. And those juristics when you plug them in, if you exclude um total ramification away from a finite set, then those juristics predict that you're excluding a positive proportion of SP number fields. So certainly we're excluding an infinite set because you can make field extensions uh totally ramified. You can make number fields totally ramified at any prime you want. So for each prime outside of this finite set, there is some number field that doesn't fall in there, but then the juristics predict that it will actually be a positive proportion, not just a density zero infinite set. So that's some some context. And uh just for comparison, so Mazur and Rubin, the statement is um says that they exclude infinitely many. I have not gone through the proof carefully enough to be able to definitively say they don't get something stronger. They might they might get proved something stronger, and it's not stated in the theorem, but the theorem just tells us we exclude infinitely many. Okay. Okay, so then the next thing I promised you is this and let's see. So this integer M, it can be computed in terms of what's called a torsion packet. So I'm going to talk more about torsion packets later, but for now this is just something that already has a name and has been interest uh uh studied before. And so there is an algorithm to compute the torsion packets that we're interested in for genus two curves. That was um made by Bjorn Poonen and he has pari code to do this. So, we ran this code on this this LMFDB. This curve of conductor 76 76, which satisfies all of our conditions, and the algorithm tells us that it's going to be two. So, actually for this curve, and that that's smallest possible for hyperelliptic curves. For any hyperelliptic curve, it will always be at least it will be it will be even. Um And so, what we get is that total ramification of odd degree is unlikely on this curve. Okay, and we there's uh so, we went Something went wrong with my Okay, there we go. Um we went through the LMFDB and we ran this algorithm on many other curves that satisfy our examples and these are the first 20 that we found ones that have n equal to two and also our theorem applies. So, for all of these 20 genus two curves, we know that total ramification in every odd degree is finite. So, for me when I see a question like this, um you know, I just I'm like, well, what like, how finite is this finite set? Uh what does it look like? So, um the methods give an effective approach, but it's not very efficient. So, we could compute it for degrees three and five. Um so, that's what um we did for these these examples. And so, this is just to give you an idea of what we're getting. So, each row is the curve with its LMFDB label. And then in the middle column, the D equals three column, these are, if you range over all degree three points on that curve, and you look at the odd primes of good reduction that could be totally ramified in some residue field, that is what you get on the list. And odd is just because of the way we wrote the code. I wasn't careful enough to do it in a way that we could tell um whether two is in the list. Uh and then you should also show it throw in all the primes dividing the conductor, but I didn't want to clutter up the list too much. So, you can see it's it's fairly sparse. Um and then as you go to degree five, it still is, I guess, very fairly small number of primes, but the size of the primes um can be pretty big. Um and for each of these primes, you can construct uh can construct a point that is totally ramified. So, every prime in this list actually does appear as one of the the primes that's totally ramified. Okay. Great. Um so, I would love to be able to compute uh higher degrees. Right now, the way we're doing it is sort of naive and we end up with some Grobner basis and I just want want to finish in degree seven to um but if this looks appealing to you and afterwards you feel like you have some ideas, I would love to love to talk and cuz it would be great to have more more data. Okay. So, that's that's M. And the next thing I promised you was this hypothesis. So, this hypothesis is necessary in a very strong sense. So, what I mean is if you take a curve and it doesn't satisfy this hypothesis, so I think I did that negation correctly. Then, what we show is that for all sufficiently large degree, the set of primes that can be totally ramified has positive density. Okay. So, this assumption is necessary in a very strong sense, meaning for each curve that the hypothesis does not satisfy, not only is there a infinite sequence of of D as you go up that have this being infinite, but that you get its positive density and it's all sufficiently large degrees. Okay. So, another way to think about this hypothesis is that the that is really sort of the the geometric or yeah, the condition that is telling you whether total ramification should be likely or unlikely. Okay. Just a word of of context. So, this hypothesis So, for the second theorem about the the strong converse, um that always holds that the genus is at most one. So, for any curve of genus zero or one, regardless if it has rational points, it doesn't have rational points, you have a map form. So, we do get uh total ramification is likely on curves of genus zero or one. Okay. Any questions here? Okay. And then, so the last thing I promised you about this theorem is where um this hypothesis about the Jacobian uh comes in. And this really was sort of part of our initial starting for the data is if the Jacobian is finite, then one thing it tells you is for each degree, no matter how large the degree is, all of the degree D points come in finitely many complete linear systems. So they come from finitely many different embeddings and just taking hyperplane sections with the embeddings. So that's a a very nice concrete uh parameter space for the degree D points, and we felt that we just should be able to leverage that. And uh yeah, turns out to be to be the case. So yeah, this is really the key idea that they come in finitely many complete linear systems. So I I'm going to walk you through how you can control total ramification in a single complete linear system. Okay, so if we have a single complete linear system, then the points in this all come from intersecting with hyperplane. Okay, so this is my cartoon of a curve embedded in some projective space. It should be very large dimension, but my cartoon only works in P2 because that's the limit of my artistic abilities. Um so but still so anything is coming from slicing with a hyperplane, so we draw some line. You know, here I've drawn the real points, so it looks like it's several distinct points, but when you're taking a closed point, you know, there's some Galois action uh joining them, so this is a single single closed point. So what happens if this residue field is totally ramified at some prime of good reduction. So then when you look at spec of OL or you look at this evaluation ring and you reduce mod P, it's going to become non-reduced. So what that corresponds to is when you degenerate this over the special fiber, this hyperplane is going to become totally tangent at a rational point and FP point. That is the geometric picture of a point that's totally ramified at this uh prime. So what this tells you is when you take the divisor given by X and you take the limit by degenerating to the special fiber, it's going to be the degree of it times some rational point FP point on the curve. So in particular, it's going to be contained inside this one-dimensional subvariety in that component of the Picard scheme. Okay. So let's uh go through this. So if the degree of the point divided by the index, if that's relatively prime to the size of the Jacobian, then you can divide in uh in that um in the Picard scheme. So you get that this point, I can write it as a multiple of a of a divisor of minimal positive degree. So I can view this X um Oh, and now this sort of changed what Well, okay. So not only is it in some fixed linear system, but it's in a fixed linear system that's a multiple of a point of minimal positive degree. Okay, so now I'm going to take the condition from the gold box and just just uh um substitute in this this divisibility condition. So now we get this multiple of this point of minimal positive degree is in this one-dimensional variety. And you see that there uh I'm just going to rewrite degree over the index cuz that gets kind of annoying. So I'm just going to call that some D. And then I can I can pull out a D on both sides. I'll write this in terms of D and the index. Okay, and now I have a multiplication by D on both sides, so I can sort of divide by D and consider torsion points. And so what we're getting is that if you're have a point whose residue field is totally ramified at P, then you have to have a non-empty intersection of this subvariety, one-dimensional subvariety of your with the D torsion of the Jacobian. Okay, so this on the bottom line is sort of packaging this geometric degeneration cartoon. That's the condition we get. Okay, so I'm going to take this bottom line and slide it up and then we're going to deduce more more from that. Okay, so if you look at this intersection, so this intersection is happening in the Jacobian or the curve over FP. But really what's happening is that it's it's just everything is the reduction mod P of something that's defined globally. So this intersection is the special fiber of something that is happening integrally, of an integral model of the curve, an integral model of this divisor, an integral model of the Jacobian and it's happening integrally. And so I have this intersection defined integrally, so that's a closed subscheme over spec okay. And so its support will be a a closed subscheme of spec okay. So if the generic fiber is empty then it will be empty for all but finitely many primes. Just because that's the way that the Zariski topology works. So what that tells us is that if the generic fiber of this intersection is empty, so now this is happening over our number field K then total ramification is unlikely in this multiple of the complete linear system. Okay, and just to remind [clears throat] you what it means to be unlikely is that if you take all of the points, those points that are in this complete linear system that are totally ramified at some place, the set of places that can appear that is finite. Okay. So why is this condition on the bottom that's so great? Well, if we look at this intersection and I said this in words, but just to draw attention to it, we're taking a subvariety of the Jacobian intersected with a torsion subgroup. And the subvariety intersected with the torsion subgroup is exactly what the um Manin-Mumford conjecture, which was just proved by Raynaud uh controls. So that that theorem tells us if you have any abelian variety and any closed subvariety then you have basically the intersection with the torsion just comes from torsion translates contained inside your variety. So, in other words, if you take the Zariski closure of Z intersected with all the torsion points, what you get is a finite union of torsion translates that were already in Z to begin with. So, what this tells you is that if um Z doesn't contain zero, then it's hard for you to intersect the torsion of a particular order. And for the D that often that you exclude is for all D outside a finite union of proper subgroups. Okay, so what the Manin-Mumford conjecture tells us is that if our sub variety doesn't contain zero, then we should expect this intersection to be empty for all D outside a finite union of subgroups. In particular, it will be for all D coprime to the LCM of all the MIs. Okay, so these are all the ingredients. I'm going to bring them together on one one slide. So, on the left is the same theorem that we've been looking at. And on the right we're going to give the sketch of the proof. And I'm just going to assume that not only is the Jacobian finite, that it's just trivial. It has trivial Mordell-Weil group. And um that's not a super strong assumption, but it it makes the bookkeeping much less onerous. Much Yeah. Okay, so we do is we take this hypothesis and then we show that that hypothesis means that zero is not in the sub variety that we care about. It's basically a direct translation. It's a it's an if and only if. So, since zero is not in the subvariety, then as we saw on the previous slide, the Mordell-Lang conjecture, which was proved by Raynaud, tells us that this intersection is going to be empty for all D outside a um proper finite union of subgroups. And so then for all D outside that proper union of but subgroups, there are only finitely many primes where you can have a point uh totally ramified at that prime. And so that tells us total ramification is unlikely in this multiple of the linear system. But that's that's every Since the Picard group is generated by D, every closed point has to live in live in a multiple like this. So, those those are the main ingredients. Any questions? Yes, Padma. Hello. I should Maybe I should have asked this before. I was just trying to phrase my question. How hard is it to uh verify the hypothesis you place on your curve? So, if I give you a particular curve, you have the hypothesis of your theorem that as you vary over all maps to P1, you want to check that there are no totally ramified points of that degree. Is that a checkable [snorts] >> Yeah, but but it's only only maps to P1 of degree equal to the index. Yeah. So, for curves of index one, and if the curve is genus at least one, you will have no map of degree one, and so then there's nothing to check. Okay. So, if it has index one, then the condition is only that you have a rational point. It has no rational point. Okay. >> Which Okay, yes, that can be hard, but >> [laughter] >> many times we can we can verify it. And certainly on the LMFDB they they found the rational points for many of the curves. And okay, when the Jacobian is finite, so if we have the Jacobian, then this is easier to check. So you first check the Jacobian is finite, maybe you check what your uh generators are, and then it's and then it's much easier to check whether you have this. And I mean points of higher maps of higher degree of degree at least three, I mean, they typically are not totally ramified. You know, the generic case is that you just have simple ramification. So the most often cases when this fails is when you have a hyperelliptic curve and the index is two. Um I had maybe second question which maybe is orthogonal. You can postpone this to the end of the talk if it's going to take you far off far away, but it seems like uh so for for any given curve you get like the special set of like if you fix the invariant city set like D, you get a special finite list of primes that you associate to this curve. Can you use this as a signature for your genus D curve? So if I fix the genus and I just for each curve I record this finite set TD, can I go back and say which curve I came from? I I I I didn't just stare at your table long enough. I don't know what to look like for each of the different curves you produced. Look like different different collections of primes show show up in in your table, right? That's right. That's an interesting question. I think we don't know either way. Because you're not just keeping the union of the TDs, you're keeping them indexed by D, is that what I understand? Mhm. >> Yeah. Yeah, I'm not sure. Okay. I think I guess it would I think that's closely related to whether the um it's closely related to whether the curve over the finite field how that intersects with the torsion with the Jacobian over the finite field and the group structure on it. It whether that encodes the curve. So, I maybe that could be related to the L function and then that would possibly give you the curve, but that That's just the first way I have to think about it. Yeah. I I don't I don't can't say anything more definitive. In your table, you didn't have coincidences, did you? Like the sets looked different for different curves. >> No. No, we didn't have any coincidences, but that's a very small data set, so Yeah. [laughter] Yeah. Okay, anyway. Yeah. Thank you. Okay. Yeah, thanks for the question. Okay. So, I don't see any other hands. So, um I'm going to continue. Um So, when Isabel and I figured out this theorem, then one of our question was like, okay, well, how special is total ramification? Or does it does it apply more generally to other ramification indices? So, uh Yeah, that's what we explored next. But, before stating those results, we need to define how we talk about it. Let's just review what happens if we have an extension of number fields and you have some place, then when you take this number field and you tensor it with KV, it will split into a product of other field extensions indexed by the extensions of V to L. And we know that the degree of L over K is equal to the sum of the ramification degrees and the residual field degrees at these local extensions. So this tells us, you know, this this discrete data encodes something about how V splits in K in K. Oh, sorry, splits in L. So we're just going to package this as an abstract tuple of integers to be a splitting profile. And a very reminiscent of the term the terminology we use for number fields, we're going to let R be the size of the splitting profile, then the number of elements in the set that corresponds to the number of places. The degree is the sum, and then we'll often want the ramification indices or the residual um degrees, these Fs. Okay, and then given a an extension of number fields as I described below, that gives you a splitting profile, and we denote that splitting profile P sub V of L over K. Okay. So So what can we get for arbitrary splitting profiles? So on the left, this is the same theorem that I already presented, just to be there for comparison, so I've graded out a little bit so it's not so doesn't seem so cluttered, but it's there for comparison. So what uh what can we show? Well, we still need the Jacobian to be finite cuz as I said that that is um key part of our of our tools. And then we take a splitting profile that an unramified base splitting profile, so all the Fs to be one. The index of the curve has to divide, sorry. We take a splitting profile where all the Fs that they split um there's no residual field extensions. The index has to divide the ramification degree for each degree, and we want the degree of this thing over the index to be coprime to the size of the Jacobian. So, in our initial theorem, that was like taking this this base splitting profile. So, then our condition so, it's a much harder but it's yeah. So, we we put a stronger hypothesis than is than is necessary, so we don't get a strong converse for this theorem, but if you have no points of degree at most the degree of this profile, then you get something similar. So, for all D coprime to some integer that now depends on M and the splitting profile uh that depends on C and the splitting profile if you take the same F, but you bump up the the ramification degrees by the same integer that as you go up that sort of one-dimensional ray of splitting profiles you'll get unlikely splitting profiles for every D coprime to M. Okay. And the reason why the result has this form that we're looking sort of one ray at a time of splitting profiles, this is because we're using the Manin-Mumford conjecture which is about torsion subgroups. And so, basically to reduce to the case of studying torsion on an abelian variety, you need to only look at a one-dimensional ray of splitting profile. You can think of this is like the torsion subgroups are indexed by a one you know, they're just indexed by Z. So, that that is matching up with this one-dimensional ray. But so, those of you in the audience who work in unlikely intersections, you know that we expect things like the Mordell-Lang conjecture to hold more generally. And so if we assume those stronger conjectures, then we can prove a conditional result where you range over higher rank subgroups. Okay, so specifically what we need is a special case of the Zilber-Pink conjecture. So the Zilber-Pink conjecture controls intersections of subvarieties with algebraic subgroups of complementary dimension. Torsion subgroups are dimension zero. So that's where Mordell-Lang comes in. So we don't need intersection with arbitrary subgroups. We just need the intersection with these particular ones, which we call like kernels of dot products. So your Z sits inside a a self product of some abelian varieties. And then you have an integer vector of length R. And then that gives you a map from A to the R to A, basically dot product. Taking the dot product with this integer integer vector. Okay, so those those uh kernels of those maps, those subgroups we care about. And Okay, this is not stated exactly the same way that the Zilber-Pink conjecture is stated, but we we showed that it is a consequence of the Zilber-Pink conjecture. And it is this piece that we need. And then once you have this conjecture, what we can prove is the following. Um And you can So here I've restricted to the case that the Mordell-Weil group of the Jacobian is trivial. You can get away with finite, but it's just a little messier to state. But we do want now the Jacobian also to be geometrically simple. And then we fix our F vector to satisfy these these properties. And then what we show is for all ramification vectors outside of a proper finite union of subgroups. So, you take a finite union of subgroups you take Yeah. A finite set of subgroups each of that you proper but also the union of them is also proper. So, then it will exclude a entire coset of a finite index subgroup. And then we get that this splitting profile is unlikely on the curve. Which again means that the set of places where this splitting profile is achieved is finite. Okay. So, uh Yeah. I just like to end with the I think that these results together suggest uh um a implication of the more precise implication of the geometry controls arithmetic philosophy that that part of the arithmetic that should be controlled is large ramification degrees and that large ramification degrees should be unlikely unless there is some geometric reason for So, thank you for your attention.