 Thanks a lot. So I'd like to thank very much the organizers for the invitation. It's a true pleasure to be here. It takes me back to my youth, which is always good, right? When I was a PhD student, I used to spend several hours here at the library, so it's really nice to be back. Okay, so everything I want, I'm going to tell you about today's joint work with Jean-Porty and well, I just take the occasion to do some kind of advertising for character varieties over fields of positive characteristics. So I hope I'll try to convince you there's something to study there to do there. We'll see. So the plan of my talk would be like this. I'll try to review a little bit what character varieties are because I guess there are several students in the audience, so I apologize with people that are experts in the field. Try to be patient with me and not expert myself, so then I'll try to tell you why this character varieties overfields of positive characteristics should be interesting or should be studied or something and after that trying to tell you, giving you some results, well, some very basic results, I need to start with some other general results, okay, on character varieties of nots. So I'll give two types of results on character varieties of nots and use these to give well to say something about character varieties in positive characteristic and in particular ramification phenomena, so it doesn't mean much like that. I'll try to explain what that means during talk and at the very end, if I got a little bit of time, I'll try to mention something about the proofs of these general results about character varieties not necessarily in overfields of positive characteristics. Okay, so let's go. So to start, to understand what character varieties are, you have to start by understanding what representation varieties are, so you take G, a group, which is finally presented, so say you have X1, XN generators with some relations, and you want to understand, you want to consider the set of all representations of your group G into SL2, say C, okay, so what do you do? Well, you have to map your generator somewhere to some matrix, so your generator XI goes to some matrix, a 2 by 2 matrix, say I 1, 1, I I 1, 2, I I, 2, 1, I I, 2, 2, I and then, well, this mapping extends to a representation of the whole group provided that when you have a relation R, J and D in the generators and you plug in well, your matrix is there, so you have the same word, but in these matrices now, well, what you get is the identity and that's necessarily a sufficient condition to have a representation. And of course, you have also, you want these to belong to SL2C, so well, the determinant must be 1. Good, so this is a nice set, but it's a little bit big and you'd like to, you know, consider that two representations are basically the same if they are conjugated. So, you just well, you, what you would like to do is just quotient out the set that you have by the action of SL2C by conjugacy and, well, the action of by conjugacy factors through the quotient of SL2C by its center, so that's PSL2C. Okay, so that's the idea. You want something small that gets all the information you have in the representation variety. There's only some problems in general. Of course, you have a topology on this set that comes from the topology of SL2C, these sets live inside some C to the 4n, yeah. But as you know, quotient can be very badly behaved and so if you, if you think as a topologist, at least you want your points to be closed. So what you do when you take this quotient is you look at the orbits of your action, if they are close, fine. That's a point in the quotient. If they're not, you collapse. Orbits whose closures intersect. Okay, so if you are more of an algebraic geometer and not a topologist like me, you think differently. You think that these guys are algebraic sets and actually are called, well, varieties, although they're not necessarily irreducible. And you can think of them, I mean, thinking of an algebraic set is the same thing as thinking of the ring of coordinates over the algebraic set. And given the ring of coordinates over R of G, well, you just take the core, the functions that are invariant by the action. And that's again the ring. And you just consider the variety that's associated to that ring. Okay, so let me see what's next. Okay, so as I was saying, all the conditions that you have here are polynomials. And actually, there are nice polynomials that have integer coefficients. Huh, that wasn't a nice choice of coefficient of indices. So let me just change these. Okay, so this is IJK, like that. And so you have two things here. So it's clear that R of G is defined by polynomials with coefficients in the integers. It's not as straightforward to see that that's the same. That's also true for the character variety, so for the quotient. But it's still true. I mean, you have to believe me. And actually, what's even more interesting is that everything I've told you till now never uses the fact that we're working over C. I mean, the coefficients here are just, you know, we could as well work over Q or whatever. Only that if you do some algebraic geometry, you'd like to work with algebraically closed fields. So that's why you choose C, for instance. But the good point is that if you'd like to consider representations over, say, the algebraic closure of some finite field of positive characteristic, then, well, you just take the same equations, mod p, and you're done. You're good. So that's the same polynomials that give you the character variety of the representation variety will give you the character variety over fpq when you think of them as, you know, classes. I mean, the coefficients are classes mod p now. So that's good. Not necessary. It's not even, I mean, it's not even reduced here. I was telling you it's called variety, but it's not reducible in general. So that's what I was telling you. And something that you know, people that work in algebraic geometry know very well is that actually, if you look at your variety over C, or if you look at it over a field of positive characteristic, generically what you see is the same thing, meaning that if you have the same number of components, the components have the same dimensions, and so on and so forth. But sometimes for certain bad primes that are called ramified, things happen. Things happen that maybe, I don't know, dimension grows up, or some components, some irreducible components disappear, or something like that. And this happens for a finite number of primes. What I didn't say, and I add to, is that what I'm going to tell is for prime characteristic different from two. Two is a very odd prime, although it's even. I'm sorry to say so. So it really behaves badly. So it's not even clear, not even true that the equations are the same over a field of characteristic, too. So forget about two. And so primes for me are odd. Two is just a stupid number. So why would you care? Why would you consider these? Well, of course, as I described it, you might wonder that the representation variety or the character variety may depend on the choices that you have here of the choice of representation. But if you change representation, sorry, presentation, if you have a different presentation, you have tits and transforms that take you from one to the other. And these induce morphisms, isomorphisms between the character varieties or representation varieties or whatever you want. And moreover, these are defined over the integers, just like the equations. So the isomorphism class of this variety is well defined, only depends on your group. And by the way, what group I will be interested in, well, the group I'll be interested in are fundamental group are compact manifolds. And more specifically, as I told you before, I'm interested in fundamental group of null exteriors. So just think your favorite, not please choose it to be hyperbolic. So it's not well, it's meaningful in some sense. And that's your g. Fundamental group of that not is g. Okay. So the point is this character variety encodes, I mean, is itself an invariant of your group and so of your manifold or of your not. And in any case, anything that you can construct in a natural way from your character variety is, again, an invariant. So for instance, things that can be constructed starting from the character variety are things like the A polynomial for basically for hyperbolic knots or for a hyperbolic knot. You know that you have the autonomy representation in PSL2C, this representation lifts to SL2C and lifts if you have a knot on a curve, so on a new reducible component of dimension one. So you always have this component, which is the component that was used by Thurston to prove the hyperbolic denser joy theorem, for instance. Okay. And so that component, that one-dimensional component, may be non-smooth, may be singular, but there's a unique way to desingularize it. Once it's desingularized, it's just a remand surface. And you can, for instance, consider a genus, something that Peterson and Alan Reed here studied. And the genus is, again, another invariant that you can consider. And then the other information that you can deduce or you can have from the study of the character variety, for instance, color shell and theory allows you to find essential surfaces inside your manifold just by looking at curves in your character, in the character variety of the fundamental group of the manifold. And that uses something that Ian Agol was describing this morning, this Buster tree built out of rings. Okay. So I won't say much more about this, but in the same spirit, all these strange primes that can appear, okay, these runnified primes, they're also invariants. What kind of invariants are they? What are they telling you? What happens? And what's the relationship with the topology or the geometry of your manifold? That's what I would like to understand. And so far, I have to say not much is known. Okay. So why would you care? Okay. So when Zhuang and I started thinking about this, the initial motivation, which is sort of obsolete now, it's not really interesting at this point is, this, I was telling you this color shell interior allows you to find essential surfaces in the, in the, in your manifold by looking at the character variety. It happens that some surface, some essential surfaces cannot be found in this way. They are not detected by the character variety over SL2C. And so our original question was can be, can they be detected in characteristic P for one of these ramified primes P? And well, now this question is, well, not, not very interesting anymore because a recent result of Friedl Kitayama in a gel says that you can always find these essential manifolds provided that you consider character varieties over, well, corresponding to representation in SL and C. So if you increase the dimension of your representation, you're, you're good. Okay. For some N large enough. Okay. Also, this original question we have seems very hard to, I mean, yeah. Okay. So it means that, so essential surfaces correspond to ideal points of, of your character variety. And ideal points of this, of the character variety give you valuations and the valuations just like tomorrow gives you action on trees. And then the action on trees give you surfaces which are, say, correspond to stabilizers of their essential fundamental group correspond to stabilizers of edges on the tree. So if your surface is one of these is detected, but there are some other which are not. And the point is that the non-examples that are known so far were given by Shanwell and Zhang, I'm not sure that's the correct pronunciation, so forgive me, are somehow structural. So they build them so representations of are killed over them. I don't know. I don't want to enter into details, but there's something which is the construction is not something that related to C. It's just really structural to SL whatever field of you use. Okay. So it seems very difficult that all, well, for sure not all undetected surfaces can be detected in characteristic P. And even if there are surfaces that are detected in characteristic P, but not in characteristic zero, we don't have any examples. So far is we don't know. Okay. But still, I mean, it's not even that easy to understand what kind of phenomena happen. And let me see what I'm going to tell you later. Next. Yeah, I'll come back to that later. So still you would like to understand what happens, what happens, what are these primes, what's the relationship with the geometry of the topology of the manifold or not or whatever. And also give examples. So what can happen? Why it happens? Okay. And what's the relationship with the geometry and topology of the manifold? That's, that will be, I think that would be nice. And the point is that we have examples, but they're sort of artificial. I mean, we really need to have examples of character varieties in whatever, okay, whatever characteristic and use these to give examples of ramification. The point is that so far the examples of ramification are somehow, you know, built so they work. So we don't have interesting examples. And the examples we have are all based on the fact that in characteristic P parabolic elements of order P, which is kind of stupid, but it's central in what we will be constructing later on. Okay. So it would be nice to see something smarter, something that happens for a more geometric reason and not just this stupid algebraic reason. Okay. So far I don't know. And yeah, before I did tell you this, maybe I'll just add something. And why is it so difficult? The point is that the computational complexity, when you want to study character variety, varieties is soon very, very difficult to handle. Okay. Either you have very stupid examples and so nothing happens nowhere or you just can't do computations. And well, what can you do? I mean, so you need to find some generic ways to give, to find information, species of information about your varieties and exploit these. But of course, since you're generic, you never find, I don't know, very specific behaviors. Okay. So in the next slides, I just want to give you some results about character varieties of knots. The character varieties of two class of knots, very specific class of knots. And these are sort of general and tell you how they look or at least something about their structure. So the first family of knots I want to consider is a subclass of Montesinos knots. So what are Montesinos links? Montesinos links are guys that look like this. So every box here represents a rational tangle just like this one. So you plug a guy like this one in each box and you have, I don't know, n boxes or whatever. And the result is what is called a Montesinos link. And I'm not interested in any kind of Montesinos link. I just want the link to be a knot, so just one component. And moreover, I want it to be what we call of kinoshita Terazaka type, which means that basically in one of these boxes, so these boxes are very complicated tangles, but their behavior is just like this. So you have two strands here and two strands here. And then you can do three things basically. Either you're here and you go back or you're here and you go straight or you're here and you go straight across. And the other guys can't do anything else but fill in the other two dots. So we want that in one, exactly, precisely one of these boxes, the behavior is like this. So that's what we call a kinoshita Terazaka, Montesinos knot. For these guys, this is a result that we have. We can say something about their character varieties. And what we say is the character varieties of these guys have components of very large dimension. And in some times, in some cases, there are lots of components of large dimension, which are, I'm saying, non-standard. So as I told you before, for instance, a hyperbolic knot, we know by first and there's this curve inside, you also know, okay, so the list of things should be on the next slide. So there are certain components that are well known. So the first one is the component of a billion characters. So the fundamental group of the knot, the abelinization, that is the integer homology is just Z. And well, you just map Z wherever you want in SL to see, and so you have just a line. Okay, that works for each knot. You have that line there. Very nice to know. Then you have, if you're not a hyperbolic, you have this other curve that contains the lift of the allonomy of the hyperbolic structure. And if your knot is a hyperbolic Montesinos knot, you have something else. It's what is called the Teichmuller component. And this is, okay, so if you consider the quotient of your knot group by the relation that the meridian has ordered to, that's just the fundamental group of the orbifold where the knot has ramification of order two. Well, this guy is a cypher fiber orbifold. It has a base of the orbifold, which is hyperbolic. And it's a two-dimensional orbifold that has a whole space of representation. And these characters are precisely the characters of those representation. Okay. And these in general are pretty large with respect to the number of rational tangles or if you want the number of, minimal number of generators of the fundamental group of the knot. But there are more. Okay. So let me make some remarks here. These extra components that we find can be, well, you can find knots such that the number of this component is arbitrary large. And that's because that's true for two-bridge knots and was proved by Otsukairaile and Sakuma. And basically, the idea is that these representations and these characters that we consider are built starting with representation of two-bridge knots, which are related to these rational tangles that you've seen at the beginning. And also, I said in the statement of the theorem, I said there are components, there is least one component of parabolic characters, meaning that the meridian of the knot is sent to a parabolic and another one where the trace of the meridian is non-constant. And in that case, I said that the dimension of these components is n minus two where n is related to the number of rational tangles of the knot. But in fact, there are components of every dimension from one to n minus two. And as I was telling you, these components are these representations and their associated characters are built using representations of two-bridge knots. And actually, they're all representation of a specific quotient of the knot group. I'm not going to enter into details, but keep in mind that all these components that we've found come from representation of a specific quotient. And finally, of course, I'm not saying that that's the whole, that's everything. There may be other things. There may be symmetries that give you other things. I'm just telling, there's all this stuff, which is more than what was already known before, but there may be more that I'm not even able to access, understand or see. Okay, so this is everything I want to say on this result. And I won't even touch on the proof of this because I already, yeah, gave seminars on that. I don't want to be too boring. So, okay. So the next result is for symmetric knots. So what's a symmetry of a knot? A symmetry of a knot is a defumorphism of finite order of the tree sphere, which leaves the knot invariant, and you want it to be orientation preserving. And in our case, since at the end I'll be interested with these odd primes, whatever, I'm all interested in symmetries of odd prime order, which means that these guys have just two possible behaviors. Either they act freely or they act with a circle of fixed points, and the circle is an unnoticed circle, so a natural circle, a rotation around an axis in S3. So we're interested in these guys, and as usual I'll be interested in hyperbolic knots. Yeah. So this is the result we have. So you take a hyperbolic knot, and you assume that it has a symmetry, and it's periodic of order P. So it means that the symmetry has a fixed point set, non-empty fixed point set. So if you consider the character variety, it must contain a sub-variety which is invariant by the induced addition of this symmetry. And that's the variety we're interested in. And again, variety is algebraic set here. And it happens that this algebraic set contains at least P minus one over two one-dimensional components that are invariant. And these components that sit inside this sub-variety are actually components of the whole variety. They're not like intersection with, I don't know, hypersurfaces or something. They're really components of the whole variety and not just of the invariance of variety. On the other hand, if you consider knots that are free-symmetric, so they emit a symmetry without fixed points, in that case, you can find for each prime P, a knot with a free-symmetry of order P, such that the number of one-dimensional components is bounded by a constant independent of P. So what do you like to summarize this? It means that, theoretically, your character variety should be able to detect if you're not as a periodic symmetry of large order or understand if, given a symmetry, the symmetry that you have is free. Theoretically, because I mean it's much easier to understand by hand, if a knot has a symmetry, then compute the character variety and say anything about it. But anyway, and so this point of view was already exploited by Hilden, Losano and Montesinos, and so they used the existence of periodic symmetries to say something about character varieties of periodic knots because these can be seen on the character variety of two-component link, which is somehow simpler than the original knot, and so the character variety is simpler and hopefully computable. So that was their idea also. Okay, so now what I want to do is try to use these results to say something about character varieties over fields of positive characteristic. Okay, so let's use the first first result. So what do I say here? We have that, you take, I was telling you that in this result about Montesinos knots, these extra representations that we have that give you all these extra components in the character variety, they all come from some specific quotient of the knot group. So this specific quotient of the knot group, let's call it gamma, and now you can consider a further quotient where the meridian of your group, so the generator of the homology if you like, as order p. So this also gives you representation of your original group. And this specific quotient, so I'm considering now the character variety of this specific quotient, and then generically this character variety ramifies at p. What is that? So what does it mean? I mean what's the ramification situation here? Well the dimension of some component of case, some component of the representation of the character variety of this gamma p has generically dimension n minus 2 in characteristic 0, while it has dimension at least, sorry, forget about it, as dimension n minus 3 in characteristic 0, but as dimension at least n minus 2 in characteristic p. And actually, I was telling you the idea is parabolic of order p, and the extra surfaces that should be able to detect now, these are concave spheres for the orbit fold somehow, okay? So what's the idea of the proof here? So you take this, the character variety of this gamma p, and it should be, so what is this? You take, it's the intersection of the character variety, the whole character variety with the hyperplane where the, well a bunch of hyperplanes where the trace of the meridian is cos k pi over p, okay? So that's just saying that the meridian is order p. So generically, since the components of the character variety of gamma have dimension n minus 2, this guy, this extra intersection, and you know that the trace of the meridian is non-constant, the intersection as dimension at most n minus 3, okay? On the other hand, we also know that the parabolic component as dimension n minus 2, and when you reduce everything more p, well you can't see the difference between the character variety of gamma p and the character variety of parabolic, the sub-variety of parabolic characters. So on one hand your expected dimension is at least, is at most n minus 3, on the other hand is at least n minus 2, and that's generically the case. So generically p ramifies, okay? Well this is the first type of example, so you have a jump in dimension, the expected dimension grows, and the other results give you the other type of ramification, okay? So what are we considering here? We're considering a two component link, so you have to think of this two component link in the following way. You have your knot, it's periodic, so there's this symmetry, this rotation which has an axis, and now you take the quotient of your knot and of the axis by the action of your symmetry. What you get is S3, and the knot and this axis give you a two component link, and this component link is L, A is just the image of the axis, and K0 is the image of your knot. And it goes both ways, if you have a link like that where A is a, you know, an unknotted circle, and the linking number between A and K0 is co-prime with P, you just branch, take the cyclic branch cover of the three sphere along A and you lift K0 and gives you a knot, well, okay, which is periodic. So I say that for infinitely many primes P, the knot that you obtain that way, so it's invariant character variety, the part of the sub-variety which is invariant by the action of the symmetry of order P, ramifies at P, ramifies when you take reduction mod P. So try to explain why that's the case. So theorem 2 tells us that you have a periodic knot whose period as order, symmetry as order P, and so the invariant character variety contains at least, sometimes more, P minus 1 over 2, one-dimensional components. And now again, you use the same thing in characteristic P, elements of order P are periodic and vice versa, so if you take, if you consider the character variety in characteristic P, well, there's no difference between components, okay, so maybe I should say something more because otherwise it's not clear. So in this character, these components there come from, should I say now? No, I say that later. Oh, yeah, so what are these components? These components are just the intersection of the character variety of this two-component link with a bunch of hyperplanes and these hyperplanes correspond to characters that are associated to representations in which the meridian of A, meridian of A, so where you run if I, have order P in the representation. So it's just saying trace of the matrix associated to the meridian as is twice cosine of something, okay. So this is where these components come from. But again, as I was saying, telling you in characteristic P, having order P of being parabolic is the same thing. And well, what are parabolic components? These are just, well, you take your character variety of the link L and you just intersect with trace of your meridian equal to two or something like that and that does not depend on P. It's just something, some number, I don't know, 300, whatever, but it's a fixed number of one-dimensional components and if you're at P, you don't ramify, you have to see the same number of one-dimensional components, but that's fixed for almost every P while you expect them to be at least P minus one over two and that grows. So at a certain point you have to, you get ramification and you get, oh, some components have disappeared somewhere. Where? I don't know. I don't even have an idea Well, yes, I have an idea. The idea is that all these hyperplanes get crashed down over a single hyperplane and probably, probably the multiplicity of these guys is different and takes care of the fact that you have more of them, but I'm not sure. That's probably the idea, anyway. So that's what I was saying. Okay, yeah, I think I still have 10 minutes. Okay, so I'll try and say something about the proof of the general result about symmetric knots. Let's see what I want to say. I've already said something. So the idea I was saying is you can see your invariant sub-variety as a sub-variety of some two-component link. Okay, why is that? Well, you take this two-component link that I was explaining before. So it's the image of the knot and of the axis of the symmetry in the quotient by the action of the symmetry. And, of course, you have a short exact sequence here, okay, which is just a sequence of coverings. And the symmetry induces a symmetry of the character variety just by composition. And we're interested in the irreducible components of just the irreducible components characters of these that live inside these invariant sub-variety. Why is that? Okay, that's something I should have said before. So remember, I told you, quotients are very badly behaved in general, and you only want points to be closed. So you consider, well, if you have orbits which are closed fine, otherwise you have to crash down things, etc. Well, the good thing is orbits of irreps are closed. So these are really well behaved, okay? So that's why you go and consider these guys. So the rest of the characters format the risky closed set that we just disregard, because it's just, you know, very complicated. I mean, it's well, on one sense, on one hand it's well understood, on the other creates a lot of problems. So just forget about it. Not in particular that since the olonomy representation is irreducible, well, you have these distinguish, exceptional, curve, whatever you want to call it, sitting there. And it's also sitting there because the symmetry of the knot can be seen, if the knot is hyperbolic, can be seen as a hyperbolic isometry. And so it must be invariant by the action. So it's living there. Okay. Next. So the exterior of the link, of the quotient link, is a sub manifold inside the quotient of the exterior of the knot by the action of the symmetry. And actually, what's the difference? You get the second one from the first one by some singular then filling, okay, with a ramification of order P in the core of your torus. And of course, if you look at the fundamental groups, you pass from the first to the second by quotienting. And this means that each representation of second gives you a representation of the first because the second is a quotient, okay. So the character variety of the quotient of the exterior of the knot sits inside, injects inside the character variety of the link. On the other hand, if you have a character on this quotient, it induces a character on the exterior of the knot just by restriction, just by looking at this short exact sequence, right? You have a representation of the central term and so you get representation of the subgroup. But the idea now is that if you just look at irreducible characters, this map is bijective, okay? So this gives you a birational equivalence between the characters of the character variety of the quotient of the exterior of the knot without the abelian part, okay? And the invariance of variety without the abelian part. And why is that? Well, well, well, well. So assume you have an invariant character, okay? So you got this. And this is assumed that your character is irreducible. So you, what does that mean? You fix a representation here, okay, such that K of row is this one. And if you compose with your symmetry, this is a conjugate of this one. And since the action is nicely basically translated on the fiber here, okay? Well, you can find a matrix, okay, such that this guy here is induced by conjugacy with this matrix. Of course, you have in the matrix is only defined up to sign in principle because the action by conjugacy factors through PSL. But the fact that the order of phi is odd tells you that you can choose a well-determined sign, okay? So you can choose uniquely something, okay, some matrix, some element of SL2 which works like this. And you just map in that exact sequence, you just map, I should say that this exact sequence splits. So you can, you have, you can see Z over P somewhere in the central group. And you just map the generator of this Z over P that you see in that central group to this unique matrix that you've, that you've constructed here, okay? So once you've got that, you see what we are saying here is that if you just look at the invariant character variety and only the components that contain irreducible characters, that's birational equivalent to something that leaves inside the character variety of the link, okay? Okay, yeah, that's what I was saying. And now you know that your excellent curve leaves there, the curves that contain the, the olonomy. But then, well, everything's defined over Z. So, well, you have a Galois group acting somewhere. And, well, there's no, I mean, of course, geometrically, this, the image of the meridian of A is sent to something which has a well-defined angle. It's 2 cos 2 pi over P. But that's just one choice for the, for, of solution of the minimal polynomial for the cos, the cosinus twice the cosinus of 2 pi over P. The others work as well. So you have your Galois group acting there and that gives you the other components, okay? I'm sort of hand waiting here, but that's really the idea behind this. And since everything, all the construction is algebraic, well, all the properties that you see on the, on, on your excellent curve or your distinguished curve, the one that contains the actual olonomy character, well, are preserved when you look at the other components. There's no reason why they should change because everything is done over Z. So, say, Zariski tangent group is the same, whatever you want is the same, and, well, the same, the same properties hold, and in particular, those components that you see there are actual components of the variety and not just of the invariant sub-variety. Yeah, I was just saying two words about the proposition. And this is even more hand-waving, but basically, you can also, as I told you, you can always describe your periodic knot in terms of some quotient two component link. And in fact, that's true even for free periodic knots, only that the link is not at all unique. And for the periodic knots, there's a well-defined choice. In the other case, it's not, but still, you can do that. And what's the difference between the two cases? Well, in one case, you want the meridian of your, you know, around A, which is the axis, to be order P. And that, we've seen, is just intersecting the variety with a bunch of hyperplanes. And so, since the hyperplanes are different, well, you get separate different components. On the other hand, when you have a free symmetry, the dense surgery you are doing along the A, it's a natural dense surgery. It's not a singular dense surgery. Okay? So, it's just you're on the A component, you're just killing some mu to the power P, say mu is a meridian, and lambda to some K, which is prime with P, whatever. And, well, this has many chances to give you some irreducible component instead of a bunch of hyperplanes. And generically, let's say that's the case, you can construct examples where it's well behaved. And so, the number of intersections that you have only depends on L and this, well, this hyper surface, which you expect to be irreducible, and not on P, not on P in some sense. Well, and that's basically the idea. And, yeah, I'm sorry, couple of minutes too late, so thank you for your attention.