 But I'm delighted to be here and to be speaking at this celebration of Maxime. Well, I've known Maxime for many years. And I remember once we were having some mathematical conversation, and it was pretty complicated, and so I was wondering, well, how did you think of this? How do you know this? And Maxime said, oh, of course, but there's gravity. Remember that? Gravity. And so that was it. And so I've learned a lot of mathematics from Maxime over the years, and it's always very inspirational. And some of my talk will be about ideas maybe from physics and how they are perhaps useful in arithmetic. Anyway, so we're talking about K3 surfaces and, well, algebraic K3 surfaces giving algebraic equations and labeled by even positive integers, degree, degree 2 K3's are double covers of P2 ramified in a sextic, a quartic in P3, the most famous one perhaps, intersection of three quadrics, or a comer surface, blow up of the fixed points on the surface when you're quotient by the evolution. x goes to minus x, a comer surface. So here is a page from Klein's protocols from 1876, a student of Klein made models of K3 surfaces, comer surfaces, and nach den Zeichnungen von Professor Klein. So Klein did the technical drawings for that, and then there is here Das Modell here. And Klein himself gave lectures in his own seminar on comer surfaces. Here we are reading about, you know, comer surfaces and the connection to hyperliptic functions. Perhaps a reference to the Abelian surface that we've seen before. Now, so why are we thinking about K3's? Well, first of all there's fascinating geometry and usually one talks about comer, Kaler and Kodaira. And people are wondering why K3, and I have an explanation here in 1882. Again, Klein, his own handwriting, here we see K dot, well, I interpret this as a three. And then there's another one, K dot, I think. In any event, there is a reference to Andre Veil that has something to do with some mountains somewhere, but I don't believe that. In any ways, there are also interesting connections to well-strength theory, mirror symmetry, mathematical physics, perhaps our universe has something to do with K3's. I don't know. There are connections to interesting groups, and just to begin, just to warm up, so the curve counting came up in mirror symmetry, and it's quite fascinating. You write down some generating function, this growing coefficients, rapidly growing coefficients, and this is supposed to count rational curves on a K3 surface. The same as formula by Jals Asloff was established over the years. And that curve count is supposed to be connected with, well, maybe something in the mirror world, and I don't know really about mirrors, but if you have an algebraic K3, which is general, so rank-by-curve one, well, then the mirror thing, at least I can read what it is. There is a one-parameter family of dual K3's, which are of the form, you know, one elliptic curve times another by an evolution, so a comer case. In fact, a quotient of the mirror by some evolution is this comer case. Now, another interesting thing, maybe to note, there is something called the elliptic genus, a function of two complex variables, but when you evaluate it in zero, you get the other characteristic. That's kind of amazing. But then, if you expand this function into something and you see coefficients showing up, then, suddenly, these are dimensions of reducible representations of this monster group there, and the subsequent coefficients are sums of such dimensions. Sorry? Yeah, yeah, okay, much here. Now, I got this paper from Iguchi Ogury in Tashikawa as a submission to the Journal of Experimental Mathematics. I was one of, I'm still one of the editors, and I was really stunned that, you know, something like this can happen, and this was established recently by, you know, actually several groups of people, independent publications, so it's a theorem. And, well, that particular group has something to do with another group and sort of an algebraic geometry. Mukai established that, you know, if you have symplectic automorphisms, then, in fact, well, of a K3 they are embedded into this M23. Now, so this is all, you know, on the geometry side. The main questions for us, we work over non-closed fields, like finite fields, number fields, or functional fields of the curve, and the kind of questions that we want to ask are, well, are there rational curves or rational points? And when they are, well, are they dense, maybe in the risky topology or in some other topologies, perhaps potentially dense after finite extensions of the ground field? So, and what I'd like to do today is to serve with some ideas and techniques in this area of arithmetic geometry, and some of those techniques are inspired, actually, by developments in mirror symmetry and some high-dimensional algebraic geometries. All right. So, first of all, let's start with simple examples. There is this particular diagonal quartic and there was a conjecture of Oilers that there are no non-trivial solutions. Trivial would be, you know, 0, 0 and 1, 1. And that was disproved by Elkies. He produced infinitely many solutions over the rationals. And then there was another conjecture, more recent, by a certain dire, actually, that this particular diagonal quartic has no non-trivial solutions. And here, trivial would be, again, 0, 0 and 1, minus 1, 1, 1. And, well, when it came out, so it was sort of natural to try to disprove it that my postdocs and Göttingen and graduate students ran a computer and found that there is actually a non-trivial solution. And here is a solution. And then, well, they had to do a lot of work because what do you do? I mean, you can't do approximations. It's really integers. So you have to try integers, you know, from here to there from, say, minus 100,000 to minus a million to plus a million and see what happens. And it takes a lot of time. They had to reprogram basic arithmetic for that. In any way, they found no other solutions up to 100 million in size. So we don't know other infinitely many solutions. There are only these two or these, you know, the ones obtained by changing science and so on. So in other words, the arithmetic of K3s, you know, is hard even for computationally. Now, so what kind of geometric invariance will be relevant? So we have a polarized K3 and we have its picar lattice and it sits in, you know, H2 and so this is what we have, some distinguished lattice. And well, the picar rank, the picar group is just, you know, z to the rho and that rho, it can vary from 1 to 20 for fields of characteristics zero but over fp bar, we are not allowed odd numbers. We're only allowed even ones, but we can go up to 22. So this is a basic invariant and so we have an intersection form on picar and we can learn, you know, geometric information from looking at this form. For example, if in this lattice and the picar lattice, you have a class of square zero, well, then we have an elliptic vibration and then we can use it to, you know, prove something arithmetic. Now, how do we compute this picar lattice in practice? So, or even that number, even the rank? So for fp bar, we can do this by simply counting points over finite fields, computing the hash availability function and we can do brute force. We can just plug in, you know, numbers from zero to p and in any event, we have to do this computation because h2 has rank 22. We have to go to sort of the middle and in principle, we have to go up to p to the 11. So p to the 11 and if you have, you know, let's say four variables, so the, you know, straightforward brute force algorithm would require, you know, p to the 11 and then, you know, p to the 33 maybe or, you know, at best maybe, in any event, it's not practical for primes bigger than 10 and in fact, you know, some paper from two years ago said, well, for peak for seven it took two days of computer time to compute just that. So, however, there are, you know, improvements you don't have to do brute force. Kid Liar has developed an algorithm and Anton Shimmerloar explained it in the Burbanki talk several years ago how to rapidly compute points and, well, it's better than what I wrote here. Now, over Q bar, there are some at-hawk methods. If by, you know, miracle, if we have a case three where in one prime we have rank p car two and the other prime over p2 bar we have rank p car two but the discriminants of those don't match modulus squares, well, then you are forced to have p car one because of these compatibility of lattices. So, sometimes you can just find examples where, you know, not one prime of this set and Van Luik has produced infinitely many examples of case threes of the rationals, quartet case threes with rank p car one, geometric rank p car one. Now, so, Andrew Crescent, you know, Brandon Hassett and I, we were thinking about this computability issue of 4k threes and we proved there is an algorithm with a priori bounded running time to compute rank p car and actually all the generators. So, well, and at this stage you can only do it in degree two. So, what's involved? Well, we needed an effective version of Kuga-Sataka correspondence well, and the other effective versions of techniques on algebraic geometry, effective geometric invariance theory, mass of useful effective state conjecture and so on. So, and the main issue, sort of what's the main problem in extending this from degree two to other degrees, for example, to quartets is an effective construction of certain arithmetic quotients, namely quotients of bounded symmetric domains by, you know, discrete groups. Well, it is known by a general theorem of Bailey Borrell that these are quasi-projective algebraic varieties, but what we need is, in fact, equations for these varieties, effective equations. Now, there are some instances where this is known. So, the Mumford's, you know, lectures on theta functions. Mumford does it for a moduli of a billion varieties in all dimensions, and we need something similar for moduli of K3s. And degree two is very special because in this case it so happens that that corresponding, you know, moduli space, well, you can construct it via GIT, and that quotient on the nose is exactly with d mod gamma in this degree. And there are, you know, similar things in higher degrees, but they require some further factorizations, and we are not able to push it through. So, in other words, we need some effective bound on the generators of rings of automorphic functions, if you like, on degrees of these generators. But given that, we can proceed and bound, you know, compute the picarangs. Now, compute, again, so what does it mean as you can imagine, you know, with all the words on the slide, GIT, you know, Kuga-Sataka, AK-3 surface, you know, gets associated with an abelian variety of dimension 2,019. How do you compute anything on an abelian variety of dimension 2,019? But you see in arithmetic geometries, there are results like this. So in Baker's theorem, for example, in Ratzenin's theory, you have an equation, you have a priori bounce, the bounce are very large, but, you know, you plug it into something else and it's becoming useful later. Okay, so once we know, let's say we know the picar group, so we can, you know, try to explore some more geometry, what about rational curves? Well, there is a theorem that every projective K3 has at least one rational curve and, well, very general such surfaces contain infinitely many rational curves. Now, over number fields, number fields are countable and coefficients, you know, all of this is countable and very general could exclude all the surfaces defined over your number field. So it would be important in number theory to know that, well, K3's over number fields have curves, many curves. And I mentioned previously that there is a formula that counts curves or, you know, supposed to be curves. But unfortunately, the formula involves coefficients, you know, there is a dn and those coefficients grow very fast, but the formula does not prove that there are infinitely many curves, unfortunately. And so there are some special K3's for which existence of rational curves can be established, you know, directly and those are elliptic ones and those with infinite automorphisms. But we have to have picars of higher rank. For an elliptic one, we have to represent zero, so we have to have rank at least two and similarly for automorphisms, we have to have higher rank picar and particularly rank picar one, you don't know what to do. And in the Kummer case, over a finite field, there is actually, you know, the following theorem which we found that, you know, was quite striking to us. So if you look at the Kummer surface over a finite field and pick any number of algebraic points, so points over Fp bar that are not on those 16 exceptional curves, then you can find infinitely many geometrically reducible rational curves which are defined over the ground field, over your small finite field, that pass through these points. And this is some kind of version of space-filling curves. So in other words, well, so you produce many, many rational curves that, you know, even pass through prescribed points. So here was this joint theorem, Ms. Bogomolov and Brandon has said that, again, in degree two, in rank picar one, you could prove that the case three, you know, any case three of degree two has infinitely many rational curves. Here I wrote number field, in fact, any field of characteristic zero would work. So what is involved, well, in the original argument of, let's say, Mori Mokai, proving that case threes have rational curves, one starts with a special case three and then tries to move the curves, the rational curves sort of in a family and then specialize and rational curves, specialize to rational curves, so that guarantees the existence of some rational curves in the specialization. But unfortunately in that specialization, of course, you know, many curves could collapse onto one curve and then you don't know that you have infinitely many. And here, we used sort of a mixed characteristic version of that. Well, we looked at a case three over a number field and then reduced various primes and the reduction model of primes was, you know, the special case. We know that an Fp bar rank picar is at least two and rank picar two, we have a chance of finding new rational curves and then we would try to lift them to characteristic zero. And that idea worked. Sorry. But to sort of control the deformation theory and all that's needed, well, first we needed rational curves, you know, a chain of rational curves and then, of course, we had to make sure that the sum of those curves, the class is multiple of the polarizations, the only class that we have at hand in rank picar one so that we are able to lift. And, you know, one of the issues we had was transversality and also those intersections. We can't control that. We don't know what the curves are. But if all the coefficients were one, then we succeeded. And in degree two, again, by accident, we were able to produce chains of rational curves like this, where the coefficients were all coefficients were one. So subsequently, Li and Litke approved a much more general theorem that if you have a case three of odd picar rank, then there are infinitely many rational curves. And again, they used reduction mod p, but they had another idea, well, that used, in fact, conservation spaces, rigid stable maps, and, well, combined with what we did prior to that, they were able to produce infinitely many rational curves. Now, what is the problem? What is left? What's left is rank picar two. You see, not every case three of rank picar two has this elliptic vibrational or internet automorphism. So there are some cases where the problem is still open. And the main issue is, well, how do we know the rank picar jumps when we reduce module of various primes? So there's nothing that, you know, guarantees this jumping behavior. If we had that for infinitely many primes, then we could, you know, start the previous machine. So let's talk a little bit about the jumping of picar ranks of case three. Well, so we already know that if you rank picar one and reduce mod p and look at fp bar, we have to have at least two. So there is some number here. It only depends on the case three that you have to add automatically. Well, that function eta, it's, you know, if you have complex multiplication or real multiplication, there is something that you can, you know, specify. And jumping would mean that you have strict inequality here. Now, equality does actually happen for infinitely many primes. And in fact, after some finite extension of the ground field, even for density one of primes. So the jumping behavior is going to be a rare event. But nevertheless, you want to know, does it happen? Is the set infinite, for example? So, and again, the main concern for us would be rank picar two. Well, not knowing what to do in rank picar two, let's look at the Kummer case. The rank picar is at least 17. So the rank picar of a Kummer surface is, well, the 16 exceptional curves that we had. And then, of course, the rank picar of the bilion surface, well, it could be one. But in fact, if the bilion surface is a product of two elliptic curves, well, then you're guaranteed to have at least 18. And if those curves are esogenous, then it's at least 19. Well, and if in addition, they have complex multiplication, well, then you're at least 20. And so now we can look at the jumping behavior in this case. All right, so when are we jumping? Well, if you have esogenous curves, then we are jumping even only if LP is a prime of super singular reduction, actually. Well, maybe. And so does that happen? And the theorem of Elkes from 87 says there are infinitely many such primes. So there is a recent 2013 theorem of François Charles, I mean, I heard this talk in New York just a month ago or so, that even when the curves are not esogenous, the set of jumping primes is infinite. There is no paper yet, but it's totally convincing. Now, if the bilion surface is absolutely simple on the other hand, then we don't know. Well, not knowing this, I asked my student at NYU to compute those ranks and compute the jumping behavior. So he implemented this algorithm of kid liars that developed for epileptic curves in the case of quartic surfaces that allowed him to go from primes less than 10 to primes less than 100,000, which is significant enough to actually produce some statistically significant data. So what are we seeing here? Well, in the first slide, we're seeing some examples, example one, example two of case three, so geometric picard number one, and so we are guaranteed to have two, but the question is, do we have four? And we're counting primes up to x and dividing by, you know, x log x, the number of primes up to x, and this is what we're seeing, essentially one over square root of b. Well, it's not quite square root of, it's four four here and four two maybe here, but well, those are the data and that's the line that's supposed to give the square root. Now, if you're on geometric crank picard two, on the other hand, those examples, well, we're supposed to look at primes jumping to four, and we are seeing the number of those primes, and it's essentially half the primes. Well, it's not really half. Now, you could think that, well, maybe it's, you know, complex multiplication or something like this and maybe, you know, field splitting or non-splitic of primes and some quadratic extensions, so half the primes split, half the primes don't split, but it's not the case, actually, and as we can see, there is no oscillation around one half. We are going to one half sort of from above, you know, really getting one half. So it would be nice to have some, some heuristic explanation in all cases of, you know, what is the asymptotic of jumping primes for k-3 surfaces, and they're probably involved, you know, maybe the satiate group, you know, some combination of satiate and length rotor should give it. So here are some other examples. For example, rank 16, and again, you are seeing, you know, 0.56, if you like, and here is rank 17, and then we are seeing, you know, B to the 4-2 and B to the 4-6. So that's what the computer tells us here. Now, so we understand threshold curves and, you know, maybe the jumping behavior of primes, let's think about rational points. And the question is, we have a k-3 surface over some non-close fields. Are rational points dense after some finite extension of the ground field? And we looked at the two cases, the number theoretic setup, we are, you know, working over the rationals and also the geometric setup, we are working over the function field of a curve. And so here is sort of the situation that if you have a general pencil of k-3 surfaces over P1, then actually rational points are the risky dense. But here, general really means general, sort of complement to countable subset of things. And so at this stage, we are unable to overcome this. And over number fields, it's sort of the opposite case. If you look at special ones, you know, elliptic k-3s, or those with infinite automorphisms, then, again, we have, you know, potentially dense sets of rational points. And so let me just very briefly say, what are the problems? So in the first case, we need to know that the jumping primes, if you like, they're not too jumping. We need to ensure that rank picard is exactly two for infinitely many points on the base, for infinitely many primes, but not too much more. Now, if you again think about the modular space situation, so we have family of k-3s, it's like a curve in the modular space of k-3s, an algebraic curve, right, P1, all right? And so what we need to make sure is that this curve does not intersect several Brill-Neuter divisors. It's supposed to intersect just one, but not two simultaneously. So now, when you look at it like this and, you know, you pull back the curve, before the portion by gamma, this really looks like some version of unreward conjecture maybe, and perhaps techniques developed in connection with that are applicable in this context as well. So we want to say that we are passing through maybe two rational subspaces, but not three upstairs, so maybe that will work. So this is a problem over a function field, but over a number field, you simply have no ideas how to approach k-3s, rank picard is one, very little geometry to play with. And so then we thought, well, maybe, you know, okay, so here let me sketch the proof of a theorem for special k-3s, elliptic k-3s. So the statement is that the points are dense, and how did this work? Well, we have an elliptic k-3, and you already know, or let's say we proved that we have infinitely many rational curves. Well, some of those curves will be multi-sections. Not all can sit in singular fibers, there are infinitely many curves. Well, rational curves do have infinitely many rational points, and they will intersect infinitely many fibers. I mean, yeah, well, and then these points in those fibers, they, well, maybe they are of infinite order in the fibers, and so this way you can generate more points. And it works, but you have to make sure that what we are looking at are not, so to speak, torsion, that what the induces in the fibers are not torsion points. Well, this is some condition that needs to be checked. It's subtle, but it works. Dense after a finite extension of the ground field. So there is some finite extension of the rational, so that over that field the points are dense. Of course, yeah. So, now, we didn't know what to do in rank pick R1, but we thought, well, maybe we can look at zero cycles of some degree. In other words, look at points and some extensions. We fixed the degree of the extension. Let's say quadratic extensions are the points dense there. Well, that means that we are looking at rational points on Hilbert's schemes. Well, and so we have a theorem that if we have a polarized k3 of any degree, then you can, well, I don't know whether I was correct and here, then, yes, on some symmetric or some Hilbert scheme, rational points will be potentially dense. So, for example, if you have a k3 of the degree 2 times square, then already on Hilb 2, rational points will be potentially dense. And so, what's used here? Well, we use that idea of Yau Zaslo, that was then also developed by Baville. We use this Abelian Fibration. I mean, I don't know how we were aware of this paper, but so on Abelian Fibration was sort of an analog of an elliptic Fibration. So what we need here, of course, we need multi-sections. Now, this time around, we don't find rational surfaces that could surface multi-sections, but there are nevertheless surfaces with many rational points on them, potentially. And we can make these surfaces from, in fact, elliptic curves, genus 1 curves on the underlying k3 surface and then propagate the points that you obtain in this way. But this is really all we had. And well, so for many years we were working sort of on other things, but so the big success of drives of algebraic geometry, if you like, let us to think about applying some of the techniques at least in the setup of k3 surfaces. All right, so there is this notion of derived equivalent k3 surfaces and well, there are many definitions. So one would be the orthogonal complement of pika groups that transcendental lattices are isomorphic as port structures. Well, the equivalent is there is some definition in this language in the language of derived categories. There is some object in the bounded derived category of the product, so that the corresponding function here is an equivalence of triangulated categories. So we want to say that it actually works over any field. You can write down the definition of derived equivalent k3 over any field, and there is a paper of Yao or Guiza and others where you can count the number of derived equivalent k3s, let's say in pika 1. So let's look at some examples. You see in higher rank pika when the pika lattice is big the transcendental lattice is small because they have to fit into d22 and therefore in higher rank pika derived equivalence tends to imply isomorphism and so we don't actually get anything new, it's sort of the same. So for example if rank pika is at least 12 or if the k3 has an elliptic vibration visor section or if rank pika is at least 3 and the discriminant of the pika is cyclic. Now so let me give you some examples here of, I think that's the lowest degree example of derived equivalent but not isomorphic k3s. So we look at these lattices two lattices and we look at k3 surfaces where the pika groups have, I mean are these lattices and split means over the ground field there is no gallo action anymore on the lattice. This is what we have over the ground field and so then the hope is maybe there is some relation between arithmetic properties of these k3s so let's try to explore that. So, well first of all both have zero cycles of degree 1 over any field. The first one actually always has rational points again over any field and that happens because we have smooth rational curves on that surface which intersect with which of odd degree cycles intersect hyperpolarization of odd degree and therefore a rational curve with an odd degree cycle has to be a p1. So not only do we have points we have actually two p1s full of points infinitely many points on these p1s but the other k3 in fact has infinite automorphisms over some extension of the ground field and so it has a potentially dense set of rational points. What we don't know however is that first of all the first one which has infinitely many rational points has potential density that is, you know, the risky density over some finite extension of the ground field we don't know that. And we do not know that the second one which has potential density has actually points over the ground field even though it has a zero cycle of degree 1 over the ground field. So this kind of a curious situation and this would be sort of the first maybe test case of our hope that well maybe we can bootstrap some geometry from one case to another case. So what I'm describing here is joint work with Brandon Hassett from this year and the paper is on my web page. So alright, so assume we have derived equivalent k3s over a field of characteristic not equal to 2. Well then their PICAR groups are actually stably isomorphic as the Galois, K-Bauer, K-Modules. They're not isomorphic, I mean I've just shown you two rank 2 PICAR which are not isomorphic but if you add Galois and where you think there are, there are. And the other thing of relevance in arithmetic geometry Brauer groups, so you may have heard of Braum-Anian abstractions Brauer groups, so PICAR and Brauer groups as invariants that play role in arithmetic. Well the Brauer groups are the same in both cases. So again we can hope to relate the arithmetic. And then there are results that if you have a 0-cycle of degree 1 on one of the k3s you also have a 0-cycle of degree 1 on the other k3. And actually it's more general if you have 0-cycle of any degree of this degree then you'll have 0-cycle of that degree on the other side as well. So the index is preserved under the derived equivalence if you like. All right, so what do we know about finite fields? Well there is a result from 2011 that if you have two derived equivalent k3s over a finite field then they have the same number of points. In particular if one has points then the other has points. Now what do we know about the real numbers? Well if you have derived equivalent k3s over real numbers then actually those manifolds are defiomorphic and in particular one has points the other has points. So the natural question is what happens over the pietics and well finally what happens over q. So the hope is that over pietics you have the same property. Then the next hope is well therefore the deliq points are sort of the same. The deliq points are the same. You can look at the Brouwer kernel and the Brouwer groups are the same. The computation of local invariants you can also relate them. Then the Brouwer kernels would be the same and then there is a conjecture of Skorbogatov that let's say rational points are dense in the Brouwer kernel. In any event you sometimes expect that the Brouwer kernel does capture a lot of arithmetic. So the hope therefore would be that over the rational as well if one has points of the rational the other has points of the rational but it's just a hope, not even a conjecture. So now what about the pietic case? So here are the theorems. Assume that we are looking at derived equivalent k3s over pietic field of residue characteristics 7 or for tactical reasons and assume that both admit ADE reduction. So we are now looking at mixed characteristic. Then if one has points then the other has points. Now just a comment you see having potentially good reduction turns out to be a derived invariant. So there is a recent paper by Matsumoto from 2014, literally a month ago where he looks at this property of derived equivalents but for example we don't know whether or not having an ADE reduction is a derived invariant even that which would be a natural thing to suppose. Well not knowing what to do with the pietic for now so let's focus on the geometric case. So we have a family of k3s over a disk and well we don't always have a section we don't always have rational points over this field. So here is an example of a family without without a section so then you would like to ask does having a section you know is with a derived invariant you know if you have a derived equivalent k3 one has a section should the other have a section. Yes so k3 over a functional field if you are if you are derived over that field over that field so I will talk about what happens with derived equivalents integrally but now so how do we think about this well we know that after some finite base change there is something called the Kulikov model Weierstrass model a nice model of a k3 the generation which could be either a k3 surface or this you know type 2 I like to think of this as a towel you know you have rational surfaces at the ends and then you know other surfaces in between or soccer ball I guess union of rational surfaces combinatorial triangulation of a sphere so there are these three cases and you know they are well studied and so here there is sort of a technical lemma assume that we have x and y derived equivalent over the field and suppose x admits a Kulikov model without base change right over that field well then why also admits a Kulikov model and both sets are empty so you are guaranteed to have sections of course now the non-trivial part is well there is this base change involved you know how do you get the Kulikov model to begin with and so here I just want to discuss one case of you know one case that happens so suppose we have some cyclic group that a K3 surface via some quotient another cyclic group H and that group is in the automorphisms of a K3 right so then we can look at sort of an either trivial family you know take a product on the first on the on delta 2 this group G acts simply via n's roots of 1 and on here it acts with subgroup of the automorphisms and then we get sort of a twisted family and we can look at what happens here and well as you can imagine you know you can do this in the draft equivalent context and you will see the group actions you know here and there and so here is a lemma that well we know when the quotient here when this family has a section it has a section if and only if the H action on X0 has a fixed point alright so now we have to analyze those fixed points now the question then is if we have you know a K3 surface and some finite cyclic group acting on it and another one and suppose the action is compatible with an automorphism of hot structures thus having a fixed point on X0 you know imply having a fixed point on Y0 right completely down to earth alright so now what kind of cyclic automorphisms do we have on K3 so the Euler function of that has to be less equal than 20 and it turns out that all such n arise now and then you can look at those that preserve the two form and you know the symplectic ones and in fact you know the groups tend to be very small so 1 lab to 8 I mean you don't get very many symplectic ones and a lot of the fun is coming from the non-symplectic actions and the mixture of those so I thought that I should give you a table of all numbers with Euler function less than 20 as you can see you can have quite interesting automorphisms in all 66 and 50 and the way you should read this table is well if the automorphism is large then those happen to be unique the K3 carrying such large automorphisms happen to be unique in high rank Picard as well and there is an enormous literature classifying these particular K3's and so in particular you know 66 50, 44, 33 essentially 1 K3 and so draft equivalence is not really that interesting for those and then when the action is very very small you know symplectic non-symplectic it's sort of also well understood you can analyze the fixed points and so most of the fun is somewhere in the middle of this table and then again huge collection of papers and classifications and tables you can have this fixed point locus and you know some of the papers are recent, Artibani, Kyum, Kondo, Masheed, Nikola, Noguz, Asarti Taki so many people looked at this but there is no complete, I mean no simple answer that you can take off the shelf and therefore I mean this is still in progress hopefully some combination of these papers, you know intersection, you know the Joint Union I don't know, I mean some combination because there are actually some contradiction some I mean there are papers that are inconsistent but still there is hope so now, so here is a slide of you know things to do of course we would like to have some mixed characteristic version of Kulikov models, I mean it would be nice to know what happens if you work over the pedics and you look at you know the special fiber what happens how can you relate you know this drive to Kulikov and K3s and naturally we already brought it up so there is this paper by Bridgeland Makheoshe that the Furia Mukai transform of K3 families over curves it specializes and again it would be nice to have a mixed characteristic version of that for our applications and it would also be important to explore what happens in the central fiber I mean they have just results that says you know it specializes but you know what does it really mean for us and well maybe in sort of conclusion so we know very little about rational points in general rank pick R1 K3s over non-closed fields over such fields as C double bracket T or the pedics and these are of course you know much more difficult functional fields or finite fields or the rationals and that we you know now hope that techniques perhaps in Furia Mukai transform the draft algebraic geometry may shed some light on these arithmetic problems thank you it's about practical part when you layer algorithm which is any variety equations in number variables in linear prime numbers yeah so what's implemented here is essentially linear time for these K3s and yeah you can do invaded projective spaces hyper surfaces, invaded projective spaces it's really practical yes I mean it still involves large matrices I think here it's 200 times 200 matrices that is processing but it's okay I mean many of them so well if you have your favorite K3 we're happy to compute I mean I forgot to say that that slide cost 120,000 hours of CPU time yeah yeah no no no CPU time is free