 So first of all, thank you so much to the organizers for inviting me to speak. It's an honor to be speaking at the seminar. So I'm going to be talking about canonical height and the Andrea Oat Conjecture, and everything I talk about is joint work with Jonathan Pila and Jacob Zimmerman, with an appendix by Elliot Inno and Michael Groshnik. So the Andrea Oat Conjecture is about Shimura varieties. So before saying anything else, let me first give some sort of background to what Shimura varieties are. So the most basic objects that motivate the definition of Shimura varieties are complex elliptic curves and the modular curves. So a complex elliptic curve is just something of the form C mod lambda, where lambda contained in C is a rank 2 lattice. Lattice just means that it's a discrete subset of C, and rank 2 means it's a subgroup that's isomorphic abstractly to z squared. So such an object C mod lambda is it's clearly a group, and it's an abelian group. It's also a complex manifold, and it's a compact complex manifold. So it's an abelian, compact, complexly group. Not just is it a complex manifold, it's actually cut out by algebraic equations of the form y squared equals x cubed plus a x plus b, where a and b are suitable complex numbers. So given such objects, perhaps something that you might want to do is parameterize such objects. And in order to do that, you have to answer the following question. Given two lattices, when are the associated elliptic curves isomorphic? So the question is, can you say about these two lattices so that the corresponding elliptic curves would be isomorphic? That's not too hard to see that any map between elliptic curves is going to be induced, just using the theory of covering spaces, by a map from C to C. And as it turns out, any map, little t from e1 to e2 will be induced by a capital D that's linear. And of course, you have to have that t of lambda 1 has to be contained in lambda 2. And the map will be an isomorphism if t of lambda 1 is equal to lambda 2. Now, any such C linear map from a one-dimensional complex vector space itself has to be of the form scaling by some complex number. So in order to parameterize elliptic curves, the set of isomorphism classes of complex elliptic curves are in projection with, instead of all, rank two sub lattices inside the complex numbers up to scaling. So what you want to do is find the parameterization of complex elliptic curves where all you need to do is classify rank two lattices inside C up to scaling. All right? So after scaling, you can assume that your rank two lattice generated has a z basis of the form 1, tau. So given any two basis vectors, you can always scale so that one of them becomes 1. And by replacing tau by minus tau, you get the same lattice. And so you can assume that tau is in the upper half plane, which is just a set of all complex numbers with positive imaginary part. All right? And given two lattices, lambda equals 1, tau, and lambda prime equals 1, tau prime, it ends up being dissolving a system of linear equations to see that there exists some scalar alpha with alpha lambda equals lambda prime. If and only if, there exists a 2 by 2 matrix in SL2 z, such that your tau prime is a tau plus b over c tau plus b. So in order to see this, all you to do is solve a bunch of linear equations. And well, now you see that the set of complex elliptic curves is in bijection with equivalence classes of elements in h where two elements are to be equivalent if they correspond to each other by a fractional linear transformation with coefficients in SL2 z. So let me spell that out a little bit more. You actually have an action of SL2 r on the upper half plane, just the same action by fractional linear transformations. In fact, h has a natural metric. And this action is by holomorphic isometrics. The set of elliptic curves, complex elliptic curves up to isomorphism, is in bijection with the portion of the upper half plane by SL2 of z. Well, you could actually draw a fundamental domain for this action. And it's Gauss computed this. It looks something like that. So given all of this, let's define the complex modular curve, it's called a1 of c to be h mod SL2 of z. Now, here are some extremely wonderful facts about the modular curve. So a priori, this is just a complex analytic space. But just as your complex torus could be realized as the solutions to algebraic equations, in the polynomial equations, the modular curve is the set of complex points of an algebraic curve. In fact, this algebraic curve, this is called a1. This algebraic curve is defined, the equation is defined in the algebraic curve actually, actually q rational. And if you would interpret this appropriately, a1 is actually defined over z. And we've seen that the complex points of the modular curves bijectively correspond to elliptic curves over c, the same modular interpretation holds over any ring. So up to starchy issues, in other words, up to starchy issues, which I'm not going to address, if you're given any ring r, then the set of isomorphism classes of elliptic curves over r are in bijection with r value points of this algebraic curve. And it makes sense to talk about r value points of this algebraic curve because this algebraic curve is defined over z. All right, so this is a brief summary of modular curves and the objects that it parameterizes. Let me now briefly talk about special points of modular curves on the modular curve. So a point on your modular curve is said to be special. If, well, look at the associated elliptic curve corresponding to that point, look at its endomorphism ring. And we've already seen that your elliptic curve is a group. It's an abelian group. So any integer is going to give you a map from your elliptic curve to elliptic curve, namely take any point, let's say, y on your elliptic curve, take any number n, and just look at y plus, y plus, y plus, y plus, y n times. That tells you that the ring of integers is a subring of the endomorphism ring of elliptic curve. Well, suppose the elliptic curve has more endomorphisms, then we call that elliptic curve a CN curve and we call the point parameterizing that elliptic curve special. So now, when is a point special? And let's say, complex value point, when is it going to be special? Well, there's a nice interpretation in terms of the upper half plane. So we know that there's this canonical map from the upper half plane to the complex points of your modular curve, and that we define the complex modular curve in terms of the upper half plane. So let tau be any pre-image of x in the upper half plane. It's going to be special if the endomorphism ring of elliptic curve is bigger than just z. And lattice theoretically, this is going to be the case when there exists some complex number, which is not an integer such that if you were to scale your lattice by that complex number, you end up with sub lattice of that lattice. And again, figuring this out corresponds to the solving linear equations. And you see that this happens precisely when tau satisfies a quadratic equation of the form a tau squared plus beta tau plus c equal to 0, where a, b, and c are integers with GCD1. And associated with this, we define the discriminant of your special point to be negative d, which is b squared minus 4ac. It's not too hard to show in Gauss, in fact, prove this, that there's only finitely many special points with fixed discriminant. Well, there is countably many integers b, and then there's finitely many special points with discriminant negative d. And so there's only countably many special points. What else ends up being true is that if you were to take a special point and think of it as a point of the modular curve, well, it's actually going to be contained in q bar, not just in c. So every special point actually descends to q bar. So these are some facts about special points that will be useful in defining special points and such like for arbitrary Shimura varieties. So having talked about elliptic curves and the modular curve, let me now talk about Shimura varieties in generality, and I'm going to define them by analogy and by using examples. So what were the key players in our definition of the modular curve? Well, after we set things up, we saw that an extremely important role was played by the upper half plane. So the upper half plane, as I said, has a hyperbolic metric, also a complex magnetic space, and it's also a bounded space. So it doesn't seem very bounded, but it's actually bihologomorphic to the open unit disk. So it is actually bounded. And the high dimensional generalizations of the upper half plane are so-called Hermitian symmetric domains. We had this rather large V group that acted on H by holomorphic isometries. And analogous to this, there is going to be a lead group of holomorphic isometries, G of r, that acts transitively on X. Now, having SL2R and the upper half plane wasn't enough to define the modular curve. We needed the subgroup SL2Z, or actually the finite index congruent subgroups of SL2Z are enough. And then we did need this so-called arithmetic subgroup SL2Z sitting inside SL2R in order to define a modular curve. And similarly, the analog of this is going to be an arithmetic subgroup sitting inside G of r, which therefore acts an X. And just like in this setting, in the setting of the modular curve, we could choose our arithmetic subgroup to be the Z part of a group defined over Z. You can do similar things. And in fact, gamma will behave like the Z part of a group defined over Z. We defined our modular curve, the complex modular curve, just to be the quotient of H by SL2Z. And similarly, we defined the Shimura variety to be X mod gamma, at least its connected components to be X mod gamma. Just like the modular curve, which is a complex analytic space is actually algebraic, and it's defined over Q. The same is also true for Shimura varieties. But now it's going to be defined over an explicitly defined extension of Q. And showing that objects like X mod gamma and algebraic was done by Bailey and Perel. And it's work of Deline, as well as Milne and Borovai, that tells you that your Shimura varieties actually descend to explicitly defined number fields, as opposed to just being complex algebraic varieties. We saw that the modular curve parametrized isomorphism classes of elliptic curves. Now, in general, Shimura varieties aren't really modular spaces of varieties, but they do carry polarized variations of hot structures. They are polarized variations of hot structures. Namely, points of Shimura variety parameterized give rise to hot structures. And as you vary along the point of Shimura variety, this hot structure will vary in an appropriate sense. And the notion of special points corresponded to CML epic curves. And in the setting of Shimura varieties, the hot structures with a commutative monfortate group correspond to special points. In the case of the modular curve, we saw that, or because I told you that special points are defined over Q bar and actually are bends in modular. They dance in the strongest possible sense, namely not just at the Zalski dance, which is just the same as the infinitude in the case of curves. But if you were to take any complex point on your modular curve and take a small disc around it, then the set of CM points will be dense in the analytic topology in that disc. So in other words, special points are also analytically dense in the modular curve. The same is good, exactly the same good to be true for arbitration model varieties. Again, special points are defined over Q bar and are dense in S. So in other words, you should just be thinking of Shimura varieties as high dimensional analogs of modular curves. They do look like there is going to be some object analogous to the upper half plane. There's going to be some subgroup analogous to SL2Z and the quotient is going to be your Shimura variety and your Shimura variety will descend to a number field. It carries a lot of extra data, though not necessarily families of varieties. And special points are defined over Q bar and are dense in your Shimura variety. So before talking about the distribution of special points, let me just give you some examples of Shimura varieties. So you have sort of a picture to have in your head that goes beyond just a modular curve. So there'll be some of that. The modular space of elliptic curves is a modular curve and a Shimura variety. Now the high dimensional generalizations of an elliptic curve are Abelian varieties and the modular space G-dimensional polarize Abelian varieties. Let's call this A-G, A-Z Shimura variety. And just like the modular curve was defined by the group SL2 or GL2, A-G is defined by the symplectic group on a two G-dimensional vector space. And similar to the case of elliptic curves, special points are in projection with C-M Abelian varieties. Then if you were to take an orthogonal group with an appropriate real signature, you get a Shimura variety that parameterizes polarized K3 surfaces. Special points should be thought of Shimura varieties in their own right. In fact, they're zero-dimensional Shimura varieties. Strictly speaking, connected components of zero-dimensional Shimura varieties, they are associated to Torai. Special points, as I said earlier, are all defined over Q-bar. Now if Shimura varieties don't just live in isolation, they actually map to each other. In a very precise sense, whenever you look at groups defining Shimura varieties and you've got maps between these groups that's in an appropriate sense, then the associated Shimura varieties will also map to each other. And whenever your Shimura variety has a modular interpretation, these maps also agree with this modular interpretation. So for instance, if you were to look at A1, the modular space of elliptic curves, the modular curve, and if you were to look at A2, the modular space of Abelian surfaces is a natural map from A1 to A2 induced by maps between the groups defining A1 and the group defining A2. And this map has a very natural modular interpretation. So if you were to take a point in A1, that's going to correspond to an elliptic curve. Well, given an elliptic curve, how do you get A2-dimensional Abelian variety? You just look at the elliptic curve cross itself. This is an Abelian surface, it's two-dimensional Abelian variety. Let's say it corresponds to the point Y in A2, well, then just map the point X to the point Y. And such a map is actually defined in terms of the groups defining A1 and A2. And as is clear in this example, namely if you would take a CML elliptic curve, then E-crossy will be a CML Abelian surface. Whenever you have a map between Shimura varieties, you always get that special points, map to special points. And finally, let me say that while many, in fact, maybe even most Shimura varieties have modular interpretations in terms of Abelian varieties, not all Shimura varieties carry families as varieties. So not all of them have known modular interpretations. And those that don't are called exceptional Shimura varieties, largely because many Shimura varieties that don't have modular interpretations are defined by exceptional groups, not classical groups. And those that do are said to be of Abelian type, because roughly speaking, they parameterize Abelian varieties, though there's a small caveat to that statement as well. So let me talk a bit about the distribution of special points, which is basically the Andreol conjecture. So here's a question that Oat and Andreo note they posed. The question is, what sub varieties, a Shimura variety S, have a Zalski dense set of special points? Well, first of all, S itself, because I've already told you that special points are dense in the ambient Shimura variety. Now, if you were to take any Shimura variety and map it to a different Shimura variety, we've seen that special points map to special points. And because special points of S1 are dense in S1 and special points map to special points, the image of S1 also has a Zalski dense set of special points. And getting the technical here, but then irreducible components of heck it translates, of images of such maps also have a Zalski dense set of special points. And we'll define all such sub varieties, a special sub variety. Just for the purpose of intuition, just think of this case. So these are sub varieties that obviously have a dense of special points. And the conjecture of Andreo note is that these are the only ones. So let Z contained in S have a sub variety, having a dense set of special points, then Z is a special sub variety. And there's been a massive amount of past work, some of which, and this is an extremely non-exhaustive list, but some of which includes work of Andre who proved the Andreo conjecture for a product of modular curves. And Alex Ovan, who proved it for a product of arbitrarily many modular curves, but contingent on the Riemann hypothesis. Then Klingler in joint work with, I think Ulmo and F5 proved the Andreo conjecture in full generality, but assuming the Riemann hypothesis. Then Pila and Zimmerman both have a lot of past work on the Andreo conjecture. And of course, there's a lot of other people who think about this. Let me just say a little bit about the case of AG. Sorry. Yes. You mean the Zariski closure of Z or in the statement of conjecture? Oh, sorry. So Z is a sub-variety. Oh, you mean? Z sub-variety that contains a Zariski then set of special points. Then we say that Z, then the conjecture is that Z is set of special points. So you can also say that the Zariski closure obviously if Z is set of special points. Absolutely, absolutely. An equivalent formulation is exactly what you said. You could just start out with any set of special parts inside. The Shimura variety take the Zariski closure and then the conjecture would be that that is a special sub-variety or a finite union of special sub-varities. That's exactly right. Yes, thank you. Equivalent Zariski closure of special points is a special sub-variety. So let's just say special. So Jekyll proved the agreed conjecture for AG and therefore as well as any Shimura variety of abelian type and he proved it by using, by crucially using work of Maser and Zanier on isogenic estimates or on like isogenes, beautiful Kierremann isogenes and faultings are the same abelian varieties. And so using this, using their beautiful work in a very crucial way, Jekyll proved that it suffices to prove that same abelian varieties, small height, small in a quantifiable sense that I won't get into. And then used the average version of Colmez conjecture, of Colmez's conjecture which was proved by Henrietta, Goreng, Howard and Mother Pussy, as well as by Yuan and Zhang to prove that these height bounds actually hold. Oh yes, and I should say that another crucial ingredient in all of this is the Pila Zanier method. So well there are still some Shimura varieties that are not abelian type, namely the exceptional ones. And there wasn't, there was no theory of isogenes and heights for isogenes analogous to the beautiful theorem proven Maser and Zanier. However, Benyaminie, Schmidt, and Yafive, building on prior work of Benyaminie proved the following theorem. So let S be a Shimura variety associated to an adjoin group, an adjoin deductive group. In fact, an adjoin semi-simple group, G, and let T be a rational tourist mapping to G in a way that's compatible with Shimura structure. So suppose this induces a map ST to our Shimura variety where ST is a view dimensional Shimura variety associated with the tourist T. Oh, I'm so sorry. I should have said this was Maser Wussholz, I saw it on the theorem. Apologies. Thanks for the comment. Well, so given a Shimura variety associated with an adjoin group G, and if you have a special point induced by a map from a tourist T to G, then suppose you can prove that for a way height on a Shimura variety, the height of your special point is small, similar to, small in a similar sense as in this case, then the answer your conjecture is true. And what we prove is that special points as above in adjoined Shimura varieties, in fact, an adjoined exception Shimura varieties have bounded height. And so the answer your conjecture follows. All right, so let me now the last seven or 10 minutes dive into our strategy. So let G be an adjoined group and let, and suppose it induces a Shimura variety, it's called S. Now, let V be a G representation defined over Q. So then what we do is that we define a height on S using this data, but we find the following way. Suppose you have, or rather our definition needs, our definition has a following crucial consequence. Now, suppose you have a sub Shimura variety, of your Shimura variety, induced by a map of reductive groups G prime to G. Now, given this, V is also going to be a representation of G prime, just because G prime maps are G. And so you get a height on S. So our definition gives us a height on S prime as well. So then the height on S prime is compatible with the height on S. Now, in other words, you could just ignore the fact that S existed, V is a representation of G prime. So our construction gives a height on S prime. But on the other hand, now if you remember the fact that S prime maps to S and we have a height on S, that induces a height on S prime, the two different heights are compatible with each other. That's what our construction gives us. Then this height is well-defined, whenever G is GSP. So you don't actually need a group to be adjoined, it also covers other cases. In fact, it covers the case of, for any Shimura variety pavilion type. And in the case of GSP, this agrees with the faulting side. And finally, so what this means is we've defined a height that behaves well after pullback and that agrees with the faulting side on GSP. And finally, heights on special points in the original Shimura variety S can be reduced to heights on so-called partial CM types. Delhi has a beautiful trick in the second paper of Shimura varieties. It reduces bounding heights, partial CM types to bounding heights on CM abelian varieties. But this is already done using the average cold mesh conjecture while proving the unreal conjecture for AG. So this is the broad outline of what we do. Now, what I'll do is I'll briefly describe how we define these heights. And then I'll end my talk with that. So let G, S and V be as above. So work of Deline and Milne associates to this data. A lot of associates to this, a lot of data on your Shimura variety. First of all, for every prime L, you get an eta local system on your Shimura variety that defends to the number field of definition of Shimura variety. Using the classical Riemann-Hilbert correspondence, you also get a vector bundle with connection, which is filtered and that satisfies Griffith transversality. And all of this also descends to the number field. And we'll define our height by metharizing the determinants of graded pieces of this vector bundle with connection, of this filter vector bundle with connection. Now, because everything is defined over a number field and everything is algebraic, all of this data, not necessarily the local systems, but the data of the vector bundle connection and Shimura variety will spread out over the ring of integers with some large enough number inverted. Now, the Piadi-Riemann-Hilbert correspondence by Liu and Zhu and by Diao Lan, Liu Zhu in the logarithmic case, gives you a way of starting off just with this Piadi-Kita local system defined over cube, defined over a non-archimedian local field. It spits out, it starts off with this data and it spits out a vector bundle with connection to be defined over the Bayesian-Shimura variety to your local field. And they prove that the Piadi-Riemann-Hilbert correspondence applied to your local system is actually canonically isomorphic and the canonicity is important. To the vector bundle with connection, base changed to EP. To this vector bundle with connection, base changed to EP. So they prove a compatibility of their Piadi-Riemann-Hilbert correspondence with the normal Piadi-Riemann-Hilbert correspondence. Sorry, the normal Riemann-Hilbert correspondence, the complex one. So we use this Piadi-Riemann-Hilbert correspondence and this construction to metterize the graded pieces of our vector bundle with connection. And because our construction is Piadi-Chor's theoretic and this Riemann-Hilbert correspondence behaves well under restriction, our metrics behave well under pullback. And so at special points, everything is compatible with the faulting side. And when I say compatible faulting side, I mean, like once you run through Deline's trick, you can get compatibility with the faulting side. And in fact, if you were to run this just for AG, then it's going to be compatible with faulting side. Now this gives you extremely, this gives you very nice periodic metrics at every place P but the problem is that the periodic metrics might not glue well across various different prime speed to give a well-defined global height. And in fact, what we want is ideally we'd want this metric to agree with the metric defined by the spread out integral model for large enough prime speed. Well, there seems to be no way to prove that. So the fix was to use work of A0 and Grosjeck which they generalized in our appendix. Where they show that because our Shimura variety satisfy modular super rigidity, implies then a consequence of that is that for large enough prime speed, the periodic local systems that we have on your Shimura variety base change to EP are actually crystalline. In the periodic harsh theoretic sense, following faulting spontaneous LFI. And they also prove that if you were to apply the faulting spontaneous LFI correspondence, if you apply to this integral periodic local system, then you actually get the integral vector, the integral spread vector bundle with connection for large enough prime speed. So then what we do is for small p, we use the periodic P-man Hilbert corresponded to define a metrics for large prime speed and large is in terms of the ambition, whatever I T. We use a crystallinity to define the metric using the integral vector bundle with connection. And the fact that we have crystallinity tells you that things behave well under restriction. That's exactly why we need crystallinity because like periodic harsh theoretic operations behave well under restriction. Otherwise, they don't behave well under restriction. Otherwise, there's no reason that if you were to take a random integral model of your vector bundle with connection, there's no reason that the metric defined by that should give you a nice, should behave well when restricted, susceptible of limits. There's like a little more subtlety that happens while comparing these two metrics, but let me not go into that. Let me just stop by saying that once we have heights defined in this way, we can then use delin strict to establish the height bounds on the same point using height bounds on C-mobiling values. So let me stop here. Thanks very much for your attention.