 Thank you very much to the organizers. It's a great pleasure to speak here. It's my first appearance in the number theory web seminar. So I'm very happy to be doing this. So the title of my talk is about explicit class field theory and a connection with orthogonal groups. And most of what I'll be talking about today is joint work with Jan von Kahn. And since the format of the web seminar is a bit impersonal, we don't get to meet each other in person. So I wanted to at least put the picture of my collaborator here as my first slide. OK, so the idea is to try to generalize a very classical theory, the theory of complex multiplication, which concerns the values of modular functions at cm points to other settings, like the setting of real quadratic fields. And so a singular modulus is simply the value of a modular function, like the j function being the most classical and standard example, at a quadratic imaginary argument of the upper half plane. So something like a tau here of discriminant d, where d is a negative, say, fundamental discriminant. And the value of the j function at such a tau has all kinds of nice algebraic properties. It generates, it belongs to the Hilbert class field of the corresponding imaginary quadratic field. And it has an interesting highly structured factorization. So for example, the value of j at 1 plus root minus 23 over 2, an element of discriminant minus 23, satisfies this cubic polynomial with rather large integer coefficients. And likewise, for j of 1 plus root minus 31, we have this kind of value that satisfies. And these two polynomials are interesting. Their splitting fields are the Hilbert class fields of the corresponding quadratic imaginary fields. And while the coefficients seem to be very large, when you look at the constant terms of both of them, you see that they really factor into small primes, primes of small primes with large multiplicities. Now, the fact that j of tau belongs to the Hilbert class field implies, in particular, that its value at any tau corresponding to a quadratic imaginary for the class number 1 belongs, is an integer, is an actual integer. And you can see this, here's the value of j of tau and all the quadratic imaginary tau corresponding to class number 1 imaginary quadratic fields. And you see that their values are integers, which are highly factorizable. So although they're quite large, their largest prime factor is in fact quite small. In this case, the prime factor is strictly small. The largest prime factor is strictly smaller than the discriminant of the corresponding tau. And this kind of behavior persists also if you subtract from j of tau, the value of j at any other quadratic imaginary argument. So for example, here, if I translate j of tau by 1728, which is the value of the j function i, then I get the following table where, again, I see the same kind of phenomena. Here, the largest prime factor is never larger than the discriminant of the corresponding tau. So you have this. Now, that previous slide suggests that the natural quantity to study really is the difference of j values at two distinct quadratic imaginary arguments. So if we fix two such things, tau 1 and tau 2, you might want to assume for simplicity that the discriminants are relatively prime to each other. Then you might study this quantity j of tau 1 minus j of tau 2. And it was studied very classically. You'll see examples in the classical treatises of Klein and Fricka on modular functions where certain values are studied. And then, of course, this difference of singular module, I also plays a key role in a seminal paper of Gross and Zagier from the 1980s, which has sort of motivated a lot of what I'm going to talk about today. Now, the theorem of Gross-Zagier concerns the factorizations of these differences of singular module i. So if Hj is the Hilbert class field of the corresponding imaginary quadratic field, then the theorem asserts that if I take the difference, j of tau 1 minus j of tau 2, that belongs to the composite now of the two Hilbert class fields of q square root d1 and q square root d2. And if I take its norm down to q, I get an integer, and I take its absolute value, then I can write that integer, of course, as a product of primes with certain multiplicities. And the theorem says something about what that factorization looks like. Namely, it says that the primes that appear in this product always divide a positive integer of the form d1, d2 minus a square over 4. In particular, the largest prime factor is never larger than d1, d2 over 4. That's the first thing that you have. And moreover, these primes have certain constrained splitting behaviors in the two imaginary quadratic fields. They're either inert or ramified in each of q square root of d1 and q square root of d2. So these are patterns that you will see for all these factorizations. And the theorem of Gross-Zagier is more precise. It even gives a precise formula for the exponents, ML, that appear in this factorization. So it's a completely explicit recipe for the factorization. And I'll give you a little bit of a, since this is meant to be a kind of the general almost colloquium style, tough for a broad audience, I want to give you an idea of the geometric explanation underlying this phenomenon. So if we consider the difference j of tau 1 minus j of tau 2 and we consider primes that divide the norm of this difference, well, these j of tau 1 and j of tau 2 can be interpreted as a j in variance of elliptic curves with complex multiplication by these corresponding orders, corresponding maximum orders in the quadratic fields. And if l divides this norm, that means that these two elliptic curves have the same reduction, modulo l or rather modulo prime of the Hilbert class fields that lies above l. And so there's some elliptic curve over the algebraic closure of fl, which is the simultaneous reduction of these two elliptic curves. So that means in particular that if we look at the endomorphism rings of A1 and A2, those two endomorphisms inject into the endomorphism ring of the special fiber, which is therefore quite large since it contains two different quadratic imaginary orders of different discriminants. And the only way I can happen is if the endomorphism ring of this elliptic curve over fl bar is an order in a quadratic algebra of discriminant l. And so in an order that contains elements omega 1 and omega 2 having these discriminants d1 and d2. Now that, of course, places a large constraint on the kind of elliptic curve we can have. The curve has to have a super singular reduction at l. And if we now take these two elements omega 1 omega 2 and consider the pairing matrix associated to them, I mean this endomorphism ring is endowed with a natural quadratic form given by the trace form, then you can really discriminate. The determinant of that pairing matrix is d1 d2 minus t squared over 4. And because this is allowed us in a definite criterion algebra, that discriminant has to be positive. And it also has to be divisible by the prime l because essentially because the quaternion algebra is ramified at l. And that, of course, places a constraint on the prime l. Then it has to divide this expression, as was asserted in the theorem of growth since I gave it. OK. Now I want to formulate a real product. I'm sorry? Yes. May I ask, do you have a converse? If you take the intersection of all the endomorphism rings which contain all the l's such that the endomorphism contains two copies of A1 and A2, do you recover all the devices of the norm of the difference? A little bit more complicated. But as I said before, I mean, I guess you list of suspects, of course. But then in the work of growth since I gave it, there's a formula for this exponent ml that arises. And I do not think that all the integers which satisfy these two conditions, which I wrote here, will necessarily have an ml non-zero. OK. So further, it's not. I don't think you can rule out the fact that some of these l's could actually not appear. So these are necessary but not sufficient conditions for an l to appear in the factorization. Does that answer your question? OK. OK, so now what I want to talk about is an extension of this story to the setting where the imaginary quadratic fields are replaced by real quadratic fields. And the goal then is to define a quantity j of tau 1, tau 2, which is going to behave like the difference of singular moduli. And the key differences in this construction are going to be, firstly, that this, rather than having an analytic function, like j, which involves a Archimedean limits, it will be defined via a periodic limiting process. So it'll be a periodic construction rather than a complex analytic one. And the second key difference, which is very much at the heart of what we're puzzled by, is that we lack an arithmetic geometric understanding of the construction analogous to what we have coming from the theory of electric curves with complex multiplication. OK. So to describe the formula, so I want to give a very down to earth description of this function. It's a little bit less, how can I say? It's a bit less fine than the finite invariant we can define. But the advantage is that we can actually sidestep a lot of the formalism that's involved in, yeah. So I'll explain it now. So we have this local greens function, which we can associate to a pair of real quadratic arguments. So I'm going to let tau going to tau prime be the natural involution, which sends the square root of dj to its negative. And I'm going to, so to any real quadratic tau, I can associate a geodesic on the Poincare upper half plane, which is simply the geodesic joining the point tau to its conjugate tau prime. So that's like a semicircle joining those two points, which lie on the boundary of the complex upper half plane. And so you have tau 1 and tau 2. You get this pair of geodesics, gamma 1 and gamma 2. And you can consider their topological intersection on the upper half plane h. So these two geodesics, either they intersect or they don't. I mean, if they're like this thing. And the intersection number is either 0, 1, or minus 1. So it's just an intersection on that non-compact symmetric space attached to SL2R. Now we can associate a local greens function to the pair tau 1, tau 2 by considering this expression. It's essentially tau 1 minus tau 2, which is the difference of tau 1 and tau 2, multiplied by its algebraic conjugate numerator. And then in the denominator, you just normalize by this factor, which is comparatively less important, tau 1 minus tau 1 prime, and tau 2 minus tau 2 prime. But this is just some algebraic number, which belongs to the compositum of q square root d1 and q square root d2, so to this bicarbonic field. But then we weight this contribution by the topological intersection of gamma 1 and gamma 2, which means that if our pair tau 1, tau 2, if they're corresponding geodesics do not intersect, then the greens function simply takes the value 1. So it has a trivial value. And so this by its very definition, it belongs to the bicarbonic field, but then you look at its invariance property under tau goes to tau prime, and you see that it actually belongs to this distinguished quadratic subfield of the bicarbonic field. And it's easy to see from the definition that this greens function is a point-pair invariant. If you simultaneously translate tau 1 and tau 2 by an element alpha in SL2q, this expression here is essentially a cross-ratio of the four elements, tau 1, tau 2, and their conjugates, and therefore is invariant under translation. And likewise, this exponent here is, I mean, SL2q acts as hyperbolic isometries on the upper half plane, so it preserves the intersection number. So, okay, so we have this property. And now we're gonna define a global multiplicative greens function by simply taking the product of these local ones over suitable translates of these towels. So the group gamma, which is SL2z, which I'm just gonna denote by gamma, acts on the real quadratic elements, of course, by Mobius transformations in the usual way. And it acts on the set of two-by-two matrices with integer coefficients both by left and right multiplication. So I can now denote by gamma 1 and gamma 2, the stabilizers in SL2z, of these real quadratic elements, tau 1 and tau 2. These groups turn out to be of rank one. They're essentially cyclic groups of rank one with the generator of modular torsion, which is given by the fundamental units of the core, or by matrices whose eigenvalues are the fundamental units of the corresponding real quadratic fields. Okay, so we have that, and then I'm gonna define g sub m of tau 1, tau 2, to be the product over all matrices of determinant m, running over this double coset. So I take a matrices, modulo gamma 1 on the left and gamma 2 on the right, of the local greens function evaluated tau 1 and alpha tau 2. Okay, so it's clear that this expression here depends only on the value of alpha, mod gamma 1 on the left and gamma 2 on the right. That's easy to check from the equivariance properties of this expression. And so I take the product over all alpha determinant m. Now this could still be an infinite product because the indices over which it's running, there are this double coset space, consisting of element of matrices of determinant m, modulo gamma 1 and gamma 2 on the left and right, is an infinite set, but the way that I define this local greens function by decreeing that it would contribute a one whenever the corresponding geodesics do not intersect, actually cuts this down to a finite product. In particular, it's a finite product of elements in this quadratic field. So the final outcome is just an element of q square root d1, d2. Yeah, so that's what I'm saying here. It's really just a finite product really of these local greens. That's the importance of this factor, this weighting factor given by the topological intersection at infinity. And okay, so it belongs to this quadratic field. Okay, so now to make this interesting, we will need some kind of, to bring in some kind of analysis. And so we're gonna try to consider a periodic limits of these expressions. We do this by introducing a prime p, which for simplicity, I'm going to assume that it's inert in both of the real quadratic fields. So both d1 and d2 are non-quadratic residues. I could also have that p divides and describe, I'll ignore that case for now. And so this is, if this is true, then the prime p actually splits in the quadratic field q square root d1, d2. And therefore I can consider both the local greens function and this product g sub m as elements of qp by choosing a prime of q square root d1, d2, which lies above the rational prime p, okay? And now what I'm gonna consider, instead of just taking gm, I'm going to consider the product gmp to the n. So where now I take the product over all matrices of determinant m times the power of p. And the idea is gonna be to grow this power of p, okay? So this is still an element of qp. It's still given by a finite product of things, but this finite part of the force is getting larger. It's involving more and more terms as n grows. And we're gonna be interested then in these limits. We take n go to infinity of gmn evaluated on tau one, tau two. And that's what I denote by j sub m of tau one, tau two. So we wanna consider these limits when they exist. As a first theorem about real quadratic singular moduli. So I'm gonna first assume for simplicity that the prime p is two, three, five, seven or 13, which is equivalent to the fact that the modular curve level p, x not of p has genus zero or equivalently that there are no way to cusp forms on gamma not of p, okay? And then the theorem, which Yan and I, yeah, this is the theorem of Yan, Vanquin and I, is that this limit, the limit of gmn as n goes to infinity exists in qp. So it's a well-defined periodic number. And it has an interesting algebraic property. First, it belongs to this compositing of the two narrow Hilbert class fields of the two real quadratic fields up to torsion. And furthermore, if I take m equals one, this quantity j one, tau one, tau two, exhibits exactly the same sorts of factorization patterns as we saw in the difference of singular moduli in the theorem of Gross and Zagier when the tau one and tau two are imaginary rather than real, okay? So we'll see that it's the same kind of phenomena that arise that the primes that divide this difference are rather small and less than the project discriminants, essentially over p, et cetera. So there's a theorem very analogous to the one of Gross and Zagier, the governing these quantities. Okay, now I say a word about what happens for general p. So this restriction, of course, is very strong. You might wanna ask what if there are customs of level p? Well, we can consider this space of so-called weakly holomorphic modular functions on gamma not of p. These are modular forms of weight zero modular functions and we require them, we allow them to have poles at the cusps. But if so, gamma not of p, x not of p rather than modular curve has two cusps usually they're not at zero and infinity. And we still require that these functions be regular at the cusp zero. So they have a single pole which is at the cusp infinity, okay? And so that's the space which we're gonna be considering. Now, if I write, if I take such an element in M zero shriek gamma not of p I can write it as a Laurent expansion and q into the two pi i tau. And there's gonna be a sort of Laurent series part, a principal part and this other piece. And so these coefficients A, M of the principal part if I take such a thing and I assume that the coefficients aren't actually integers then I can consider a weight a kind of multiplicative combination of the GMNs weighted by these coefficients A, M that appear in the principal part of this weekly homomorphic modular form of weight zero weekly homomorphic modular function. So I do this and I call this G sub phi of N because it's governed of course by the form, the function phi. And the next theorem is that I can then take the difference of these G phi Ns of tau one, tau two and I'll call that J sub phi of tau one, tau two. And that limit exists as a piatic number and it belongs to the compositing of the two Hilbert class fields. Now the way you might wanna think about this J phi is as a kind of a piatic board shards lifts of the weekly homomorphic modular function phi. So it's a and I'll try to explain more of that when I talk about the eventual generalization of this framework. Okay, so now I wanna one thing I wanna do in this lecture is give it an idea of the ideas that go into the proof of this theorem, give some kind of outline. And it actually helps to actually recast the formula in a broader setting, which is inspired in fact by two things. So the first is a the theory of what we called rigid maromorphic co-cycles for orthogonal books. So this is something that I developed that's with Leonard German and Mike Lipnowski about a year or two ago. And so the goal is to sort of extend the notion of rigid maromorphic fun. I mean, these rigid maromorphic co-cycles are meant to play the role of a maromorphic modular functions in the setting of for things like real quadratic fields. And one can try to generalize them to general orthogonal groups. So that was the first source of inspiration for the formulas I'm now going to describe that are more general. And the second were the calculations that arose in the PhD thesis of Romain-Branche-Rô that was defended two years ago or a year and a half ago. So those are the two things that I and now I'll try to explain that a little bit. So the idea is to think of the previous formula as having something to do with a quadratic space of rank four and signature two-two. So now I want to look at higher rank quadratic spaces. So the basic object behind the construction is going to be a free Z module of rank two R equipped with a bilinear form from V cross V to Z a Z valued bilinear form. And the basic assumption is that the tensor product of this lattice V with the reals is a real quadratic space of signature RR which means that when you write the quadratic form in a diagonal form over R you have R plus signs and R minus signs. So it's a quadratic form of that signature. And I'm going to assume for simplicity this is not an essential assumption it just makes statements a little bit cleaner that this lattice has discriminant one. So the pairing actually identifies V with its Z linear dual. Now the final group of this quadratic lattice is defined over Z and it's even smooth over Z because of this assumption on the discriminant. And the main case that was of interest in the setting of real quadratic singular moduli as I sort of said at the beginning is that it's the case where the quadratic space M2 of Z, the space of two by two matrices with integer entries equipped with the trace form. So A paired with B is the trace of AB star and B star is the so-called adjugate of B the inverse of B times the determinant if you want. And there's a natural action there's a SL2 cross SL2 acts on M2 of Z by simultaneous left and right multiplications. So UV acts on A by UV A V inverse. And that group action determines a homomorphism from SL2 cross SL2 to G, which is defined over Z and this. And so the framework I'm about to describe now when you specialize this setting you will recover the formulas that I wrote down earlier. Okay now this quadratic space of rank two R comes as it has naturally associated to it two different kinds of symmetric spaces. So firstly there's an Archimedean symmetric space which is very standard in the theory of Chimora varieties or spaces associated to orthogonal groups. One can consider X infinity to be the collection of all maximal negative-definite plane R planes R dimensional subspaces of VR. So because of the signature the maximum dimension of the negative-definite subspaces are and you can look at this collection of all these spaces. And so it's a kind of subspace for the Grasmagnian of R dimensional subspaces of VR. And it turns out that that space is a real has a natural structure of a real manifold of dimension R squared. Okay. And there's also a periodic symmetric space associated to V which is now going to be a kind of rigid analytic space. So this space we're going to denote by XP is identified with the set of isotropic lines in the tensor product of V with CP. So I take CP which is the completion of the algebraic closure of QP and I tensor V by that to make it into a vector space of dimension two R over CP. And I look at all the lines in this quadratic space which are isotropic. And I subtract from that the set of lines at the set of lines which are spanned by vectors which are orthogonal to some QP isotropic lines to some vector which is isotropic, but defined over QP. So I remove all the orthogonal compliments of isotropic lines defined over QP. And so that's my symmetric space. And that turns out to be a rigid analytic space as natural structure of a rigid analytic space of dimension two R minus two. So you can see that the way that these two symmetric spaces are defined is a rather not parallel. The dimensions are quite different in the way that we define. So there's a feeling that perhaps the construction is not, I would be a bit hard pressed to give a very strong conceptual justification for why we made these definitions. They seem to work in what we're doing but one could imagine perhaps other things. And the formula I'm gonna present are very much going to exploit the interplay between intersection theory, topological intersection theory on this archimedes symmetric space which is the kind of thing that appears actually Roman-Brancheral's thesis, for example. And something happening with the analytic, the rigid analytic functions or meromorphic functions on this piatic symmetric space. Okay, so I wanna associate a certain quantity to this orthogonal group together with a maximal torus in the orthogonal group G. So I'm gonna call T such a maximal torus. I assume that it's R split, it's split over R. So these real points is just isomorphic to two R copies of, we're sorry, to R copies of R star, okay? Now, for such a torus, there's a general classification of tori in orthogonal groups. The Q points of such a torus are identified with L one star where L is a totally real ital algebra. So product of fields of finite extensions of Q equipped with an involution, lambda goes to lambda prime. And the L one star, this subscript of one denotes the elements which are of norm one relative to this involution. So lambda, lambda prime is equal to one. Now, if I take the sub algebra of L of this field of this ital algebra, which is fixed by the involution, I'll call it F instead of lambda, lambda is equal to lambda prime. So L is a quadratic extension of this F. So to speak. And, but now the assumption I'm going to make about the torus is a strong condition, strong assumption about how the torus is actually globally non-split. I'm going to assume that L is a totally real field, not just any ital algebra, not a product of fields but really just one field. And even more than that, I'm going to assume that there's a prime P for which the tensor product of L with QP remains a field. So this prime P then has to be, there's a unique prime of L, which lies above P. Okay, so that's a rather strong assumption on the torus being non-split at P. Okay. And so this is an essential assumption. The next one is just to simplify things a little bit. I'm going to assume that the sub ring of L which preserves the lattice of E. So V remember was a Z module, right? And that in general would be a no order in L. And I'm going to assume that it's the maximum of the ring of integers of L. Okay. Now the action of this torus, the irrational points on the torus T, it becomes diagonalizable over the tensor product of V with the field L. And of course there the action splits completely and decomposes VL into a direct sum of one-dimensional eigenspaces. And I'm going to fix an eigenline, subspace of dimension one, which is preserved which is fixed by the torus action in VL, okay? And so this C is isotropic and therefore it gives rise to an element of XP after choosing an embedding of L into CP. So then you just base extend the change scalars to CP by this embedding. And that gives you an isotropic line and the condition of L being a field after tensing with QP sort of ensures that this C is never orthogonal to any QP rational isotropic line. So it really belongs to XP. It's a really a point. So that was one of the justifications for making this assumption on L being non-split at P so that I get a point in this symmetric space. And in addition on the Archimedean side at infinity I have two R real embeddings because L is a totally real field. So I have two R, it's a totally field of degree two R and these real embeddings of L, they come in pairs because I grouped together an embedding and the one which differs from it by composition with this automorphism prime. So I have then two R vectors, two R lines rather C1, C1 prime up to CR, CR prime which are contained in the tensor product of V with the real numbers and our isotropic lines there. And these lines are all mutually orthogonal to each other except that Cj and Cj prime pair to something non-zero. So they give a natural decomposition of VR as a direct sum of our hyperbolic planes, okay, over R. Okay, so now to these I'm gonna now describe certain Archimedean cycles which are associated with this torus and those are, they generalize the geodesics on the upper half plane that were associated to the real quadratic tau, okay. So now if I have an R tuple of positive real numbers I can consider the R linear span of the vectors T1, C1 minus T1, C1 prime. So here I made an abusive notation I'm just choosing a vector in that isotropic line which I'm calling C1. And I take this thing and then up to TRCR minus TR inverse CR prime. Now these R vectors are mutually orthogonal to each other and they all have negative inner product of themselves. They have negative length. And so this plane which is spanned by these R vectors is negative definite. So I get a collection of negative definite planes of dimension R, one for each R tuple of real numbers and so then I can then use that to define a cycle, a cycle dimension R, which is the collection of all these pi C of T1TR as T1TR range over the positive real numbers. So that's a subspace of a cycle in X infinity of dimension R. So this delta C is an R dimensional topological cycle in the Archimedean symmetric space. Now, give it a vector of positive length. I'm sorry. Yeah. You impose here the TR to be positive, but then that depends on the choice of the embeddings. XI, I, because XI, I, XI, I prime. You're completely right. Yeah, so here maybe I should have said this. I said this, but then I actually kind of normalize the, I take these vectors that's in these lines, but I normalize my choices so that these inner products are always positive. No, but it depends more on the order than on the positivity of that thing. Like who is CJ and who is CJ prime? Yeah, so CJ is a vector in the line and CJ prime is its conjugate. Yeah, but what I mean is you could have chosen the conjugates and then in your next definition, when you impose the TI to be positive, it exchanges to, so in the next line. I make the choice. Okay, so you're right. So you fix the choice once and for all, and then for this choice you define these. Yeah, that's another way of saying it exactly. So you, I mean, I choose the sign of these Ts in such a way that the corresponding vectors are going to be a negative length. Okay. So if C1, C1 prime was negative, right? I would put a plus here, not a minus. Okay, okay, so you would adjust them so that they are all positive. Okay, thanks. I mean, these sign things to some extent are a little bit a matter of convention because I could have defined my symmetric space as the set of all positive definite subspaces as well. If I have a negative step in something, I think it's orthogonal complement that's going to be a positive one. So the two spaces would be a natural bijection to each other. So in the literature, you will see both conventions being used more or less, it's not very essential. Okay, we'll just have to be consistent, of course, with this. So I choose in such a way that is negative definite and then I get this cycle. Now in parallel, I also want to associate certain cycles to the positive vectors in my lattice. So if I have a V and V with positive length, I can consider the negative R planes which are orthogonal to V. So V is positive, this orthogonal complement is going to be a signature R minus one R. So we will contain certain negative definite subspaces. And it's one can show it's not hard to see that this delta V is of co-dimension R in X infinity. So as dimension R squared minus R. And in particular, these cycles delta V and the cycles delta XE, they are of complimentary dimensions. There are some of dimensions as R squared. And therefore we can talk about the topological intersection. It should be some met. And it's always turns out to be always zero, one or minus one. And it's completely well-defined. There are no issues of a non-transverse intersection of degenerate intersection in the setting which I'm describing now where L is a field. You always have a well-defined, very simple kind of intersection number. Okay. So this is gonna allow me then to define certain piatic class invariance attached to this torus T in the setting. So I let V of one over P be simply the lattice V tends it with Z of one over P. So I turn it into a Z of one over P lattice. And the Z of one over P lattice, so it's a rank two R Z module, as sorry, Z of one over P module free over Z of one over P of rank two R. I can also write it as a direct limit an increasing union of the Z lattice is P to the minus N times V. And this expression of the Z of one over P lattice is an increasing union of Z lattice is gonna be useful shortly. Okay, so then I consider the following expression. I take the product over all vectors in the Z of one over P lattice of a given length M and I look at the orbits under the Z, the integer points of the torus. So I look at T of Z. The non-split assumption implies that T of Z and T of Z of one over P are essentially the same. So it doesn't really matter whether I choose the Z points or the Z of one over P points. So I look at these orbits of vectors of length M and I look at the simple expression. I take the inner part of V with XC and V with XC prime and I divide by the inner product of XC and XC prime. Now this expression here, it depends our priority on the choice of a vector in these isotropic lines, but you can see that it in fact does not because here XC prime is the conjugate of XC. And so if I rescale XC, this entire ratio will be unchanged. So it's a really canonical invariant attached to XC and V. And I take this element and I raise it to the power of the topological intersection of the cycle delta V of the previous slide with the cycle delta XC. Okay, so that's my JM of XC. And again, because of this factor here, I mean, this set is infinite, but this factor makes the number of terms appearing in the product which are not equal to one finite. Oh, no, sorry, sorry, no, I take it back. I mean, I'm taking now the product over V of one over P. So if I've been taking the product over a Z lattice, I would get a finite product, but having inverted P, this becomes an infinite product. But I define it, one has to be careful and define this infinite product. I define it to be the limit of the JM ends where JMN is the product over this thing, the P to the minus N V mod T of Z. And now this product here is a finite product. Okay, and but I take, allow N to increase and then to infinity to get my JM of XC. Okay. And if you, so in addition to these JMNs, it's useful to introduce a J phi N attached to any weekly whole morphic, wow, here 20 formal powers, Laurent series in Q, okay. If I have a fee like this, then I define J phi N to be the weighted product of the JMNs, raised to the power AMs, these coefficients that appear in the principal part of the, of the fee. Okay, so that's my JM, J phi N of XC. Okay, and then the theorem is basically going to be an algebraicity result about the value J phi of XC. Which generalizes the previous result on real correct senior module. So the theorem is if fee is a weekly whole morphic modular form of weight two minus R. So two minus R is kind of the complementary weights to weight R and level P, then the following limit exists in QP or not in QP, but in, in F P in the completion of F at P. So this J phi of XC is just a limit as in goes to infinity of the J phi Ns. And it belongs, so I prioriates just the periodic number which belongs to the periodic completion of F, but it actually belongs to the narrow Hilbert class field of L relative to a suitable embedding of that Hilbert class field into the completion. So those just narrow class Hilbert class field up to torsion. Okay, so the case of, as I alluded to before, the case of real correct singular module can be recovered from this more general results by setting R equals two, V to be again this four-dimensional space M2 of Z, L to be the bicodiotic field Q square root D1 square root D2, F is this quadratic subfield of a bicodiotic field and the isotropic line or the isotropic vector is this one here, which belongs to this tensor product of V with CP. So that's the vector of that. So if you plug that was into the formula, you will recover that earlier formula I gave you. Okay, so now I'm going to say, give you a little bit of an idea of what goes into the proofs of these results. And it turns out that there's even some conceptual clarity to be gained by treating the general case. It doesn't make the proof much more complicated and even make simplifies it on some conceptual level. Cause yeah. So the main thing, the main object in the proof are certain Hilbert modular theta series, which are associated to finite order HECA characters of the field L. Okay, so I remind you a little bit, the notation involving finite order HECA characters. If I have, I consider Isabel to be the group of fractional ideals of L and PL to be the group of principal ideals having a totally positive generator. So the quotients of IL by PL classically is what we call the narrow class group of L. And a character of this narrow class group, we say that it's a mixed signature. If it's value on principal ideal is the sign of the product. Remember that we have these two R real embeddings of L into R, but we group them together according to when they differ by this prime. And so we chose for each prime of, for each Archimedean place of F, we chose a real embedding of L that lies above it. And I call A1 up to AR, the images of A under these are distinct real embeddings. And they take the sign of the product. And if that's, if PSI satisfies this, we say that it's a mixed signature. So it's sort of, even an odd at each of the real places lying above any real place of F. Now, if PSI is a mixed signature, the reason why these characters are useful for us is that when we look at the induced representation from L to F, we get a two dimensional representation of the Galois group of F. And this two dimensional representation is odd at all the Archimedean places of F. So it's an odd, it has eigenvalues one and minus one when evaluated at each of the Frobenius elements attached to the R Archimedean places of F. And we know that such odd Galois representations of F, Archimedean representations odd to correspond to a holomorphic Hilbert modular forms of parallel weight one. And in this case, the forms that correspond to these induced representations are the classical Hilbert modular theta series, which I'm gonna describe now quickly. So we can define a function on the integral ideals of F depending on this HECA character PSI by saying that C PSI of an ideal A of F is the sum of the PSI of J, where J ranges over all the ideals of norm A. Okay, so that's the function on ideals. And then we can write down certain generating series, which are indexed by the elements new in a certain, the totally positive elements of certain ideals of F. And so I write theta PSI of I to be the sum over the news of this ideal of C PSI principal ideal, generated by new times, sorry, times I inverse times Q to the new, which is just a shorthand, of course, for the usual thing you see in the theory of Hilbert modular forms. Okay, so this is a Hilbert modular form of parallel weight one on a certain congruent subgroup of SL2F, which is essentially of level one under my assumption that OL preserved the lattice, et cetera, and that the lattice was of a discriminant one. Okay, now I wanna bring in some Abelian extensions of L, so for the next part of the proof. So if I have an Abelian extension in this context of level one, where everything is un-ramified, so to speak, this H would just be the narrow Hilbert class field of the extension L. So this would be an Abelian extension of L, but it would also be in fact, a Galois over F, right? It's Galois, but not necessarily Abelian over F. And I'm gonna write G for the Galois group of H over L and G plus, so the evolution in Gal L over F induces an evolution on G, which I again denote by prime, and I let G plus be the subgroup of G, which is invariant into this evolution and G minus the quotient of G by this G plus. So you have the following diagram of field extension. So you have your F, which is the base field, totally real, this quadratic extension L corresponding to the torus in the orthogonal group. And then you have this G, the Galois of H over L, which decomposes as a subgroup G plus and a quotient G minus. And this H minus is actually Galois over F and its Galois group is a generalized dihedral group, where this evolution acts as minus one on this G minus. And so we're focusing particularly on primes P of F, which remain inert in L over F. And so the Frobenius in Gal H minus over F, which is associated to this P, is actually an element of order two. It's a reflection in this dihedral group, which means that this prime P has to split completely in H minus over L. And then the splitting behavior of this primes in H over H minus is immaterial. It sort of splits in some way, but what really matters in the construction is that we have this behavior of P being totally split in this extension H minus. Okay, so the crucial results that really allows us to prove things is the theory of deformations of these modular forms and their associated Galois representations. So I fix a prime P, which is in inert in L over F, and I consider this theta function. And the main result is that this theta function admits the first order, ordinary pietic deformation of the following form. So here you recognize the coefficients. I hope that you people see I have my cursor moving over the slides, I hope you see it, yeah. So this is just a coefficient of the original Hilbert modular theta series. And then you add in this infinitesimal term, so the epsilon is just a short hand for this element of square zero, and you just add this pietic logarithm of the corresponding element in U times Q to the U. So this is kind of not terribly interesting kind of deformation of the modular form, but then there are these extra coefficients which crop up, which only occur at the coefficients of the original theta series, which were zero. So one of the features of theta series is that when you look at your Q expansion, it's there fairly lacunary. Any element in U which corresponds which is divisible by an ideal which is inert, for example, the corresponding Fourier coefficient has to vanish. But in this first order deformation, you get this interesting sometimes Fourier coefficient. And what characterize these S psi of U inverse is that, so this is a function on ideals. It can be written as a linear combination of S G of A's as G ranges over the Galois-L over L. And the crucial point is that the coefficients that arise here are pietic logarithms of algebraic numbers in H minus star. So in the multiplicative group of this abelian extension of L. So this is the crucial thing, the crucial phenomenon because we see the algebraic numbers that we were trying to construct appearing as the first order, as the Fourier coefficients of first order deformations of these Hilbert theta series. That's the source of the algebraic numbers. And I won't give you the formula for these S GA's is quite involved in general, but as a special case, S G of P Q, if I take a product of two primes, P and Q, which are both inert in L over F. Then I can take the recipe for it is that this is going to be the P prime attic logarithm of a unit U of Q prime. This U of Q prime is an element of H minus and it's chosen so that it has valuation one at Q prime. So Q prime is just some prime of H minus above Q that has valuation one there and valuation zero everywhere else. And I take the P prime attic logarithm of that and these P prime and Q prime are pairs of lifts. They're chosen to have a certain compatibility relation, which is that if I look at the two Frobenius elements attached to P prime and Q prime, these are canonical elements in my dihedral group, the Gal group of L over F. And I want that this element G, which occurs here, is the product of these two reflections. So this is now an element of Gal H over L, okay? And so that's the formula. I'm sorry, do you have a common logical interpretation of this deformation by saying that the trivial deformations are like boundaries and the non-trivial are sort of co-cycles? I'm not sure. One of my goals is to give this lecture without saying the word co-cycle once. Okay. Yeah, I mean, of course, the mechanism behind the proof of this result is the theory of deformations of Gal representations. So the specialization of this data series, of course, it's associated Gal representation is this Arten representation, which is particularly simple. It's a dihedral type representation. And when you look, when you analyze this first order deformations, you're led to consider Galois homology, the homology of the Galois group of F with values in the adjoint representation of this Arten representation. And out of that, you extract, out of really class field theory for H minus, in some sense, you see appearing these logarithms about to break numbers. So these very interesting coefficients really emerge naturally from the deformation theory of the Galois representation. So the log of the units are kind of the trivial deformations. And then the ones in H star minus are the non-trivial ones. I don't know. Yeah, well, there's the deformations of a, when you're a Frobenius element at the prime is a reflection. So, yeah, so these ones correspond to deformations where the Frobenius with a prime splits and therefore the elements. So this Galois group of H over H minus over F is a dihedral group. And if the Frobenius lands in Gal H minus over L, yeah, over L, right? Then we get these relatively straightforward terms coming up in the deformation. And then something interesting happens when the Frobenius is a reflection and you look at the deformation, you pick up some information about an extension class between two Galois, between two one-dimensional Galois representations, which gives you these numbers. Okay, so I'll leave it at that because I'm running out of time but that's sort of what's going on. Okay, so now that I have this, I can just describe in one slide the proof, the remaining proof. So I first define an ideal, a fractional ideal in L by taking the inner part of the Z lattice V with C, this L vector. That gives me a fractional ideal well-defined up to homotasy. And I can look at and make an ideal of F by taking the norm from L down to F of this ideal divided by this C, C prime inverse. So this is actually canonical, not depending on the choice of the vector in C. And then I define a function on ideals by sending alpha, an ideal alpha to the sine of A if alpha differs by this S by principle elements, by principle ideal and zero otherwise. This function is not a character, but it's an odd function. It's a kind of mixed character function, I should say. And therefore it can be expressed as a linear combination of characters. And that's gonna be enough for me because I've analyzed the deformations of all the theta C's for C characters, mixed signature characters. I can then recover something about this thing. And I'm gonna then consider the theta series associated to this function. And it's Q expansion attached at the ideal I, so to speak. And so I get this first order thing by taking this theta psi S tilde and I take its derivative, or in other words it's epsilon term, the derivative respect to epsilon. And so I get this. So the crucial thing is I take the Hilbert modular form and I look at its diagonal restriction to get a classical modular form over Q. And now this is just, I can write down it's Q expansion, now the Q expansion is indexed by integers M, okay? So that's my diagonal restriction. And so this is actually, because it's the diagonal restriction of this derivative, taking the derivative means I only get a piatic modular form, not a classical modular form. It's a piatic modular form of weight R and tame level one, okay? And then the main point of the proof is just the calculation of the Fourier coefficient solution. It turns out that if you calculate the Mp to the end Fourier coefficient of this piatic modular form, you get two contributions. The first is the piatic logarithm of this finite product, Jmn of Xe. And the second contribution is the piatic logarithm of an R of M, where R of M belongs to this H minus star, tensor Q. And this R of M, of course, is coming from when you take the diagonal restriction, all these new Fourier coefficients in the deformation where the original coefficients were zero in the Theta series. And the crucial point, the crucial feature of the calculation is that this R of M depends on M, but it does not depend on the exponent N. So when you multiply M by powers of P, you do not change this algebraic contribution, okay? So then the key thing is to take this piatic modular form of weight R and look at its ordinary projection, where you just iterate the UP operator. And if I write this now as a few expansion, this modular form is now a classical modular form of level P because of how the ordinary projection behaves. And so this A of M is just a limit of A of M, P to the N factorial. And then, of course, here, I just recover the piatic log of JM of C plus the piatic log of this R of M. But now this A of M is the Fourier coefficient of a classical modular form of weight R and level P. So if there are no classical cost forms of weight R and level P, then I'm done because this coefficient has to be zero and I recover the algebraicity of the log of JM C from this. But more generally, the space of forms of weight R and level P emerges as an obstruction to the algebraicity of these expressions. But that obstruction is controlled by serduality by the principal parts of weakly homomorphic modular forms of the complementary weight two minus R. And that's what explains the formulation of the result I gave before. Okay, so I'm gonna finish now. I was actually at a further slide where I mentioned rigid homomorphic co-cycles, but since I had kind of promised myself to not try it out to mention co-cycles at all, I'll just skip that slide. And thank you for your attention.