 Okay. Hello everyone. Good morning. Yes. First of all, I would like to thank the organizers for the multiple conference and for inviting me and if I may add for the amazing food. So today, I would like to talk about limit periods and an arithmetic interpretation of it corrected by also combinatorics. I cannot promise combinatorics, but I will try to get to the final reading. So what we are after, the main subject is limit periods. So let me set that up. We're going to have a family, hers, and it's simple. So it's going to be over a complex disc. And here are some of the properties of this family. First of all, I don't write it, but as usual, it's going to be proper flat family. And then my fibers are smooth, proper, or e not zero. And so t equals to zero is going to be the singular fiber. Hence, this is a degeneration, x zero, that's the zero fiber is irreducible with a single node. And the degeneration to this node is simple. Essentially, I'm talking about the simplest possible degeneration one to cook up without being completely trivial. And for later purpose, we can also draw some pictures. My x t looks a little bit like this. And it degenerates to x zero. And topologically, this is an accurate picture. So the genius can be higher, but one of the arms always shrink to a node, you can arrange for this to happen. So that's what would be my x zero. And now that I've defined x zero, there's another, I mean, I can recreate x zero using smooth curve. So I will do this. I normalize. Now I have smooth curve genius one lower. And to mark points. Let's call them key and q. So if I were to glue these two points, I get x zero. So let me write this c is the normalization x zero and key q in c mapped to the node. Now what's interesting when you have a singularities, I'm not in general, especially depending on the degeneration, but in odd dimensions, whenever you have such a nodal degeneration is that the periods on x t, they do not specialize continuously at equal zero, but some of them blow up. There is in particular one problematic loop. So let's call this maybe alpha. So this is the loop that passes through the node in a special fiber. And here it will give me a path connecting key and q. Now, again, just for our amusement, we can say the following. I will give an alternative description of this, but it's nice to have somewhat explicit description of what's the problem here. Why it's not immediately trivial, what's happening. You have holomorphic forms here. One of the holomorphic forms on x t will specialize to something that lifts to a metamorphic form in c. So a hollow form, not all, but one of the holomorphic forms on x t, let's say x eta t, specializes to a metamorphic form, eta on c. So it's going to be a fiber of the relative dualizing sheaf and that can be identified with metamorphic forms. So I can determine what eta looks like and then choose eta t. Anyway, so the upshot is if you were to compute this integral, this blows up and it blows up not in a kind way. So it was blowing up metamorphically, let's say the function of metamorphic function of t. Then maybe I could take some kind of projective limit and so on and then this would not be so interesting. So what makes this story interesting is that it has a logarithmic expansion and then there's the constant here and then there's something that goes to zero. Now you see the problem is of course physicists do this all the time, divergent integrals never stop the physicists and you just write down an asymptotic expansion, you kill all the terms that blow up and take the constant term and they'd say this is what nature intended. So of course there's a theory of limit mixed structures and this is defined in a super fancy way but that's what it is that you take a logarithmic expansion, kill all the logarithms, read the constant terms. But the problem is what if my parameter was 5t as opposed to t? Then of course it would look like this and now my new constant has changed. That means the constant term in an asymptotic expansion is dependent on what I've chosen at my coordinate. Therefore to define limits I actually need to fix an infinitesimal rate of approaching the origin. If you fix this then your limits are well-defined. So let's say if you had a t in mind then take the differential at t, otherwise you fix just an element, the differential at zero, the disk and your limits are well-defined. So this gives, so for any t non-zero the periods specialize plomortically and you just, the periods determine the hot structure on x t, on the homology of x t. So here what they determine after you read the constant terms is that it still determines something and it gives you a mixed hot structure and what's called a limit mixed hot structure which I will write as l psi and I write this as the limit, but of course it's a fancy limit where I take an asymptotic expansion and read off the constant terms of my periods determines the limit mixed hot structure. Good. Now I mean you might be surprised a little bit why I'm doing this baby case and you think I'll start doing something serious and this was just a precursor to it, but in fact this super simple thing has not been properly analyzed in the sense that one sort of technological item was missing and I'll point this out to you. So our goal is to understand interpret maybe I should say interpret l psi from the point of view of the central fiber and in particular of this triplet cpq. Now this sort of thing is called this subject of interpreting limit mixed hot structures from the point of view of central fiber is called asymptotic hot theory. Now the subject has progressed and even in the 70s we knew almost everything there is to know about this limit and I will now tell you what we knew and what was missing. Okay first of all the limit mixed hot structure this is called by extension type with central piece the h1 of c so what that means is I mean there's a topological so topologically determined weight filtration on l psi and the graded pieces are well we can write it like this so there's a simple piece that doesn't seem to be doing anything h1 of c and then another simple piece but this is all a determined topologically I won't draw these pictures but you see for example h1 of c if I'm taking loops around these two donut holes they seem to specialize and give me loops at this donut hole that's essentially the explanation for the presence of h1. I can take a loop around p that gives me a loop around one of the arms here that explains well depending on homology or homology one of these pieces and that's path explains the other piece so there are two kind of special loops here the one that p would normally contract and this path would not appear in the homology of c and those are the two special paths so what this already tells me is that even if I ignore the extension itself and just read off the pieces the piece the middle piece so the periods here determine the curve c by forl all right so so this is the kind of thing that we want to do there are other pieces here the extension periods we want to understand what they mean and the other pieces are also interpreted so this let's say I have the weight one piece gives me that extension over here so now here I broke that extension but normally that extension is present and in fact you can interpret what this means so in this case this will be the relative homology of c with respect to pq similarly the other extension is this one so that it will give me that extension over there similarly I just write down what the extension is so this is now the homology of puncture curve c and you have a residue map that is constant c well okay what do these mean so the extensions by the way these these two are dual are dual extensions and what they do is the following so if I were to take the extensions of let's say I guess I want to take this this is the Jacobian so the extensions of this h1 is isomorphic to the Jacobian of the curve and in general it's always there's some Jacobian and so when we talk about these extensions really I'm talking about points on p on the Jacobian of the curve and what could they be so these extensions so the two extensions I have so they both go to the point let me write about the Jacobian image of p minus 2 in here so what that means again something you can see the holomorphic forms that do not have a pole in the limit the holomorphic forms in the end will want to eventually integrate from a path from p to q and the integral of holomorphic forms on the curve c integrated from q to q is the definition of the abel Jacobian here and that's one of the periods here so that's give me the following if but if c is not hyper elliptic then in fact the abel Jacobi image p minus q determines both p and q all right so that means the first piece the the greatest pieces gave me c the extensions gave me p and q so l psi determines we can assume it's not hyper elliptic otherwise interesting information anyway determines c p okay this is so we've gone up to 70s basically this was known even then but there's one one other piece so I have a bi extension so I've studied the pieces the extensions now the bi extension and the issue is that there is a one parameter family family of bi extensions over the two given extensions so the of course the converse is true if I have l chi it determines cpq but otherwise if I have my family then I have cpq and that determines the extension pieces so I thus far is everything determined geometrically so this last bit is to be understood there's a bi extension emerging these two extensions and let's maybe borrowing future notation these will be the bi extensions of h1c specializing for these two extensions so bi extensions that specialize p minus q p minus q um this thing is non-canonically so non-canonically isomorphic to c star so it's a c star force and what we are now saying is what would be the meaning of this bi extension everything else had a meaning what is the meaning of this bi extension and now although this is nice what's so this is c star force which means this identification is not canonical so the corresponding number is not canonical but there is something that's going on which I will talk later on it's called the height of the l chi and this lands canonically into the space of real numbers it's the obstruction splitting the corresponding real mixed heart structure the obstruction space is canonically the reals so now I have something like this and for our convenience it's not so bad to say this is if you do do everything correctly this height is essentially the real part of the constant of my asymptotic term here this what you need to do right is to make sure your form specialized to an eta that has purely imaginary periods I won't mention this again it's just there's a way to express this sign purely as this constant and an asymptotic expansion of an integral now you might also realize why this interesting question this sort of integrals of course come up quite often in maybe more complicated the generations and somehow you recognize zeta threes in your limits and you want to understand where they come from or to prove that there's zeta threes and this is kind of a start oh no this this is really the goal of the book and now I was going to say the what what is known about this number so I will talk about higher dimensions at the very end this I mean this zeta three is not going to appear in this talk except it's one of the source of motivations that you you say observe numerically as zeta three you want to be able to prove that it's a three and that would mean interpreting your periods from this point of view of the central fiber okay so what is the meaning of that uh limit period essentially or the height of the by extension so what we can actually go to literature as I say this is heavily studied example one of the simplest examples um so let me get my quotation right so they have a book um well Clemens Miller stuff it is they have a book on uh mixed structures called period method and period domains yes Carlson did I write it wrong I think I even have it in thank you very much so Carlson Miller stuff Peters have a book called period mappings and period domains and period domains and in the introduction of this book to get us warmed up they consider this example and they write down the period matrix for this by extension they start interpreting the pieces just like I've done they say this piece is totally determined to see these these are extensions they are the belgium Jacobius of p minus p and so on and then they come to this uh value of star the regularized period and then they say they have a different symbol for it but they say the value of this number star has no significance and this talk is about the significance of this number but they have a good argument right so they essentially prove in some sense that this value cannot have any significance um they say that well as we've observed in the sport that I just erased unfortunately um that if you scale so the argument is this if you scale your rate of approaching to zero psi by lambda then well psi changes by by log of lambda and in particular the heights will change by the log of absolute value of lambda and since no psi is determined I could have chosen any other psi um and then that means yes no zero is allowed so this is not zero otherwise you you cannot regularize as well so this means I have a number which I it's which it's well defined up to logarithms of complex numbers which means it's uncompleted undefined it could have been any number yeah so this star is had no cannot have any significance or it just reflected your choice of psi but one observation that we can make is I mean the statement implicitly has the following assumption that no psi is distinguished so if x i was distinguished maybe this fact would change okay so any questions thus far so I'll first then give our statements and then we will interpret it and refine the statement a little um so essentially at the same time we started the project and ended the project more or less at the same time with valence in so um we are able to interpret this number now and the only hypothesis you need is to say uh assume that the strip that's cpq is defined over a number field uh number fields k so then let me be wait for a few minutes then there exists rational choices so you want k rational choices for the rate of deformation for psi and that's also already pretty good so here I had an uncountable indeterminacy now I have countable indeterminacy and for those for those choices we have here's the what I called was the height of l psi so that's essentially a limit period or the real component of it you arranged it correctly uh this height is equivalent well I have to deal with the k indeterminacy so this is the weak form we'll make stronger later on it's another height but that depends only on my data the central fiber it's the Neuronthate height of p minus q uh modulo the logarithms of rational numbers so it's actually um so the norms of your scaling uh appears here so it's you end up with an indermed c of q star now I'm not going to define what a Neuronthate height is but let me just say that it's a measure of complexity this is a measure of arithmetic complexity at this point p minus q in the Jacobian which can be embedded into some large uh projective space but it's I mean what you would know you could do is choose an embedding into projective space and then read off the coordinates of p minus q that is gives you some bounds on how complicated those would be but you have to do it you know without choosing coordinates and this is a canonical kind of bound okay um so what's in is that this depends only on my central data I no longer even need the family of course I need to sort out this indeterminacy at some point so uh so there by the way we did the slightly different thing so we make this formula precise then leitron balancer essentially says that there exists such a formula it does not give these formulas but he says there exists such a formula for the kind of degenerations that give you bi-extensions so then these are odd the odd dimensional degenerations with a nodal central fiber I'll come to that later okay good I think our first order of business is to uh fix the second sentence where I say that there exists k rational choices for cosine so let me make that a little bit clear and then we'll interpret this statement a little bit more conceptually um well remember that we had this how much do I want to call here okay let's briefly recall since my symbols have disappeared I had my family x and then here I had my nodal elements which um I normalized would see and I have p and q big okay so there's a rather easy lemma this is a baby form for codius tensor but for singular degenerations rather than smooth degenerations uh there exists a canonical map held on by kappa that sends differentials at the disk centered at the origin to something that relates to c so it will send to this tensor product of differentials at p tensor differentials at q and you see already what advantage this gives me so if c p q is defined over let's say k a number field and embedded I think let's take it embedded in complex numbers if I have my data defined over there then the right hand side right hand side has a k vector space inside namely it's the vector space of k differentials tensor k differentials so I so there's q so this is isomorphic k not non-canonically isomorphic k and this is in the right hand side so that means my rational choices psi let's say psi is k rational if kappa psi lands in this k vector space um so if we started with a base that was uh to say for the project of curve defined over k for this much we could have actually just taken um the k differentials defined over your curve that was a thing and this two ideas would have matched but I can do a little bit more and maybe I'll do this at the end that's I I can put put an integral structure on the right hand side by choosing a model for c and so this is a side remark this won't be so important right now but um if you had a k an integral structure here and an integral structure here they don't typically match so you really need to take the k structure that comes from here and then use codire Spencer to pull it back here but really this codire Spencer is the right book um okay so that explains what uh differentials were taking and now the ball is to interpret this a little bit more conceptually this will also explain what Baylentz and did and some of the implications of this of this formula and towards the end I would like to give the correction to that's a little bit combinatorial even tropical image we'll shift gears a little bit and talk about by extensions so if there are questions this is a good time to ask them okay so let's define by extensions in general I mean half generally I will just talk about by extensions of h1 so a mix of structure b is a by extension h1 of c so here you can take any purehood structure of odd weight you just have to shift the weights uh oops the by extension of this if the weight-created pieces of p are identified this data so that this identification is part of your data and you see I have these two extensions that will be discussed and this is a by extension that extends the central piece in two directions and so now we look at what kind of by extensions there could be especially this is to interpret the existence in this relation between these two uh meaningful numbers uh that means some by extensions are essentially distinguished arithmetically and we would like to explain what that could mean so first this is a theorem from the 90s so Hain crude that's the modularized space of by extensions is canonically identified with the Poincaré bundle bundle over the Jacobin of the Kirk so in general there's some kind of Poincaré bundle over the associated intermediate Jacobin if you want so that looks like the following let's make this precise or more explicit so I have let's say my this would be my moduli of by extensions and here I have Hain's isomorphism to the Poincaré bundle so the Poincaré bundle comes with two projections to the Jacobin and in principle the dual of the Jacobin Jacobin itself here and what we had we have an analogous map here indeed uh my by extension consists of two extensions so I put map here to the two x groups I won't write down the the x so x of the central piece with set and I've already mentioned that this is standard that these extension groups are canonically identified with the Jacobins um so it completes this picture that the Poincaré bundle is the thing that sits over this product and my c star I can explain to you the c star of a possible by extensions for example if I fix here p minus q p minus q obviously the center the fiber here so this is a c star also the isomorphic to c star but not canonically and of course the same is true over here so you can say it again oh no no no no it's in it's possible that uh it's perfectly allowed and it happens that the two extensions are not sell through it's just that the one I'm interested in lives over this yes so there are many many extensions and essentially almost all of them have been understood and studied except for the one if you're talking about right now um oh just sorry this is I should have removed the zero so there's no canonical thing not even zero okay um maybe briefly I say here what height means very kind of strangely I can tensor my mixed structures over the reels or equivalently I forget the integral structure and what you get again this is if you want a side remark so I we and then I would be talking about real by extensions so it's exactly the same way to replace all the z's with r's and this is a Haynes definition of height uh because what happens is that the extension groups in the setting this is zero and this is becomes canonically isomorphic to r and basically the map of hayne is defined in the following way you take your by extension you forget the underlying integral structure and then you uh evaluate this canonical isomorphism this is the obstruction map splitting the underlying real mixed structure and that's called Haynes height okay uh so what I mentioned this Poincare bundle for a few reasons maybe I say this now but if my triplets cpq from before is defined over k again then in fact whenever I take a k rational point of the Jacobian my fibers will have k structure my fibers I mean the Poincare bundle will be defined over k then the fiber fiber of the Poincare bundle admits sub k star forces so it was used to be c star but there's a k star inside now so that distinguishes that distinguishes then a bunch of by extensions yeah I have a k star torso by extensions and there's a theorem this is why Haynes essentially did all this work and if I would like to put this again so if you work with by extensions defined over k okay in this sense like coming from the Poincare bundle then uh you're computing that on pay types the height computes Neuron tape heights and in particular this uh the height map on on the k star torso uh will evaluate to so to this set Neuron tape height of t minus q and what I mean this is from the 90s uh and basically reading this you automatically would believe that therefore the height for limit mixed height structure should be the Neuron tape therefore it was uh onjecturing the statement that the height of the limit mixed height structure should be the Neuron tape height and is very believable there's a problem over here so the so on the by extension side these let's call them k elements are without putting too fine point on it are motivic in the sense that they relate to the cohomology of the curve c so they all look like this so here's the key restriction here so if I have two divisors in the Jacobian so these are clearly the points over here then the by extensions you can construct purely from c you subtract from c the support of one of the divisors and then consider the relative cohomology with respect to the other divisor this works only if the divisors are disjoint and when the divisors are not disjoint then this construction fails you don't get a by extension and our limit mixed height structure certainly is not of this form whenever you try to interpret it you realize that you have to subtract t minus q you have to subtract pq and want to take the relative cohomology with respect to pq of course that doesn't uh that collapses this relative thing doesn't give anything new um what you would have to show is that somehow the limit mixed height structure constructed in a very very analytic way ends up doing something that's geometric in nature motivic if you want to call it in the statement you want to prove or this is really rather a corollary of our work we did not approach it from this direction so however you want to call it this they are more corollary says that the the map which we constructed in the following way i take a differential at the origin and i don't i don't want to take the zero differential sorry this is not remaining so the differential that's zero minus zero differential and then i have i have differentials that q and minus zero zero so they remember this was an isomorphism so this map um maps the codyrus cancer k star torsor on the left hand side to the k star torsor on the right hand side so if we had first proven this then um by heinz result you would have gotten the other theorem that says our limit mixed height structures end up computing the net on tape i in this case we've done the other one we've gone the other way around we computed the period is essentially the net on tape i and then we i'm using heinz theorem to explain this sort of conceptual interpretation so what that means is that the limit mixed height structure can maybe be done however this is kind of well known that the limit mixed height structures with our current understanding cannot be done or a number field it's very analytic in nature therefore it's actually not known in general that your limit mixed height structures if you've used information coming from a number field would give the correspond sort of an arithmetic data on the right hand side so this was also one of the motivations and valence and approach the theorem directly from this direction so i should say venison's theorem is that for bi-extensions okay so modulo constraints and so let me say in this case the elk sign can be partially defined constructed in the category of in a category of motives he almost constructs elk sign with this limit mixed height structure so he cannot define it so it we don't know how to do this but he says this partial information is enough to compute the height this partial information construction computes the height or actually determines the height so his version would say but the thank you um are there any quick questions comments for the speaker currently we don't currently have any questions so let's thank others i'm sorry i didn't see by that and Okay, so let's thank to speak again, we're going to take a picture. Thank you.