Daniel Litt, University of Toronto

Thank you so much for the invitation. It's nice to be here. So, if you have any questions or have any problems with my handwriting, please just feel free to interrupt me at any point. And everything I'm going to say today is joint work with Josh Lam, and joint work with a couple other people might show up depending on how far we get.

Okay. So let me start with what the animating question is. So I think this was kind of an animating question for a lot of mathematics in the 19th century and 20th century, and hopefully 21st century, which is: how do you understand, if you're given an algebraic differential equation, when it has an algebraic solution? So I've attributed this question to Fuchs in 1875. This is kind of the first place I was able to find the question stated in this generality in print. But it was clear people were thinking about it long before that. So Kummer asked for a classification of algebraic solutions to hypergeometric differential equations in the 1830s, and then Schwarz answered that question in 1873. And I think that's what motivated Fuchs to ask this question in general.

And maybe just to fix notation, when I say an algebraic solution, I mean a solution which satisfies a polynomial whose coefficients are rational functions over the field C(z). And here you can see a picture of Lazarus Fuchs, to whom I'm attributing this question.

Great. So I think quite early on people realized that this question was maybe kind of of an arithmetic nature. So just to give you the first example. So here, this is maybe, I don't know, the first or second differential equation you might try. So d/dz minus a over z applied to f of z is zero. Here a is a complex number. So I guess this is a question in, this is a conference in differential, or sorry, a seminar in differential algebra. So probably someone can tell me what the solutions to this differential equation look like. Does anyone want to unmute and say it, or should I do it?

Oh, I'm just asking what are the holomorphic solutions to this differential equation?

Okay. So they're a constant times z to the a. And this is algebraic if and only if a is not a random complex number but actually a rational number. So I think this tells you that this question has some kind of arithmetic nature. It at least seems sensitive to the rationality properties of the structure constants of your equation.

Okay. So at this point, I think maybe it shouldn't be totally surprising that there's a conjectural arithmetic answer to the question of when does an algebraic differential equation have an algebraic solution. And I think the standard answer for linear differential equations is due to Grothendieck and Décats. So they make the following conjecture. So suppose you have a matrix of rational functions with rational coefficients. So here it's an r by r matrix with entries in Q(z). So then you can consider the differential equation d/dz minus A applied to a vector f is zero. And the conjecture is that all solutions to this differential equation are algebraic if and only if the following arithmetic condition is satisfied. So you take d/dz minus A, you raise it to the p power, and you reduce it mod p, and you ask that this differential operator be zero identically mod p for almost all primes p.

And this makes sense because I started with a matrix of rational functions with rational coefficients, only finitely many denominators show up in the coefficients. So for almost all p you can reduce it mod p. So as I've written it, this is kind of a special case of the Grothendieck-Décats p-curvature conjecture. It's the Grothendieck-Décats p-curvature conjecture for open subsets of the line over Q. But in fact the general case can be reduced to this one. It's not too hard to see that. But I'm trying to give a kind of concrete statement of the conjecture.

Okay. So this conjecture is very open. I think the main cases that are known are Katz proved it for Picard-Fuchs equations or Gauss-Manin connections. And then the Chudnovskys and Bosch and André proved it in the case where A is upper triangular. So the case where you have solvable monodromy. And that's not quite a complete list of known cases, but it's pretty close to a complete list of known cases.

Okay. So I don't think this, even if you take this conjecture as true, I don't think it's a complete answer to the question at the beginning of the talk. So first of all, this is about linear differential equations. You can ask what about nonlinear differential equations? And second of all, I mean the question is kind of something you can pose to a smart high school student, and I think the answer looks pretty wacky. So you could ask, can you just make a more naive statement? And so the goal of this talk is to kind of make a more naive statement, which is going to be a priori at least much stronger than the p-curvature conjecture. It will apply to nonlinear differential equations as well as linear ones, and then I'm going to explain some cases where we can prove it, so both in the nonlinear case and the linear case.

Okay. So let me kind of get to the motivation here. So where does the statement come from? So here's a theorem which I'm calling Eisenstein's last theorem. So it's a theorem of Eisenstein, and it's the last theorem he published. So he published this in a little three-page note in 1852 before he died of tuberculosis at the age of 29. And so I'll state something slightly weaker than what he actually proved, which is the following. So suppose you have f of z which is a power series with rational coefficients. So sum a_n z to the n, and suppose it's algebraic over the field of rational functions. So it satisfies a polynomial with coefficients in this field. Then what Eisenstein showed is that there exists an integer n such that all of the coefficients of this power series lie in Z join 1/n.

So another way of saying this is that only finitely many primes appear in the denominators of these a_n. And here's a picture of Eisenstein. Let me zoom in on it so you can see.

So there's kind of a mystery here actually. So in Eisenstein's paper he says something like, I found some amazing applications of this theorem to the theory of differential equations which will appear in future work, and then he died of tuberculosis and we have no idea what those applications were. I have some speculation, you can ask me after the talk if you want. So anyway, it's kind of nice to try to give an application of this theorem to the theory of differential equations, and that's what I'm going to do.

Okay. So you can ask a basic question, which is: does this property that you have a power series which have all of the coefficients in Z join 1/n, so only finitely many primes appear in their denominators, does this characterize algebraic functions among solutions to differential equations? So it clearly doesn't characterize algebraic functions, right? Like there's uncountably many power series with this property even with coefficients in 0,1. But only countably many algebraic functions over Q(z). Okay, what about among solutions to differential equations? Well, maybe the answer is less obvious there. Does anyone know if it's true that if you have a solution to an algebraic differential equation with this property, is it algebraic? Or does anyone have a guess?

You should feel free to type in chat if you're feeling shy, by the way. Okay. So the answer is very well known to be no. So here's an example. So you can take f of t to be the following function. So it's 1/pi times the integral from 0 to 1 of dx of the root of x times 1 minus x times 1 minus tx. So this is an elliptic integral. It's a function of t. So we're changing the elliptic curve we're integrating over. And here's the Taylor expansion of this function about zero. So it's the sum of 2n choose n squared times t over 16 to the n. Well, 2n choose n is an integer. And so only powers of two appear in the denominators of this function. But it's not very hard to see that it, well first of all, it's not algebraic. It's not even a finitely branching function. And it satisfies some differential equations. So this explicit differential equation, this is actually, it turns out to be a hypergeometric function. It's like 2F1(1/2, ...) if I remember correctly.

Okay. So, okay, this is a well-known example. I think if you Google is the converse to Eisenstein's last theorem true, or type it into ChatGPT, it will say no and give you this and some other big collection of examples. And there's a whole study of these objects. They're called globally bounded G-functions.

Okay. So, that's the end of the talk. The converse to Eisenstein's last theorem is false. No, I'm kidding. Okay. So you look at all these examples and they all have some kind of interesting property, which is, well, if you look at this differential equation, you see it's not monic, right? There's a leading term here, and this factor of t tells you that it's singular. This differential equation is singular at zero, the point we took the Taylor expansion around. Okay. And just every example has this property. So the natural conjecture is that Eisenstein's theorem should characterize algebraic solutions to differential equations if you take the Taylor expansion at a non-singular point.

Okay, so that's the conjecture I want to talk about for the rest of this talk. So I'll give a rigorous statement in a second, but are there any questions before I do so?

Okay, so here's some kind of concrete version of a rigorous statement. It's a bit long, but you can ignore this condition three for now. I'll talk about it in a second. Okay. So here's what the conjecture says. So suppose I have a power series f, it's sum of a_n z to the n with rational coefficients. And so suppose it satisfies an algebraic differential equation. The way I'm going to formalize that is that I want g to be a rational function with rational coefficients. And the type of differential equation I'll consider is the nth derivative of f is what you get when you plug z, f, and the first n minus one derivatives of f into g. Okay. So this is some kind of fairly general form of an algebraic differential equation. And then I want to assume that the point I'm Taylor expanding around, namely zero, is a point where g is defined. So this means, so here, just to be explicit, g is the ratio of two polynomials. I want the denominator not to vanish at this point. So what this is saying is that we're Taylor expanding around a non-singular point of the differential equation.

And then the conjecture is that the following are equivalent. So first of all, f is algebraic. So it satisfies a polynomial with coefficients in this field. Two, the conclusion of Eisenstein's theorem is true. Namely, that there's some integer n so that all the Taylor coefficients live in Z join 1/n. So another way of saying that is only finitely many primes show up in the denominators of the Taylor coefficients. And then three is a bit more complicated. So it says that actually you only have to look at, for each prime, you only have to look at the first few coefficients. So the way we're formalizing that is that there's some function from primes to integers which grows faster than any multiple of p. So call that function omega. So that the first omega coefficients don't have p's in the denominator for almost all p's. So think omega is like p log p or something like that. And the statement is just that the first p log p coefficients of the Taylor expansion don't have p in the denominator for almost all p.

So this is, we've added this condition. Well, one, because in all our cases that's all we're really using, all the cases we can verify. And two, including this condition makes the Grothendieck-Décats p-curvature conjecture a consequence of this one. Okay. And I'll say more about that going forward. Are there any questions about the statement?

So maybe let me just give these conditions names so I can refer to them. So I'll call condition one algebraicity. Condition two is integrality. So you should think the coefficients of the Taylor expansions are basically integers. And then I'll call condition three omega-p integrality. So it says the first few coefficients are p-integral.

Yeah, go ahead.

Yeah. So the conjecture is that if there exists one, then the function should be algebraic. In real life the function is always like about like p squared minus one or something. So that's what you get from p-curvature vanishing. So if a differential equation has p-curvature vanishing, then you can take omega to be like p squared minus one. But yeah, well, the statement is just if there exists such an omega, then f should be algebraic. And in every case we can verify, that's all that's necessary.

Yeah. Okay. So maybe I'll make some remarks on this. So some version of the conjecture that one is the same as two is attributed to André and Christol, and in work of, I've seen it in a couple places. I think it's not exactly clear whether either of them have actually made this conjecture, but they were clearly thinking along similar lines. So a priori this is much stronger than the p-curvature conjecture. So maybe let me kind of say why that is. So the p-curvature conjecture, it's some condition like the p power of a differential operator is zero mod p for almost all p. So that's related to Taylor expansions of solutions because that p power shows up when you write down the p-th Taylor coefficient of a solution. So that says something like the first p squared minus one Taylor coefficients don't have p's in the denominator. But it says that for the Taylor expansions of all solutions around all points. And so this conjecture says actually if you just find one solution whose Taylor expansion around one point has this property, then that solution is algebraic. So this is much stronger a priori than the p-curvature conjecture, and in particular there are cases where the techniques we can use to check the p-curvature conjecture don't seem to apply to this conjecture, although in some cases we can. I'll tell you some cases where we can check this conjecture.

Okay, and maybe let me just say I wrote down some kind of concrete version for kind of a fairly special looking class of algebraic differential equations, but if you look in the paper we make a version of the conjecture for arbitrary non-singular algebraic integrable foliations. So integrable foliations just means closed under the Lie bracket, and it has the form that a leaf of the foliation should be algebraic if and only if the formal leaf descends to some finitely generated Z algebra. So the Taylor coefficients live in some finitely generated Z algebra. Think like Z join 1/N. And that's the same as also some kind of omega-p condition which I'm not going to tell you.

Okay, and maybe let me just remark, so there's these three conditions here. So algebraic implies that the coefficients live in Z join 1/n. So this is Eisenstein's theorem. So two implies three: only finitely many primes appear implies that well, p doesn't show up in the first, you know, p level, p coefficients. This is just trivial: for almost all p, p doesn't show up at all. Okay. And then any backwards direction seems to be hard.

Okay. So for the rest of the talk I'm going to tell you what we can prove about this. Before I do that, are there any questions?

Okay. So maybe let me just start by giving you an example of a differential equation for which the conjecture is true. So this is theorem zero. Like everything else in this talk, it's joint work with Josh Lam. So here's an explicit differential equation. It's actually the one I wrote down earlier where there was a sort of counterexample to the naive converse of Eisenstein's theorem, and the conjecture is true for it. Okay. So what does this mean in practice? Well, this differential equation doesn't have any algebraic solutions. So to check the conjecture, you have to check that when you take the Taylor expansion of a solution, infinitely many primes show up in the denominators of the Taylor coefficients. That's actually a very concrete statement, right? To write down the Taylor expansion of a solution, you solve some four-term linear recurrence, as it turns out. And so it's saying you get some sequence of rational numbers and infinitely many primes show up in the denominators. We don't actually prove it that way. I mean, the proof uses sort of fancy arithmetic geometry, but at least the statement is saying you have some explicit sequence of rational numbers. Infinitely many primes show up in their denominators.

Okay. And the statement says, so concretely what are we saying? So we're saying take the Taylor expansion of a solution to this differential equation about a point where it's non-singular. So that's a point other than zero or one. And then you look at the subfield of the complex numbers generated by the Taylor coefficients. And what we prove is that's never a finitely generated Z algebra. So in particular if these are rational numbers, like you choose the initial conditions to make that true, infinitely many primes show up in the denominators.

Okay. Are there any questions about this statement?

Okay. So at least the conjecture is true sometimes. So this we can really, I mean I wrote down just some concrete differential equation, it's really a special case of something a bit more serious, which I'll just state for the experts. So it's that if you take a differential equation coming from a family of elliptic curves, so the fancy way of saying that is a flat connection in the tensor category generated by Gauss-Manin connections of families of elliptic curves, then the conjecture is true for those. So that includes things like elliptic integrals, like f was, the f from before was an example of an elliptic integral, but it also includes things like solutions to Picard-Fuchs equations associated to modular forms and this kind of thing. So it does cover some kind of cases that people are interested in.

Okay. So I would say this isn't quite the complete class of linear differential equations for which we can verify the conjecture at all initial conditions, but it's pretty close. So there's some other classes of differential equations where we can handle all initial conditions, but they're sort of of a similar flavor.

Okay. So for the rest of the talk I want to explain some cases where we can handle sort of certain classes of Taylor expansions. So Taylor expansions around certain special points. Okay. So let me kind of philosophically explain what I'm going to do for the rest of the talk. So what does it mean to check a case of the conjecture? So it means that you have some differential equation and you have some point, and if the solution to your differential equation when you Taylor expand at that point is not algebraic, you have to show that infinitely many primes show up in the Taylor expansion. So I'll identify some classes of differential equations and some classes of points we can Taylor expand around where we can show that. Okay, are there any questions before I begin doing that?

Okay. So let me start with the class of kind of linear differential equations that we can understand algebra-geometrically. So those are called Picard-Fuchs equations.

It's really a conjecture about any given solution. So this is some kind of distinction you don't see when you think about linear differential equations because the space of algebraic solutions is like a subspace of all the solutions. So you never have kind of an isolated algebraic solution. For nonlinear differential equations, and we'll see some examples soon, you can have an equation like the Painlevé VI equation or something where a generic solution is wildly transcendental, but there are some isolated, very special algebraic solutions. And so one thing you can think of as you think about this conjecture is that for nonlinear differential equations it's supposed to give you some kind of classification of those sporadic algebraic solutions.

So it's really about a single, I mean I think the thing that makes this much stronger than the p-curvature conjecture a priori is that it's really about a single solution. And so for nonlinear differential equations there actually was a pre-existing conjecture by Eidal, Shephard, Barron and Taylor which was supposed to classify the situation where all solutions are algebraic, but that's really very unusual. Like that kind of conjecture has sort of very, I think it's a beautiful conjecture, but it has kind of minimal applicability, whereas this thing is supposed to apply to things like Painlevé VI where there are both algebraic and non-algebraic solutions.

Okay. Okay. So let me kind of describe some linear differential equations where we can check the conjecture, and then I'll spend the rest of the talk on nonlinear differential equations. So let me just recall what is a Fuchs equation. So the situation is the following thing. One has a family of algebraic varieties over let's say some open subset of the complex plane. You can make it smooth, proper, it's not really necessary. And then there's a way of making a function out of this. So you choose a family of algebraic differential forms omega_t. So here omega_t is a differential form on the fiber X_t, and you choose a family of k-cycles. So here all the fibers of this family are homotopy equivalent, so picking a cycle at one point gives you a cycle on all the nearby fibers, and this is what I mean by family of k-cycles. Okay, and then you can define a function as follows: you integrate your k-form along your cycle. So you're integrating an algebraic differential form along kind of a topological cycle, and this will give you a function that depends on the fiber you're integrating along. So it's a function on this base, and this will always satisfy a differential equation called a Picard-Fuchs equation, basically because the cohomology of the fibers of this family are finite dimensional vector spaces. So that's sort of why there's a relation between the derivatives of this function and the function itself.

Okay, and because of this, there's a sort of nice way to identify the space of initial conditions for this differential equation. It's identified with the homology of the fiber. So sorry, I think I said, I switched between i and k. So here this was a k-form and this was a family of k-cycles. So the space of initial conditions should be the cohomology of the fiber.

Okay. So here's the main case in which we can identify a class of linear differential equations and initial conditions for which the conjecture is true. Okay. So suppose you have a solution to a Picard-Fuchs, so one of these functions, and the initial condition you're expanding around. So that's f of 0. So it's an element of the cohomology of your fiber. It's in the cohomology, and it's actually the class of an algebraic cycle. So this is a point of the cohomology of a fiber, and I want to assume that it actually corresponds to a subvariety of codimension i. So then the conjecture is true.

So what does that mean? It means the following things are equivalent. Your function is algebraic. The Taylor coefficients lie in some finitely generated Z algebra. And this omega-p integrality condition, which I won't write down again because it's kind of technical.

Okay. Yeah, so here we can't verify it for arbitrary solutions to Fuchs equations, and I think there's a good reason for that. That would imply, for example, the p-curvature conjecture for some ends of Gauss-Manin connections, which is a kind of famously open problem. Even though Katz verified it for Gauss-Manin connections, he didn't do it for some ends. But okay, we can check it for Taylor expansions around points which have some kind of geometric meaning, they're cycle classes. So if I have time I'll say more about the proof, but the input is Hodge theory and Fontaine theory. So in particular, I would say the proof of this theorem uses only mathematics from the late '80s. But we'll see soon a theorem that uses more recent stuff. But I do think somehow this, when Katz was verifying the p-curvature conjecture for Gauss-Manin connections, the technology was not yet available to prove something like this.

Okay. And the remark, I mean again, this is a very special case of the general conjecture. It works for arbitrary Gauss-Manin connections over arbitrary bases as long as you're Taylor expanding around a cycle class.

Okay. Okay. So in the remaining, I think I have what, like 13 or 14 minutes. I want to explain the nonlinear differential equations where we can check the conjecture. So they're differential equations coming from isomonodromy. So let me start with an example. I'll just write down some formulas and then I'll explain what it really means. So it's really some kind of nonlinear version of this story.

Okay. So here's an example of an isomonodromy differential equation. It's called the Schlesinger system. He wrote this down in 1912. So you take a bunch of matrix-valued functions A_1 through A_n of n complex variables lambda_1 through lambda_n. And I'm going to impose the condition just for convenience that their sum is zero. So I'll explain what this means in a bit. And so here's some differential equation you can write down. So here you have the same number of matrices and variables. If you take the derivative of the i-th matrix-valued function with respect to the j-th variable, and i is not j, you get this thing. So you get the commutator of A_i with A_j over the difference of lambda_i minus lambda_j. So this is a nonlinear differential equation because of this commutator. And then also you impose the condition that this sum condition remains true. So that's the condition that the sum of dA_i/dlambda_j is zero for all j, the sum over i. So here, in other words, if you want to figure out how to take dA_i/dlambda_i, you plug these expressions into this equation and solve for the last remaining expression.

So just as a concrete example, if you take the case where n, the number of matrices and variables, is four, and you're looking at 2x2 matrices with trace zero, this is up to change of coordinates the same as the Painlevé VI equation.

Okay. And just to remark, I mean, I'm going to say something about algebraic solutions to these differential equations. This isn't just some random thing. This is something people have been studying since maybe the '30s. So there's a long history of trying to classify algebraic solutions, beginning I think with Garnier, and then this really picked up with Hitchin in the '80s. There was a partial classification in the case of Painlevé VI by Dubrovin and Matsumoto. Then Boluch and others found a bunch more solutions. So in fact the complete list, and it was proven to be the complete list by Lisovyi and Tikhomirov with a computer-aided proof in 2014.

And then in the case of more general Schlesinger systems, so still when r is two but n arbitrary, there's now a complete classification of algebraic solutions obtained in joint work with myself, with Lam and Lenzmann. And then there was one remaining case we left open that was finished by Bronstein and Marazzi last year. So we now know all the algebraic solutions to the system when r is two. I think it's a very interesting and probably very difficult open problem when r is three.

Okay. Great. So okay. So I want to tell you first a result about this specific nonlinear differential equation, and then I'll tell you the more general result. It's a special case of... So what does an initial condition look like? Well, an initial condition is a tuple of r by r matrices. So I'll call them A_1_0 through A_n_0 whose sum is zero, and then a bunch of complex numbers which aren't equal to each other. Okay. So you should think this is, A_1_0 is the value of the function A_1 at this point lambda_1_0 through lambda_n_0. So this initial condition is the same thing as a differential equation. So let me point that out. So given this data you can write down the following differential equation which has regular singularities at the points lambda_i_0, and this condition that the sum of the matrices are zero tells you it's non-singular at infinity. Okay. So an initial condition for this Schlesinger system is the same thing as a differential equation, well, Fuchsian differential equation on P^1.

Okay, so here's the sort of baby version of the theorem about nonlinear differential equations I want to explain. So suppose this initial condition, this differential equation, is itself a Picard-Fuchs equation. So then the conjecture is true for the solution to the Schlesinger system with this initial condition. So in other words, if you take the Taylor expansion of the solution to the system with initial conditions A_i and lambda_i, it's algebraic if and only if the entries of these matrix-valued functions you have only finitely many primes showing up in the denominators, if and only if this omega-p integrality condition holds. So you can think of this as a classification of algebraic solutions to the Schlesinger system which are Taylor expansions around sort of special points. So Picard-Fuchs equations. Okay, this is, I claim this is exactly analogous to the previous result about Taylor expansions of solutions to Picard-Fuchs equations around cycle classes.

Okay. So the analogy is you should think about this as some kind of analog of a Picard-Fuchs equation. I'll explain that soon. And this being a Picard-Fuchs equation itself is some analog of being a cycle class.

Okay, so let me explain the real statement. So this is some very special case of what we prove. I want to explain the real statement, and then I'll just say a couple words about the proof. Okay, so where does this differential equation come from? So here's some general picture you can have. So just like before, Picard-Fuchs equations came from families of algebraic varieties. These kinds of isomonodromy differential equations also come from families of algebraic varieties. So given a family X over S, so you can let's say smooth and proper, if you want you can equip it with a relative SNC divisor. And that's the situation where this comes from. You can associate to this a nonlinear differential equation. So let me explain how. So associated to this is a space called M_Dconn X over S. And it's the moduli space, so really stack of flat connections on the fibers of X over S. If you have a SNC divisor, you can add with regular singularities at infinity.

Okay, so this is some kind of algebra-geometric object, and the fairly deep fact is that this algebra-geometric object carries an algebraic foliation called the isomonodromy foliation. So I'm not going to give you a description of it, but I'll give you a description of the leaves of the foliation. So the leaves of this foliation are families of flat connections whose monodromy is locally constant. So what do I mean by that? So here you have a family of algebraic varieties. They all are, you know, if you take two nearby fibers, they're homotopy equivalent to each other by Ehresmann's theorem. So if you have a flat connection on one fiber, it has some corresponding monodromy representation. If you move to a nearby fiber, you have the same pi_1. So there's some other flat connection on this nearby fiber with the same monodromy representation. And this foliation is kind of giving a description of how to vary the connections on your fibers to keep the monodromy representation the same.

Okay. It's a fairly deep fact that this, I mean the description I just gave of this isomonodromy foliation is analytic. It used the Riemann-Hilbert correspondence. It's a fairly deep fact that it's actually an algebraic foliation. So in fact, in order to make sense of solutions, whether solutions to it have denominators or not, one needs to describe this foliation in mixed characteristic, and as far as I know that was first done in the appendix to my paper with Josh. So you can look in appendix A or appendix B of our paper, where we, I guess appendix B is where we describe explicitly this foliation in mixed characteristic.

So the real statement, maybe let me just, so I don't lie. This is not a family of smooth varieties. So what does a foliation mean? This is a stack. So what we really do is endow this stack with the structure of a crystal over S. Okay. And but it doesn't actually, this is just technical stuff. I mean the actual output of our theorems is results about algebraic solutions to explicit differential equations.

Okay. So just as an example, you can take the family M_0,n+1. So this is genus zero curves with n+1 marked points. Forgetting the n+1st marked point gives you a map to M_0,n. So this is a family of varieties which you can equip with a very natural relative SNC compactification. And this gives you the Schlesinger system we saw before. So this explicit differential equation. So here M_Dconn you should think is the space of connections on a fiber of that map. That's P^1 with n+1 marked points. Sorry, P^1 with n marked points. And that's what we saw before, it's exactly this differential equation is a point of the space M_Dconn.

Okay, so what's the real theorem? So the real theorem is that the conjecture is true in this generality. So it's true for the leaves of the isomonodromy foliation through Gauss-Manin connections, which is what we call the Picard-Fuchs equations when you're working over a more general base. So another way of saying that is that if you have a leaf of this isomonodromy foliation through a Gauss-Manin connection, then the following are equivalent: the leaf is algebraic; the formal leaf descends to a finitely generated Z algebra. So what I mean by that is the Taylor coefficients you use to define it all live in some fixed finitely generated Z algebra. And again there's some omega-p integrality condition that I won't state explicitly unless you ask me to.

Okay, so I'm just about out of time, so let me just say a word about the proof and make one comment about why I think this is interesting and related to classical questions. So the proof really uses 2020 mathematics. So it relies on non-abelian Hodge theory over the complex numbers, non-abelian Hodge theory in positive characteristic of the work of Ogus and Vologodsky, and then some really recent work of Sano and Degrave on a p-adic version of non-abelian Hodge theory called the Higgs-de Rham flow. So the main inputs here come from p-adic theory and complex Hodge theory.

And maybe okay, let me just finish by relating this to the p-curvature conjecture. So I think even if you're not so excited about algebraic solutions to isomonodromy differential equations, and I would ask you why you're not if you aren't, they're extremely cool. This kind of conjecture has some real implications just for algebraic solutions to linear ODEs. So here's a proposition which is not too hard to prove, but I think it's not totally obvious. So namely that just verifying this conjecture for isomonodromy foliations on the moduli of flat bundles on the universal curve over M_g, so for all initial conditions, I guess we've already verified it for some class of initial conditions, the Gauss-Manin conditions, implies the Grothendieck-Décats p-curvature conjecture in general.

So I wouldn't go so far as to say that this is real progress towards the Grothendieck-Décats p-curvature conjecture. Maybe it's just another point of view, but I think it's a useful one. And the reason why is these spaces have a huge amount of extra structure you don't see if you just work with some random local system. So there's some hope I think of exploiting that structure to say something more concrete about the Grothendieck-Décats p-curvature conjecture. And well, I don't have time to say the current directions we're going in that, but maybe stay tuned for another talk a couple years in the future. Anyway, that's all I wanted to say. Thank you for your time.