It's really nice to be here. Let me reiterate Alina's request that you ask lots of questions. I always find Zoom seminars a little bit depressing if I'm just talking to the screen. So yeah, please, please ask lots of questions. Everything I'm going to say is joint work with Josh Lamb. And let me just remark, some of this work has appeared in print in a paper from this January, but some of it is work in progress. And I want to mention that because I'm very worried that recent developments will lead people to talk less about work in progress and increase secrecy in our profession. So I'm very consciously trying to talk about work in progress. And what I'm thinking about, I encourage you all to do this. I think it's important that we keep talking to each other about mathematics even in the face of technological change.
Okay. So I want to talk about the arithmetic of differential equations. So let me just start by introducing what I think is maybe the basic question that motivates the subject. So in the 1870s, Schwarz classified algebraic solutions to hypergeometric differential equations. So hypergeometric functions that are algebraic. And I think this caused folks to ask just in general, like, can we tell, like, when does an ordinary differential equation, let's say with algebraic coefficients or rational coefficients, have algebraic solutions? So in other words, you write down some differential equation, you want to know when does it have a solution which is in the algebraic closure of the field of rational functions. So when does it satisfy a polynomial with polynomial coefficients?
So my, like, very, like, if I had to explain this question to a grade schooler, I would just say, like, the basic question is: someone writes down a differential equation, can you explicitly write down a solution? Where 'explicitly' means as an algebraic function. So let's just, like, maybe it's not clear to you why I would be talking about this question in a number theory seminar. So let's work out an example. So here's, like, a very basic differential equation. So the derivative of f is equal to a times f over z, where a is a complex number. Okay. So I assume many people in the audience have taken a calculus class before. So what are the solutions to this differential equation?
Does anyone want to type it into chat or something? Oh, there's someone has something. Oh, yes. Okay, great. Julian Rosen says constant c to the z. That's right. So, constant c to the a. Okay. So, obviously this is algebraic if the constant is zero. But for c non-zero, this is algebraic if and only if the structure constant a is a rational number.
And that's kind of easy to see. So if a is a rational number, you take the solution to the power given by the denominator, that's a polynomial. So that gives you a polynomial satisfied by this solution. If a is not a rational number, it's not hard to see that this function isn't finitely branching, and algebraic functions have finitely many branches, so it can't be. Okay, and what I just want to point out is that this is an arithmetic condition on the structure constants of the original differential equation. So this suggests that this question, which seems like maybe it's like a question in theory of just like, I don't know, analysis or something, is actually a question of number theory.
Okay. Great. And I think there's now kind of a standard conjectural answer to this question of Fuchs, which is the Grothendieck conjecture. So let me kind of give it in a simple form. So let's say A is an r by r matrix of rational functions with rational coefficients. And then we want to study the solution d/dz minus A times some vector of holomorphic functions is zero. So in other words, the derivative of f is equal to A times f. And so the Grothendieck conjecture tries to ask the following question or answer the following question: When are all solutions to this differential equation algebraic functions? Meaning all the entries of a solution vector f are algebraic.
And the conjecture is this: all solutions are algebraic if and only if some arithmetic condition is satisfied. So you take this differential operator to the p power, you reduce it mod p, and you want it to be identically zero for almost all p. And this is called the p-curvature conjecture. The reason why is that this operator d/dz minus A to the p, reduced mod p, is called the p-curvature of the differential equation. So there's many formulations of this conjecture. One can formulate it in the generality of flat vector bundles on a variety defined over a finitely generated Z algebra. I'll discuss some version of that formulation later, but it's equivalent to this one that I wrote down. So this really is some very elementary statement about linear ordinary differential equations.
Great. Are there any questions? Oh, yeah, maybe let me just remark before I continue. So why does it make sense to reduce this mod p? So this is a matrix with rational functions with rational coefficients. Only finitely many primes appear in the denominators of those coefficients. So for most p it makes sense to reduce mod p. It looks like there's a question from Mushfiq Goldbury. You should feel free to just unmute yourself and ask the question.
