 So, all right, this is what I'm going to talk about, and let me start with what I'm really happy to be here to honor Luke. I've known him and admired him for quite a while now. And maybe in another half century, we can do it again. Okay, so what I'm going to talk about is joint work with Pamhatiap and Antonio Achas-Leon. And let me start with reminding you of Avianca's insight, which goes back to around 1957. So, if you have an open Riemann surface of genus G, then the fundamental group is a free group on 2G plus one less than the number of punctures free generators. Actually, I tried to track down exactly when this became a theorem. And it's a little bit hard. I mean, it's, by 1930, it's written down somewhere. But I couldn't find anything before that, but I'm, I can't believe that it took that long for this to be known. In any case, so that was it. So Avianca says, well, what happens if you're in characteristic P. So again, an open Riemann surface. But you're in characteristic P. So he says, well, the problem is about the characteristic P. So he says, take the subgroup generated by all the PC law subgroups for your finite group or commonly by all the elements of subgroup generated, the normal subgroup generated by all elements of P power order. Or equivalently, it's the subgroup, it's minimal subgroup such that if you divide out by it, you'll get a group of order prime to P. So now Bianca's idea is that the condition that a finite group G be a quotient of pi one of this, this open curve in characteristic P is that if you divide it. It's, so to speak, P part this G sub P. That should be what you get. In other words, if you have a finite group, which divided by its P part is generatable by the correct number of elements. Then it should occur as a quotient of a finite quotient of pi one of this open curve in characteristic P. So if it's the alpine line which over the complex numbers is simply connected, then it's saying that the condition is that G should be equal to the subgroup generated by its PC laws. If you're on the multiplicative group, it's the groups that if you divide by this P subgroup, you just have a cyclic group, etc, etc. So that the key case was the case of the alpine line. And that was proven by Renault and then was extended to the general case by Harbader and the different argument by pop. So it's that's something we know. Okay. So then the question becomes suppose we're in characteristic P. And we're given a group that according to the theorem can occur in characteristic P. So take it take some complex, take a faithful complex representation of it. And then you say okay it's a finite group so this complex representation of course lands in GLN of some number field, and you embed the number field into some allatic field, then you can compose. You can think of the group inside this is GL, and you can compose with the fact that the PI one maps on to this finite group. So altogether you get a homomorphism from this PI one to some allatic GLN. And tautologically, a continuous representation of PI one is what we call a local system. Yeah, a least shape a least allatic shape of rank and on this open curve. Okay with this extra property that it's monodromy group is actually this finite subgroup of the GL. And I mean, L here, when I've said so far L could be the characteristic that that doesn't matter it's a true statement. If we want to use a lot of homology which we do, then we'll always take will will choose an L which is different from the characteristic. Okay. So that's, that's the story that we want to investigate. So now I want to talk about so to speak working backwards. Instead of starting from a group, let's talk about local systems that we can somehow try to deal with. So first of all, our curve over the algebraically closed field will be interested only in cases where it comes from something over a finite field. We'll want to have a local system on the curve over the finite field. Take it to be geometrically irreducible. And we'll also take it to be pure of some weight and in fact, using LeForg the pure of some weight is automatic, but I'm going to assume it explicitly. And in the at least only in the examples I'm going to talk about will know the trace function will know what the traces of Frobenius are at any point in any finite extension in this local system. And now it means that we can compute it and some we can compute it on a computer. Have a formula in that sense. And at least in the cases I'm going to talk about the trace will a priori line, a cyclotomic integer ring. In fact, the only P part will be that we have a piece of unity, and then we have some root of unity, and we can call q minus one to being in a power of P and so but basically cyclotomic integer ring where the P part is just from P through its immunity. Okay, so then some general business. Whenever you have a local system on on a curve over a finite field. And you look at its determinant so that's a rank one thing. And because you started over a finite field, this rank one thing geometrically is a finite order. So we know that. And so if we do what a suitable constant field twist. We can make this determinant and not just geometrically a finite order but arithmetically a finite order. And the cases we're going to look at. Are when the, the way you the constant field twist you need to choose, you need to twist by is just the correct power of the square root of q, the power of the square root of q that makes the way come out to be zero. This twist by the right power of the square root of q will always make the thing pure of weight zero, but it may or may not make the determinant arithmetically a finite order. But we're going to talk about cases where the square the correct power of the square root a few does the job. So assuming that we're in that case, then we can pay attention to the, the geometric monodromy group which means the zariski closure of the image of the geometric monodromy the image of the pot by one of the geometric curve for this local system. Of course, the geometric group doesn't care whether you did the twist or not. But what does care a lot about what twist you did is the zariski closure of the image of the, the arithmetic pie one, the literal pie one of the curve over the finite field. And basically by by the global version of the local monodromy theorem, this g geometric group is actually a semi simple group meaning its identity component is semi simple. And that's, that's the theorem that if you if you start, say with a geometrically irreducible local system. So the tautologically this g geometric is has given with a faithful irreducible representation so our priority it's a reductive group. And what's keeping it from being semi simple is that maybe you could have a unified radical. And the, so to speak, the global version is that the unified radical is trivial. In other words, if the group is reductive and it's semi simple. And then you look at the relation between G geometric and G arithmetic. So if the smaller group. The arithmetic group is finite since it contains the geometric group the geometric one is finite. And conversely, since the arithmetic group normalizes the geometric group. So the arithmetic group itself is going to be finite because you look at how it normalizes and the information you lose is the center has to be scalars because we're irreducible, but there can't be many scalars because the determinant is arithmetically a finite order. So the situation is that we have these two groups. One sits in the other but they're finite or not together and the the finiteness that's most accessible is that of the arithmetic group. There's a so to speak number theoretic criterion that the condition for the arithmetic group to be finite is that all the traces offer bane is traces are algebraic integers. So it's obvious that if the if if the group is finite, then what we're looking at is bane is elements are these conjugacy classes in this finite group. So of course they have traces which are some kind of cyclotomic integers because they're they're traces of elements of a finite group inside an ambient GL slightly subtle part is that if in fact these for bane is traces are all algebraic integers, then then you have finiteness and the argument there is that if these Eigen if these traces are algebraic integers, then the for bane is Eigen values are going to be algebraic integers, but by purity these algebraic integers are going to be roots of unity. So that means that every for bane is has Eigen values that are roots of unity. Now where are these roots of unity. Well, this a lot of system lives in some finite extension of ql so in some usual a lot of fields. The characteristic polynomial for bane is has some the rank of the system degree and over that field. So these Eigen values are in a fixed finite extension of this a lot of fields so they're in a fixed other a lot of field but the roots of unity so the roots of unity of some bounded order say 700 just to pick a number. So now you know that the 700th power of every for bane is is unipotent. Now the for bane I are dense. So you know that every element in your arithmetic group has the property that it 700th power is unipotent. Now the arithmetic group contains the geometric group. So in the geometric group the 700th power of every element is unipotent. But the geometric group is a simple group. So I'm in a maximal Taurus, you would know that every element has 700th power which is unipotent, which means there is no maximal Taurus, or rather it's trivial, and then the geometric group is finite and therefore the arithmetic group is fine. So that's how you go back and forth. And now, if we look at what the trace is in terms of our local system before we did the twisting. We're computing that trace which is some kind of cyclotomic integer. And then for that by the twist we're dividing by this power of the cardinality of the finite field. So now that that power is just it's some power of p. So in other words, what you need is the p integrality of these divided traces. So the condition for the finite g arithmetic or probably finite g geometric is it's a piatic condition on these traces. So if you're in a fixed number field and for each piatic place of that number field, you need, you need this inequality. So step one and the sort of general program I want to describe is, okay, find interesting local systems. And when you find one. The arithmetic group is finite or prove that it's not. In other words, decide if it's finite or not. And step three, if it's finite figure out what what the groups are. If it's not fine and also figure out what the groups are. Okay, so it's sort of a three step program to keep you busy. All right, so now I'm going to give three different instances of some interesting irreducible local systems on some open curves. So, in general, we're going to, it's going to, and we're going to have systems whose trace function involves an additive character psi and a multiplicative character chi. At the beginning of the discussion will fix a non trivial additive character of the prime field. And we'll make it a lot of just by embedding the field of P through its immunity into ql bar. Then we get this so called orange wire she felt psi. It's trace function on finite extensions is you just do the obvious thing. You compose your additive character with a relative trace and that gives you an additive character of the bigger field. And that's how you get values. One extension fields and Chi is a multiplicative character of some multiplicative group of some finite extension. This gives you a comer she felt some Chi. And again, in a completely analogous way you get the traces in extension fields by composing with the norm. You have a character of the of FQ cross and you compose with the norm you have now a character of the K cross. Okay, so when I write formulas. That's the sense of the psi and the Chi. Okay, so let me start without being very specific here. If if we're on the multiplicative group. We have these so called hyper geometric shapes, which are described by a list of upstairs characters. Another list of downstairs characters there should be fewer downstairs and upstairs. And the condition on for irreducibility. Is that nobody appears both upstairs and downstairs. And it can be quite delicate to decide if one of these things has finite monadromi or not. But there's an a priori statement that if it's going to have finite monadromi, then using the correct power of the square root of Q will be the right thing to have used. And meaning that the, if you do this twist, the determinant will be arithmetically a finite order. But that doesn't have to be true. That's not a generally, that's not a statement that's true in general for hyper geometric shapes. Okay, and so there's a tremendous amount that we know about these, which I'll only refer to implicitly a bit later. So second example on the affine line. We can look at local systems right now I'm going to tell you explicitly what the trace function is. So this is the trace function. From a fancy point of view, it's the Fourier transform of an L psi of f of x tensored with an L chi of x. But because it's the Fourier transform of some rank one thing, it's geometrically irreducible for free, if you like. At least if we restrict the chi to either be the trivial character, which is saying, don't write it down at all, or the quadratic character, which only makes sense if we're not characteristic, then we know that. So this is going to be weight one, and we divide by root Q. Then that's the right thing to have done. Again, it doesn't have to be always true, it could be it can be false, even if it's x squared plus tx and you take some other chi. The arithmetic determinant won't be right, won't be a finite order if you use root Q. So that's basically saying that the product of Frobenius eigenvalues doesn't have to be just the power you think of root Q in general, but it's okay if chi is trivial or the quadratic character. And then a third case that one might try to look at is So take a hyper elliptic curve of this y squared equals a nice polynomial like this. So what I've written is the equation of the curve minus the single point at infinity. So it's pi one is a free group on two G generators. They occur in pairs, the product of their commutators is what local monodrome is around this missing point. And so the affine coordinate ring of this curve is just polynomials and x and y, but subject to the relation y squared equals this F. So the the the regular functions are either polynomials and x or y times polynomials and x. And if we want to have something that's actually on the on the curve and not just a trivial pullback, then we should have a y factor. So that's why I write y times g of x. And again, for essentially trivial reasons, this is going to be a geometrically irreducible local system. And again, if we at least if we're careful about either having a trivial character or quadratic character, we can use root q. And it doesn't it doesn't matter what character you take in general. Okay, so before going on, I want to talk about an open problem almost the kind of computer science problem. And we want to decide if the these groups are finite. We have to look at Frobenius traces. And we want Frobenius traces after the twist to be P integral, equivalently to the algebraic integers and the question is, so you start computing, which is say over various finite extensions of the field, you started life over you compute all the Frobenius traces. If they're all P integral. And so the theorem is that if, if every single one of them is P integral, then you win. Now, in practice, if you just have some local system and you're thinking, well, I wonder if it's possibly finite, I'll do some computations. So you compute for a while. And gee, all these prices are coming out to be P integral. Maybe it's that you're in characteristic two and you've gone up to the field of two to the 20th elements, that's a million. And say, well, you know, how could it not be finite. If you if you have that much data. So, okay, so here's a theorem. But it's a useless theorem, let me explain. We looked at these situations where we're over a fixed finite field and either we had a hyper geometric sheaf where the characters were ordered dividing q minus one and they're at most and upstairs and fewer than and downstairs, or we had a polynomial of degree something and maybe a character, or we had y times g of x. G had a bounded degree the genus is bounded says only only finally many things to look at. And what I'm saying is that there is some bound so that if you've gone up far enough. And then you know they're all going to be algebraic integers. And the proof is kind of idiotic. So they're only finally many of these systems that you're going to look at. Now, some have finite arithmetic group and some don't. And for each one that doesn't. You know that you're going to get an unintegral trace somewhere. So look how far you had to go for each of them to get an unintegral trace. Okay, the worst degree that you needed to get an unintegral trace will tell you well if you've gone that far and you haven't gotten an unintegral trace, then you're not in one of these bad cases, and therefore you win. But as I say it's useless, because I mean implicitly it's saying, you know in advance what the answer was. Okay, so the real question is. In terms of the input data of the size of the field rover and and the degrees of some auxiliary polynomials. How can you can you explicitly bound this this this thing in terms of the input data. I mean, even something completely horrible triply exponential something. The ideal thing from a computer science point of view would be have a polynomial in in something like n times law q basically the number of bits in the input data. And boy do we have no idea about this. So one's experience is that. Well, so in this in this collaboration with with roughly on and and yeah. The experience has been that if we have a system which we suspect might be finite. And we do some calculations of p integrality of traces, then either. We can trace right away which is say we get an unusual trace right away, or we, we compute for well, and it looks very good. And then we ask Rojas Leon, you can actually prove this, and he has 100% accuracy rate. And giving him things which he then is able to prove are actually finite. Okay, so let me now talk about what we know, and what we don't know. So let me just be briefly talking about the hyper geometric case, where, so I can summarize it by saying that the three of us. We for instance so let me let me start by saying an initial sort of interest of looking at these things was to wonder if we could get any of the sporadic groups as monitoring groups for the hyper geometric sheaves. So we've first of all figured out which of the sporadic groups could possibly occur and I'll explain in a minute what what could keep them from occurring and every time we can prove. Well, every time. It's not on the list of exclusions, then we've been able to find a hyper geometric sheet that realizes it. And the other thing we've done is looking for finite groups of lead type. Which ones can you get, and if you can get them. We got them. So let me just let me just say a little bit, because it's kind of cute about the, the sporadic group case. So anybody who knows what a sporadic group is. You know, you say well I'm interested in hyper geometric sheaves and getting interesting monitoring groups, right away I say well how about the monster. So the problem with the monster is this, it's smallest dimensional representation is something like 1986, 63. So we would need a hyper geometric sheave which upstairs has a hundred and ninety some thousand characters. Okay, now when you have a hyper geometric sheave. Unless all the upstairs characters are distinct from each other, you can never have, it can't have finite monitoring because already you'll have non trivial Jordan blocks. If a character upstairs occurs a few times, there'll be a Jordan block of that size. So you need all the characters upstairs to be distinct. Now, if all the characters upstairs are distinct, then there must be the order of the local monitor me. The generator sustained situation has to be at least that hundred ninety six some thousand, because if it were lower than all the characters would be of order, dividing that lower number, and they can't then all be distinct. Okay, so we would have an element in the monster whose order is 196,000 some. The biggest order of any element in the monster is 100 and something. So, you're not going to get the monster. And the same condition. I mean what it says is that if you're going to get. Well, if you're going to get any group in in saying an n dimensional representation from a hyper geometric in the n dimensional representation, there better be an element of order at least then. So that's. That's pretty restrictive and it rules out a lot of the sporadic groups, and there's one sporadic group that isn't ruled out by it. The Higman Sims Higman Sims group gets ruled out for another reason which I won't go into. So, but the remaining ones. The one that we haven't ruled out. We find things that realize them. Okay, and maybe I'll say more about this later. Okay, so let me talk about the case on the affine line. So, so here we're looking at some of rex psi of f of x plus Tx with a character. And if you say well what happens. If the input is some general polynomial laugh. The answer is, we don't know we know a few. We know some families of polynomials which have this extra and very special property that all the coefficients of the polynomial to separately be independent variables. So you have a big multi parameter family. If that multi parameter family has finite monitor on me. Then of course the polynomial you started with us just by specializing the coefficients to the ones of your polynomial. And we do have. We do have a complete understanding of that situation, but that almost never, so to speak statistically it almost never applies. And, but right now, I think it's fair to say that in the general case, that's the only source we have f's with more than two terms in them that we can prove something, but the case where we can prove something is if if this f is just a single monomial. So it's just a power of x plus Tx. Okay. And so I'm going to tell you exactly what happens, but it's hyper geometric sheaves which are hiding behind these things. Because so let me just say in words, you have one of these hyper geometric shapes, the upstairs characters, which, if we want finding us have to be all distinct anyway. They have some common order, say, capital A. If you do the Cumber pullback by capital A power to this hyper geometric thing, then you've trivialized the local monodromy at the origin. In other words, you've created some kind of local system on the affine line. And this kind of local system they're running down here with x to the A plus Tx with a character even on. There are local systems that have the property that they're in fact suitable. Cumber pullbacks of hyper geometric things and it's really from the hyper geometric story that we know that we're able to prove the things I'm going to say now. But the statements are sort of nice in the right way. So there are two cases in this x to the A plus Tx with the character where we get sporadic groups. You know where these numbers come from, it's sort of a mystery. So in characteristic five you can have x to the seven plus Tx no character. And you get this group called 2J2. So it's a finite subgroup of the symplectic group sp6 and it has various special properties like it mocks the full sp6 and having the same second, fourth and sixth moment. Okay. There's one other in this x to the A plus Tx with a character case where we got a sporadic group characteristic three x to the 23 plus Tx with a quadratic character, we got this Conway group co3. And now this 2G2 case was discovered by Rojas Leon. He was looking for when x to the A plus Tx without a character was going to be finite. And he found this case doing a computer search and checking integrality empirically first. He found this case right away. And then I think he went up through characteristic 11 or 13 and degrees up to a million and he didn't find any more. And then he figured out a proof technology where he was able to prove that the group was actually finite and then we were able to prove exactly what it was. But this was one of the starting points of the realization that you could get interesting groups with kind of accessible, easy to remember local systems. And then, of course, a natural question. If you're looking at this, well, you know, what else can you get and the answer is, therefore, if you like for infinite families, you can have the degree is q plus one over two, and you're talking about SL2q. You can have degree to Q minus one in the quadratic character, and it's the alternating group of size to Q, you can have these funny fractions. Qn plus one over q plus one where the end is odd. So the q plus one divides Qn plus one, and the character has to have ordered a binding q plus one. And you're talking about the special unitary group SUNq. SU3, too, is a little tiny group and the statement would be a little different for that. And then there's when the degree is q plus one. So this goes back to the actually the mid 1980s, Dan Kubrick had a technology using what became what we started to call his V function, actually he called it as V function, and he proved fineness, for instance for this. And at the time Richard Pink was spending a year or two in Princeton, and he came up with the proof that it would be a P group. And then much more recently, I asked my guest then student, Will Sawan, if he could figure out exactly what this P group was, and he figured it out that it's actually the Seisenberg group. And I think, too, it's a little bit different. And then there's this very degenerate case where you think it's you think of one plus p to the app as one plus one by taking F equals zero. And then it's just, it's the cyclic group of order p. So if you like it's the, it's the Heisenberg group of order pq squared where q is one. Okay, so those are the infinite cases. And then we also determined what the group is going to be if it wasn't finite. So the idea is you look, you look at your A, you look at your character, you look in your characteristic so that, and you have to look at powers q of p, and you see if you're on this list if you're on this list you're fine if you're not on this list, you know you're infinite so what are you. Okay, so the statement is it's very simple it's a symplectic group. If it's x to an odd power and there's no character. It's a diagonal group. If it's an odd power. And you have the quadratic character, except there's something special with seven, because when it's seven. It's G to which is this exotic subgroup of so seven. And if a is still if it's odd but but you haven't used the quadratic character or the trivial character so you're not in one of the first two cases. It's SLA. And it's SL sub a minus one, if a is at least four. And then there's some special case if a is to and then I'm not trivial character, but basically the point is we know exactly what these groups are, and in some sense, this list is kind of disappointing because so we're getting the standard standard classical groups on orthogonal groups and plectic groups special linear groups. One, I'm exceptionally group G to sort of disappointing that we didn't get any elder exceptional groups, but that's life. Okay. So in the hyperliptic case can be described very quickly we don't know anything. Can we even can we get by the kind of baby systems I wrote down psi of y times g of x plus t times x maybe with the character. Can we even get finite groups. Or suppose you just to fix ideas take. Take g of x. A fixed polynomial g of x, but vary the polynomial f defining the hyperliptic curve. So if you like reach different hyperliptic curve with a fixed psi of y g of x plus tx with character. So each one you're getting some group but does the group know so to speak which curve you're on. If you got if it came out finite for one should it be finite for the others or is that something special about that curve. We just don't know anything. Okay, so let me summarize by saying much remains to be done. And I thank you for your attention. Yeah, may I ask a couple of naive questions. Please. So you said that Leon had methods of proving finiteness, can you say a little bit more about that. And in particular, I'm wondering, again very naively if you look at the so called little crystalline comrades of these representations which should be accessible since they're all so explicitly constructed and possibly relate stuff to peak or richer. So, let me answer the first question first. Right. So, in the case of hyper geometric sheaves and also in the case of this psi of a polynomial. If the polynomial has all its coefficients independent variables so it's a multi parameter system. In terms of hyper geometric sheaves in both cases, you have a criterion, which is completely in terms of pediatric ordinals of various products of galsons galsons with different characters. Well, what Kubrick realized in the mid 80s was that if you use. Well, Kubrick just use sort of formal properties of words of galsons that followed from classical identities of galsons to be able to prove his thing. There's something called Stickelberger's theorem, which is a formula for the pietic order of a galson in terms of some. I'm hearing some strange noises. Am I, can you hear me in there. We can hear you. I hear the noise. Yeah, I hear you because the noises are minimal. Don't worry about it. Okay, fine. So, basically, the Stickelberger ideas this. You take your multiplicative character of an FQ. And you express it as a power of the type of the character. So that power is a number of my Q minus one. And you, of course, represented as an integer between zero and Q minus two. The Stickelberger formula is in terms of the pietic digits of that integer. Okay. And what what Rojas Leon has is a, if you like it, an incredible ability to turn this. Stickelberger formula, which expresses things in terms of digits and see what happens as you. As the finite field grows in such a way that he can get statements which in every case where it succeeds, he manages to find an inductive structure. So to speak with these digits that allows the, the finiteness to be proved. But we've, he and I have discussed whether, whether you can actually describe it as an algorithm. So relative to the relevant to this computer science question. And the answer so far is no that every time you sort of speak never know when you're going to find a point where you can start having an inductive structure. That's how he describes the situation. So while it seems like it should be true. It's just wide open now. And now the second question. The answer is. Hey, I don't know, and be. It never occurred to me to even think about this. But it's an obvious question to ask. And I don't know. Any other questions. If not, then let's thank the speaker again.