Kiran Kedlaya (March 24, 2020)

R. Pries: Great okay, so I’m very happy today to introduce Kiran Kedlaya, who’s talking about the Sato-Tate conjecture and it’s generalizations. Oh and Kiran, is it okay if we post your talk on Youtube? K. Kedlaya: Yes, it is All right, so let’s go ahead and get started. So, thank you to Rachel for having the foresight to start a seminar that could handle a global emergency, although this is maybe not the one she’s thinking of, and Drew for helping get us over the 100-participant barrier I see there are about 150 people signed in. So it’s wonderful to see so many people gathered in a virtual sense Let me start by saying that you should have on screen my slides You should have the screen share visible that shows the title slide I should also mention that you can download these slides either from my website, which is listed here in the middle of the slide or from the VaNTAGe seminar webpage.

01:09 - You might want to do that in case you want to look at a different slide than the one I’m talking about Or you can also get the slides in the handout mode where the overlays don’t appear if you prefer to look at them without the overlays I’m gonna display them with the overlays, but if you prefer to look at them without, you can download from either of those places, the handout version One thing you might do with those slides is click on photos. So you’ll notice that I’ve embedded a thumbnail photo in here. This is me, obviously It’s a link So if you actually have the slides available, I don’t know if you can see my mouse over here, that if you click on this it should open in your browser to the original where I pulled these off of the Internet Don’t copy these without figuring out whether you have the right to do so But I figured this is a nice way for for me to show you what all the people actually look like in this talk Okay, so let’s move on then. There are going to be five sections My plan is to pause for questions after each one although you can also interrupt with a question in the chat at any point. I can either take them as they come in or I can take them to the breaks But let’s go ahead and start with section one.

So, let me start with a quite elementary 02:39 - problem and describe a hopefully familiar fact from algebraic number theory that gives you the solution for this problem So let f be a polynomial with integer coefficients. I’m going to assume it’s square-free, so it has no repeated Irreducible factors, and it’s primitive, that means that Its coefficients have no common divisor greater than one Let d denote the degree and to make this interesting, let’s make sure d is nonzero, so it’s a non-constant polynomial. I think everything I’ll say is going to remain valid for the constant polynomial but let’s not mess with that So for p prime Let N_f(p) be the following. So I’m just going to count the number of residue classes mod p for which when you evaluate f at that residue class you get 0 mod p So in other words, the number of elements of the finite field F_p at which when you evaluate f at that element you get zero in the finite field So because of my conditions, this will be some integer in the range from 0 to d. When I reduce the polynomial mod p, I get a nonzero polynomial and it’s degree is at most d So it has most d roots in the finite field But the question I want to ask is if I fix f, this integer polynomial, and I fix, say an integer i, and I look at primes for which N_f(p) = i, well what is the probability that this happens for a given p? The primes don’t form a nice probability space So to clarify this let me define this in terms of natural density Let pi be the prime counting function, so pi will count all the primes up to N and so if I take all the primes up to N for which N_f(p)= i and I divide by pi(N), this is the proportion of primes up to N for which N_f(p) = i.

And so the question is, does this 04:49 - quantity have a limit as n goes to infinity and if so, what is it? Now in all the examples I’ll describe, the limit does, in fact, exist. But I should emphasize if you’re trying to prove a theorem about something like this, proving the limit exists is an important step. It’s not automatic that such a limit exists So to remind you of a trivial example if f splits into a product of linear factors then with finitely many exceptions when you reduce mod p, you’ll see each of those linear factors and each of them will have exactly one root and those roots will all be distinct modulo p. There might be some exceptional primes for which they come together, but since the polynomial is square-free, generically, they won’t come together. And so you’ll get exactly d roots every time That’s a slightly boring example, let’s get to a slightly more interesting example, let’s look at a quadratic polynomial So if I take this quadratic polynomial and again, I’m assuming it’s primitive ax^2 + bx + c, where a, b and c have no common factor greater than one Consider the discriminant b^2 - 4ac, and let’s ignore p = 2 in what follows because square roots mod 2 are a little bit anomalous So then the count just comes down to, by the quadratic formula, counting the number of square roots of the discriminant mod p and so that’s equal to 1 if the discriminant is 0 mod p and it’s otherwise 2 or 0 according to whether the discriminant is a quadratic residue mod p and nonzero, or not a quadratic residue mod p So this middle case occurs only for the primes dividing the discriminant And of course, there’s an exceptional thing that happens if the discriminant is a perfect square then the polynomial is actually reducible.

But if it’s irreducible 06:43 - then N_f(p) takes the value 0 and 2, each with probability ¹⁄₂. And that follows from, well on one hand, we know quadratic reciprocity implies a formula for N_f(p) in terms of the reduction of p modulo something, basically the discriminant, possibly times 4 And then, on the other hand, Dirichlet’s theorem plus the Prime Number Theorem implies that you have uniform distribution among all eligible congruence classes modulo that magic modulus. So for example when delta = 5 then the quadratic reciprocity gives you a 5 on the bottom and so N_f(p) okay, except for the cases p = 2 and p = 5, you get 0 or 2, depending on whether or not, p is a quadratic nonresidue or residue mod 5 So that’s a clean answer. It clearly implies the probability statement But of course it’s not typical that you get a clean answer And so this is where the the algebraic number theory enters the picture. So in general, the way the answer is going to be formulated as best you can, is to take a splitting field of f and define its Galois group to be G There’s Galois So now the the kind of answer that you expect to get, for the structure of N_f(p) depends a lot on the shape of the Galois group And so the best possible answer you might think is the one that looks like one in the previous slide, where N_f(p) is determined by the residue of p modulo some fixed modulus That can only occur when G is abelian.

And so 08:25 - that comes down to Artin’s reciprocity law This is Emil Artin, not Mike So the Artin’s reciprocity law implies that when G is an abelian Galois group then you do get a formula that generalizes the one that I showed on the previous slide But in general you can’t necessarily give such a clean formula for N_f(p) in terms of p, but you can still answer the probability question So the probability question has an answer that you can describe in the following way: My polynomial f splits completely in the splitting field L by design Let alpha_1 through alpha_d be the roots, remember f is square free, so these are actually distinct algebraic numbers and G acts on them by permutations And so now what I’m going to do is I’m going to answer my probability question in terms of a different probability, a more sort of combinatorial probability question Namely let’s look at a random element of G So G is a finite group so it makes sense to pick an element at random just from the uniform distribution and Let’s look at the probability that a random element of G has exactly i fixed points for its action on alpha_1 to alpha_d And so what the theorem of Chebotarev says is that, it says more than this, but it implies in particular that the probability that N_f(p) = i is computed by this this quantity c_i, this group theoretic quantity that sort of detects fixed points Quick consistency check: notice that N_f(p) cannot be d-1 Maybe I shouldn’t say never here Maybe we have finitely many exceptions because maybe one root is double But in any case, certainly N_f(p) = d -1 with probability 0 and that’s consistent with the fact that a permutation on d letters cannot have d - 1 fixed points: If it has d - 1 fixed points, then the other point is also fixed So that’s a quick consistency check Another fun fact that you can extract from this is that If d is is very very large and the Galois group is as large as possible, the symmetric group on d letters, then the probability that N_f(p) = 0 tends to 1/e This is a consequence of the derangement formula which tells you what the probability of a random element of the symmetric group has no fixed points And if d is large then this probability is computed by something the c_i which is approximately equal to 1/ e But of course there’s more in the Chebotarev density theorem than what I just said, so let me peel off another layer if I look at prime powers, I can also look at N_f(q) So N_f(q) counts the number of elements of F_q for which f(x) = 0 This of course agrees with the previous definition when q is a prime And the same theorem implies If you just apply it in a different way, it implies that You can do the following thing So let’s consider the probability that N_f(p) takes a certain value N_f(p^2) takes another certain value and so on So I fix this sequence c_1, c_2, etc. And I want to consider the probability that for a given prime p Counting roots over that associated sequence of finite fields gives me this sequence of values and that probability is again computed in a group theoretic way it’s computed by taking a random element of G and determining with what probability g has c_1 many fixed points, g^2 has c_2 many fixed points and so on So this more refined separation also has a group theoretic interpretation and it asserts more information than the previous version because for example if G is the symmetric group S_5 acting on five symbols then a 5-cycle has no fixed points and this combination (1 2 3)(4 5) is a 3-cycle and a 2-cycle, it also has no fixed points But these are obviously different conjugacy classes in S_5 - they have different orders This is order 5, this is order 6 But before I was lumping them together in the same category because they are permutations with no fixed points But this count actually separates them If I take g^2, this permutation squared still has no fixed points, this permutation squared now has 2 fixed points in fact if you think about that for a while, I’ll come back to this on the next slide, but So let me before I come back to what I just said about symmetric group Let me tell you the correct statement of the Chebotarev density theorem I’m gonna say this in a slightly more complicated way than you might be used to but this is a prelude for what’s going to happen later So let’s view G as a probability space for the uniform distribution I already did that But now I’m going to consider the set of conjugacy classes of G as a probability space For what’s called the image distribution or the pushforward distribution So concretely that just means each class is weighted proportionally to its size But I’ll show you where the image or pushforward terminology comes from later on the slide I have a measure on the set of conjugacy classes of the Galois group and then for each prime except for ones that ramify in the splitting field, I can define an associated Frobenius conjugacy class So what this is is you pick a prime ideal of the ring of integers of L above this p and that defines a Frobenius element It’s the thing that induces the absolute Frobenius on the residue field But because I want this to be associated to the prime in Q not the prime in L I don’t get a well-defined element, I only get a well-defined class So here’s my well-defined class and I get one of these for each prime number And so I get a sequence of classes and the theorem of Chebotarev says that these classes are equidistributed for this image distribution Now, what does that mean in a fancy way What it means in a fancy way is that if I take a test function on the set of conjugacy classes, It could be real or complex valued, it’s equivalent, and then I take the average of the test function over the Frobenius classes for all the primes up to N and Then again take the limit as n goes to infinity That’s just the same thing as integrating the test function with respect to the measure And notice that this integral here could be computed by pulling g back viewing it as a function not on conjugacy classes but on actual group elements So I could pull the test function back to the group and integrate it there and I would get the same answer So that’s why I’ve been calling this the image or pushforward measure So that’s a fancy way to say something much more concrete which is well if I just take the characteristic function of a singleton set, I recover the usual statement of Chebotarev density which is that each class occurs in proportion to its cardinality So that’s probably the version you’re familiar with, but I want to emphasize while it’s still a kind of an elementary statement that this statement translates into something that looks less elementary But this formulation is precisely what’s going to be needed when I get into the situation where I want to do something similar with an infinite group So I’m gonna have something like G later that’s not a finite group and I’m going to have to use this kind of measure theoretic formalism to talk about distribution All right, so one final note and then I’ll maybe pause for questions So, let me come back to what I said about the symmetric group For an element of the symmetric group counting the number of fixed points of the element and its square and its cube and so on is the same thing as specifying its cycle structure So that’s an easy exercise if you haven’t thought about this And the cycle structure has the same information as the conjugacy class So if G is the full symmetric group then the final version of Chebotarev density is the same as the second version the one that counts powers But for a general subgroup of S_d even if you assume it’s transitive there’s a well-defined map from conjugacy classes of G to conjugacy classes of the symmetric group, but it’s not in general injective This map experience is what’s called fusion where classes in G come together in S_d Just as a trivial example if you take a cyclic group acting by the regular representation then Of course, it’s abelian. So it has no crushing of conjugacy classes the classes of the same as the group itself so every element is unique in its class but in S_d all the generators become conjugate to each other So there’s definitely fusion going on And so the point here is that the final version of Chebotarev density is carrying more information than even the second version So we’ll see something like this later where the density statements we want to state there’s a sort of natural way to formulate them which is kind of easy to read but is not, in fact, the most intrinsic formulation The more intrinsic formulation will be some statement directly on classes of the group itself All right, so that’s a section boundary. So let me pause for questions. I don’t see anything in chat Let’s see, maybe give 30 seconds to see if anybody wants to unmute and ask a question R. Pries: It seems like it’s going really well. I like the example K. Kedlaya: Okay, I’m not seeing any questions yet, so maybe most of you already knew that Let’s continue into some possibly newer material the Sato-Tate conjecture for elliptic curves.

Many of you have probably seen this also, but 18:56 - This talk is meant to provide set up for the next ones in the series I deliberately am doing some review here Let’s step things up a little bit. Let’s talk about elliptic curves now. Let me stop working exclusively over Q Let me think about a general number field K. so E is gonna be an elliptic curve over an arbitrary number field K I’m going to write o_K for the ring of integers of that number field and I’ll write \mathfrak{p} for a prime ideal of the ring of integers And I’m always going to assume that p is being chosen away from the primes of bad reductions So there are only finitely many primes at which E does not reduce to an elliptic curve - they divide some discriminant And since I’m asking statistical questions, I’m asking questions about averages as you consider all primes up to a bound that’s going to infinity It’s completely harmless to ignore finite sets of primes So I will just ignore primes of bad reductions throughout this talk If you actually care about computing special values of L-functions then you mustn’t ignore bad primes But in this talk, I will So I’m always going to assume p is a prime of good reduction and The L-polynomial of E at p is the polynomial that contributes to the L-function It’s a quadratic polynomial with The quadratic term is just the norm of of p and so the only interesting term is the linear term the constant term is 1 the linear term carries the information of the number of points on E in the finite field o_K/p So this q + 1 - #E(o_K/p) is the so-called trace of Frobenius of the elliptic curve There are examples of this dating back to the final entry of Gauss’s notebooks In the final entry of Gauss’s notebooks if you massage it it gives a formula for L(E_p,T) for this particular elliptic curve y^2 =x^3 - x over Q(i) And it does it in terms of quartic reciprocity Now, of course with a modern optic we recognize this as an example of elliptic curve with complex multiplication by this imaginary quadratic field and Eventually, this was generalized by Hecke to an arbitrary elliptic curve with complex multiplication over whatever number field And this description involves what are called grossencharacters Essentially involves class field theory. So it’s related to the Artin reciprocity law that we saw earlier One point I should emphasize: so in this example, you see that the curve y^2 = x^3 - x already has a model over Q So I can ask well what happens if I look over Q rather than over Q(i) Well in general if you have an elliptic curve with CM by an imaginary quadratic field that is not contained in the base field You can see here that this can happen Then a_p(E) equals zero for all the primes that remain inert In the compositum of the base field with the quadratic field. So in this example a_p(E) = 0 for all the primes that are 3 mod 4 in Q.

This is an 22:21 - elementary fact that I think was also known to Gauss But nowadays we like to think of it in terms of ordinary vs. supersingular reduction And we’ll see how this corresponds in the Sato-Tate conjecture in a moment But let me turn to the case where E does not have complex multiplication. So this is the typical thing Most elliptic curves do not have CM and the ones that do are quite rare When E does not have CM then a_p(E) depends on p in an apparently mysterious fashion although there is some method in it because of modularity theorem But from a 19th century point of view Gauss was not going to be able to get any traction on what this number is Maybe Jacobi might have had it prior Anyway, this was not something that you could resolve by simple congruences or anything but Hasse was at least able to show that this number is bounded in absolute value by 2q^¹⁄₂, so In other words, the L-polynomial has two complex roots rather than real roots So then you can ask well If I renormalize this number to be in the interval [-2,2] If I think of E as being fixed and p is varying, this renormalized quantity is some sort of random variable And I can ask well, what is the distribution of this random variable And so the conjecture that emerged in the early 1960s by some combination of theoretical and numerical evidence it was sort of first observed by Sato and then formalized by Tate is that these quantities are uniformly distributed in the interval [-2,2] for what’s called the semi-circular measure and you’ll see why I call it that in a moment Of course in the 1960s, numerical evidence was a little bit hard to come by. There’s this beautiful talk by Tetsushi Ito that he gave some years ago, maybe five or six years, no, maybe it’s closer to ten years ago by now Let me show you his slides. So I’m gonna have to pause this screen share and move it to another window here Give me a moment to do that I highly recommend this talk for the history of the Sato-Tate conjecture This is maybe the best history of the Sato-Tate conjecture that I have seen around It includes a lot of details that are missing from the standard treatments But let me go to slide 41, so this is a page This is a letter from So Sato enlisted the help of a logician friend to do these computations This is not the distribution that I promised you, this is the distribution of the arguments of the roots If you take the roots Somebody posted that link in the slides, thank you, in the chat This is the distribution of the arguments of, say the root in the the upper half-plane This is sort of some breakdown by angles.

You see there is about sixteen hundred 25:44 - sixteen fifty data points, so that was a lot back then And you know he did some more theoretical work to try to understand what’s going on There’s this letter from Sato to Namba. Unfortunately. I don’t read Japanese very well So I’m not going to be able to parse this for you in real time but you can have a look at this at your own leisure Let me switch back to my slides Nowadays there’s much stronger numerical evidence available If I switch again and take you to one of Drew’s pages, you can see that first of all we can you know we’re not scared of elliptic curves with large heights This I believe is the Elkies’s rank-28 example Drew, is that correct? A. Sutherland: Yes, that’s correct K. Kedlaya: Yes, so this is Elkies’s famous example of an elliptic curve over Q, which has rank at least 28 And so it takes a long time for the distribution to settle But when you plot this random variable that I told you it does obligingly converge to a semicircular distribution This is pretty robust evidence for the conjecture, in case you needed it Of course it’s a theorem in this case, but we’ll come back to that Let me switch back Okay, as I just said the Sato-Tate conjecture is a theorem in the case I just showed you and in fact, it is a theorem in many cases Now if I had given this talk you know 20 years ago, this would have been completely conjectural but a lot of work has been done on the Sato-Tate conjecture and I’ll explain why any of these theorems have to do with it in a moment But for K = Q, the big breakthrough came from papers by some combination of Clozel, Harris, Shepherd-Barron and Taylor Shepherd-Barron is not to be confused his father who worked on one of the first ATMs for Barclays Bank Then this was generalized to totally real base fields by Tom Barnet- Lamb, Geraghty and Gee And then for CM fields which include, for example, imaginary quadratic fields This is famous paper of ten authors: Allen, Calegari, Caraiani, Gee, Helm, Le Hung, Newton, Scholze, Taylor and Thorne These are all results on the potential modularity of Galois representations. These are basically results that build on the set up that was introduced by Wiles to prove, to resolve Fermat’s Last Theorem I’ll tell you what automorphy modularity results have to do with the Sato-Tate conjecture a little bit later But I do want to emphasize that the Sato-Tate conjecture is in fact still a conjecture for fields that are not covered by one of these things so You know you can pick What’s a good example Q(zeta_8) or something It would be.. No maybe that’s a bad example You can pick an example of something that’s neither totally real nor CM like a mixed cubic extension Maybe more progress will be made in the coming years But we still have a lot of work to go just to prove this Sato-Tate conjecture itself Now I should mention you can also average in the other direction For example, you can fix a finite field F_q and consider all the elliptic curves over that field So Birch proved that for the prime field you actually get something that looks like converges to the Sato-Tate distribution.

29:44 - What would you have to do is you have to average over the elliptic curves for a given finite field and then take the cardinality of the field to infinity For prime powers, this is covered by Deligne’s equidistribution theorem which is part of his proof of the Weil conjectures, so that’s now the late 1970s More recently, Katz and Sarnak wrote an amazing book that gives lots of either conjectural or provable generalizations of statements like this in many cases The proofs are based on Deligne equidistribution They also talked about versions of the generalized Sato-Tate conjecture that I’ll mention in a moment Okay, so this is another pause point. Let’s see if there are questions here R. Pries: Thanks for finding those slides from Japan, those look great K. Kedlaya: I’m still not seeing any questions in the chat. Maybe as we get closer to the end, we’ll start to get some more Just a reminder, the chat window you can access you might have to hit the ‘More’ button to get to the chat window Drew and various people have been helpfully posting some of the links that I’ve been issuing as we go along so you can look at some of those even if you don’t have my slides opened Okay, so let’s now talk about a more general version of the Sato-Tate conjecture Let’s talk about an arbitrary smooth, proper, algebraic variety over my number field K Now, again, there are only finitely many prime ideals of o_K at which X fails to have good reduction I will discard those in this discussion and I will write X_p for the reduction so this will be a variety over a finite field - the residue field of this \mathfrak{p} The zeta function of this reduction then factors, according to the Weil conjectures which, according to the resolution of the Weil conjecture is the zeta function of this has this shape where it’s this alternating product that’s indexed up to twice the dimension of X Each of these factors is an integer polynomial whose degree is one of the Betti numbers of the complexification of X The constant term is one and all of the roots of one of these factors have the same complex absolute value namely q^-i/2 For example, if X is an abelian variety then If L is the reverse characteristic polynomial of Frobenius, so for an elliptic curve, this is the same L that I had before So this L_1 should be an L, I mean L_1 is L but So L_1 is equal to L. And then the other L_i’s are equal to these so- called exterior powers of L_1 Namely the roots of L_i are i-fold products of roots of L, without repetition unless L itself has repeated roots Thank you, Bjorn Yes, so this L_1 should be an L Let’s fix X and i.

And let’s do the same sort of normalization that I did before 33:24 - I’m going to renormalize so that I’m considering polynomials which have roots of complex norm one This means that if I view these as, say vectors of real numbers they sit in a compact subset of R^d And I can ask how are they distributed? Is there a measure that predicts how these things are distributed? Serre formulated a conjecture of this form in the 1990s This is very much inspired by Tate’s formulation of the Sato-Tate conjecture There are also some intermediate versions of this that I won’t get into I think maybe some of them are described in the talk by Ito Anyway Serre’s conjecture is that you can find and I’ll talk more about how you find it This is how you define it: There’s a compact Lie group G inside the group of unitary matrices of size equal to the degree of the polynomials in question such that these polynomials are equidistributed for the image of Haar measure on G via the characteristic polynomial map. So in other words these normalized polynomials have the statistics of random matrices in the group G and let me just remind you this has come up, maybe in the previous VaNTAGe series that these models where you got a model number theoretic phenomena using random matrices. These have a long history in number theory One of the earliest versions of it is there’s a famous conversation between Hugh Montgomery and the late Freeman Dyson Yes, we lost Freeman Dyson two weeks ago A rather famous conversation between Montgomery and Dyson at the IAS common room where Dyson was asking Montgomery what he’s interested in he’s talking about you know the fact that he’s considering zeros of the Riemann zeta- function. He gets his weird distribution and Dyson points out oh this comes up in physics also There’s a long dialogue between number theory and physics about the use of random matrices to model number theoretic phenomena So this is in some sense an example of that, although it’s a bit different in flavor In general in spirit it’s related and I should mention that If I take K to be Q and X to be the Spec of a number field then Chebotarev density actually implies the.. Ah I should have said you take i = 0 in this case so when X is not geometrically irreducible, the conjecture is already interesting For i=0, it basically becomes Chebotarev density You can see Chebotarev density embedded in this conjecture by taking X to be the Spec of a number field and Then what happens is the Galois group becomes embedded in a unitary group via the permutation representation and then the statement here specializes back to the statement of the Chebotarev density Okay, what’s going on for elliptic curves So how does how does this conjecture specialize back to what we’ve seen already? So for an elliptic curve with i=1 if X has no CM then the Sato-Tate conjecture is equivalent to Serre’s conjecture where the group is SU(2) So that semicircular metric turns out to be the distribution of the trace of a random matrix in SU(2) and this has other classical consequences For example if you look at the moments of the distribution which you can see in that animated gif by Drew, the moments look like Catalan numbers.

That’s also a classical fact about SU(2) 37:11 - If X has CM by a subfield of K, then Hecke’s formula implies Serre’s conjecture where the group is SO(2) So it becomes abelian. Maybe I should linger on this point for a minute Here we have an abelian group This is sort of a positive dimensional generalization of what happened we saw with Artin reciprocity, With Artin reciprocity, we saw the Chebotarev density becomes something very simple when the Galois group is abelian Well here the Galois group is not finite, but it’s abelian and you are getting some very simple recipe for the Frobenius traces Finally if X has CM by a quadratic field not contained in K then Hecke’s formula implies Serre’s conjecture where G is now a different group It’s the normalizer of SO(2) in SU(2), which is not connected. This group has two connected components One of them is SO(2) and the other one is another component where the trace is identically zero I won’t click to this now because I’m running a little bit low on time. But if you go back to Drew’s page you can see examples of all of these distributions and you can see that they’re quite different The distributions for the CM cases are much more concentrated at the boundary than at the middle except that in this third case there’s a delta spike of area ¹⁄₂ corresponding to the trace 0 and that’s very visible in the gif Now as with Chebotarev density, the true conjecture of Serre is more precise I’m gonna leave about half of this slide unmentioned. The footnotes are really just for experts Let me just say that there is a specific candidate for G This is some sort of motivic Galois group associated to G to use Serre’s terminology.

Here I’ll call it the Sato-Tate group 39:13 - There is some sort of specific candidate for G and a specific sequence of conjugacy classes in that group that have the matching characteristic polynomials and then the conjecture is that those classes are equidistributed which implies equidistribution for characteristic polynomials but is again a higher piece of information So this is not just covering my basis from the Chebotarev example where it’s easy to make cases of this You can actually make cases of this in higher dimensional geometry as well For example, it’s possible to have abelian 3-folds that have different Sato-Tate groups but have identical distributions and so you can’t see the difference from the distributions Roughly speaking, the point is that G is not just an abstract compact Lie group It’s an embedded compact Lie group. And so the distribution is measuring G with respect to a particular representation whereas this equidistribution property is measuring G with respect to all possible representations of that group So you have to choose other representations of the group in order to see the differences that you cannot see in the standard representation and You can think of this as what’s going on in the Chebotarev case as well. The permutation representation is not the Is not the only representation that you have to look at in some cases There’s a question about how this motivic Galois group is constructed. I will say a little bit about it I’m not gonna say much about it. I think maybe David Zywina have a little bit more to say when he gives his lecture.

But I’ll say a little bit in the next couple of slides 41:02 - The clarification what category of motives am I using Let me not answer that question now, maybe hold that question for the end if you could That opens up a can of worms that I’d like to postpone until the end In the meantime, let me just tell you a little bit more about the the structure of this group Because of it’s a compact Lie group, it has a connected part which is a normal subgroup And the quotient by which is the group of connected components So you have this canonical exact sequence with a connected kernel and a finite discrete cokernel This connected part is geometric in nature, it depends only on the base change of X to Qbar You can even read it off from the base change of X to C you can read it off from the Hodge structure or a more arithmetic way to do it is to read it off on the Mumford-Tate group because this conjecturally controls the action of Galois on etale cohomology. We’ll see this much more in David’s talk later in the series But I want to mention: this finite part of the Sato-Tate group is in fact a genuine Galois group It’s canonically identified with the Galois group of some Galois extension of K This description is compatible with base extension, in the sense that if K’ is a number field containing K and you replace X by its base extension then you leave the connected part the same and then you think of the Galois group of this compositum over K’ as being a subgroup of Gal(L/K) and then you just take the preimage so you take the pullback of G So you reduce this Galois group in the quotient and that gives you a finite index subgroup of the original group and that turns out to be the Sato-Tate group of the base extension. So one case where you could see this quite strongly is for abelian varieties This implies In general if X is an abelian variety then this finite extension L will contain, and it is often equal to but let me not go there, it contains the the minimal field of definition of all of the endomorphisms of F In particular, you can use this to derive upper bounds on this which Guralnick and I did building on work of Silverberg But just to give you a concrete example if you take an elliptic curve with CM by some imaginary quadratic field F and L are both going to be equal to just the compositum of K with the quadratic field. So in this case if you think back I told you that the group was disconnected indeed it’s disconnected and its component group is order 2 - it’s the Galois group of the quadratic extension that’s required to achieve this. Question: Can you say a little bit about how do you make pi_0(G) to be isomorphic to the Galois group? It needs to be some Lie group to begin with right? K.

Kedlaya: Yeah, so that has to do with 44:07 - so that gets actually to the question of how this Galois group is constructed It has to do with the compatibility of Serre’s construction with with base change So this is in the papers with Banaszak we kind of explained this in some detail. This compatibility with base change essentially tells you that you’re forced to have this stuff So you prove that there’s some finite extension of K for which the Sato-Tate group is connected and that uses the interpretation in terms of motives or Galois representations and then once you do that and you know this compatibility with base extension Then that sort of produces this interpretation of pi_0 as as a Galois group Given the time maybe I should hold Unless there’s maybe a short question about that. Maybe I should deal with more questions about that at the end But since this is a pause point, let me stop to see if there are any quick questions Okay, so I’ll continue again, please feel free to insert questions in chat as we go along so I promised you some explanation of how all that modularity stuff tells you about the Sato-Tate conjecture This has to do with the role of L-functions in the study of the Sato- Tate conjecture and this gets into Well, it starts with the proof of the Prime Number Theorem. So the Prime Number Theorem The original proof of the Prime Number Theorem by Hadamard and de la Vallee Poussin uses the fact that the Riemann zeta function has a simple pole at s = 1 with residue 1 and no other zeroes or poles in the region Re(s) >= 1 And you extract using some some basic complex analysis, this asymptotic which you can then convert into the usual Prime Number Theorem I guess you get it with summing over the prime powers as well and you You take those out because those don’t contribute to the asymptotic Establishing a power saving error term here amounts to proving a zero free region for the Riemann zeta-function You know in some right half plane which we don’t know how to do unconditionally, but the Riemann hypothesis implies that we should be able to get the error term down to N^(¹⁄₂ + epsilon) And numerial experiments bear this out. This is what you actually see if you do plots So Serre observes that similar logic applies to the generalized Sato- Tate conjecture The idea is ,say, this is a purely group theoretic statement, but it’s obviously set up to handle number theory So let’s say G is a compact Lie group and I have a sequence of elements of the conjugacy classes of G indexed by prime ideals in some number field.

47:31 - But the number field doesn’t really play a role here, except It plays a bit of a role These elements are some a priori random-looking elements that that run through conjugacy classes and they’re indexed by prime ideals. So I’m going to assume the following for every non-trivial irreducible character of the group G, I’m gonna think of that character as a class function, so every non-trivial irreducible character, viewed as a complex valued class function, I form this associated L-function. So this is like an Artin L-function except it’s not a priori to do with the Galois representation, but it looks like an Artin L-function it’s got 1 minus this character value times Sorry, this is not written correctly. I should put rho here for the associated representation I’ll correct this when I update the slides. This should really be the associated representation rho if I compute this determinant this of course does not depend on anything other than the associated conjugacy class. So it gives a well-defined formula.

But this is slightly meaningless the way I’ve written it 48:58 - Hopefully you understand what I mean You form this Dirichlet series and you want this to be holomorphic and non vanishing on Re(s) >= 1 Notice that I took out the case of the trivial character which would give me a pole at s =1 coming from the Dedekind zeta function But we already know the the analytic properties of a Dedekind zeta function So I don’t need to worry about the trivial character, I just need this for the nontrivial characters. If I know this statement, then that implies that these conjugacy classes are equidistributed for the image of Haar measure So this is a sort of template for how you can use statements about analytic continuation to prove density statements In particular it generalizes one of the ways you normally prove Chebotarev density And it is in fact the approach that’s used to prove Sato-Tate in the known cases. The known cases are theorems about potential automorphy that imply analytic continuation for symmetric power L-functions associated to elliptic curves And the symmetric powers of the standard representation are the irreducible representations of SU(2) This precisely gives you what you need in the case of SU(2) to prove the equidistribution It also gives you some idea of why the generalized Sato-Tate conjecture might be much much harder in cases other than when the group is SU(2) because SU(2) has this nice one parameter family of irreducible representations If you think about a Lie group of rank h, then it will have an h-dimensional so to speak h-dimensional family the irreducible representations are parametrized by h integers highest weights in an alcove and That there’s an explosion of cases you have to deal with that’s much much harder than just symmetric powers This gives you some idea of the limitations of what we can hope to prove unconditionally But if you’re willing to assume both analytic continuation and the analog of the Riemann hypothesis for the appropriate L-functions then you do get error bounds like in the Prime Number Theorem. And again you can confirm these numerically I meant there’s there’s some numerical calculations by Mazur and Stein that I didn’t list on the slide that bear these things out For Chebotarev density, Lagarias and Odlyzko have a famous paper that makes everything effective and then There’s work of Kumar Murty Bucur and myself, Bucur, Fite and myself that makes this more explicit I mentioned most of this largely because Alina will talk about this to some extent in her lecture in this series But now we’ve hit another stopping point. So let me pause for questions Drew is assuring me that not everybody maybe will disappear quite at 11 o’clock, but I’m going to try to wrap as close to 11 o’clock as I can to leave plenty of time for questions after so those of you can stick around for questions after I do plan to to be available for questions after but again, if there are quick questions now, please feel free to ask them, say in the chat or out loud R. Pries: It looks great Kiran K. Kedlaya: All right. So let me go ahead and wrap up.

I only have a couple of more slides 52:48 - There are three slides of references at the end that I won’t show. So I really only have a couple of slides left This is largely to set context for the talks that Drew and Francesc are going to give These are about these are about computation and classification So let me start with this a slide about classification of Sato-Tate groups The classification of Sato-Tate groups is known for abelian varieties of dimension up to 3 So in dimension 1, we’ve already seen the three cases: no CM, CM within the base field, CM not within the base field That classification is relatively easy In dimension 2 it turns out to be more interesting so Francesc, Drew and I in collaboration with Victor Rotger showed that there are 52 possible Sato-Tate groups up to conjugation That includes six options for the identity component, which I think was known previously And all of these can also occur for Jacobians of genus 2 curves or for genus 2 curves themselves There’s no difference between the Sato-Tate group of a curve versus its Jacobian In dimension 3 this is a much more recent piece of work This is actually in some ways still ongoing Francesc, Drew and I announced about a year and change ago that there are 410 possible Sato-Tate groups of abelian 3-folds which include 14 options for the connected component We have an announcement paper about this that you can find on the arXiv, again it’s listed in the references there are many questions about dimension 3 that are not settled compared to dimension 2 For example, do these occur for curves? What fields of definition are allowed? This is much more mysterious than it is in the dimension 2 case which means that there are many open questions to straighten out Some of which are being worked on and some of which are not yet being worked on Dimensions 4 and higher are going to be more complicated for reasons that Well, there are many reasons why dimension 4 is going to be complicated One reason is that the Mumford-Tate group and therefore the Sato-Tate group is not purely determined by endomorphisms There are some famous examples of Mumford and Shioda, two different sets of examples, that illustrate the difficulties involved with reading off the Mumford-Tate group from endomorphisms And so this creates some complication in the classification There are of course higher dimensional cases that are not related to abelian varieties as directly A nice intermediate case is K3 surfaces which are a little bit related to abelian varieties The Mumford-Tate groups for K3 surfaces are in some ways understood by the work of Tankeev and Zarhin Banaszak and I are trying to extend some of our previous work making Serre’s predictions explicit: For what motives of even weight which are needed to cover the case of K3 surfaces? So that’s about what I can say about classification right now There are lots of cases where the classification is very much an open problem So there’s lots of room to explore in that space. I think maybe we’ll hear more about this later in the series Let me say a little bit about Heuristic versus rigorous computation So if you have a given X in what sense can you find the Sato-Tate group or the Sato-Tate distribution? One of the motivations for this work is the fact that it actually turns out to be possible to compute L-polynomials efficiently in many cases This is a topic for a separate talk or maybe even a separate series You know there’s now almost two decades of worth of work on this topic about computing L-polynomials efficiently But let’s say you have some method for computing these L-polynomial then you can, of course, try to match the distribution that comes empiracally out of these, we had a candidate Sato-Tate group however, this you can’t really do unless a classification has already been done for you and And even then you will often have trouble matching/ pinning things down This won’t generally give you a way to prove that the Sato-Tate distribution is something other than the cases when it’s the sort of most generic, the biggest it can be given the circumstances And then I mentioned earlier the possibility that distinct groups can give you indistinguishable distributions That’s another issue One possibility is to compute more structure For abelian varieties of dimension <= 3, you can compute the Sato- Tate group by computing the endomorphism algebra There’s work on making this rigorous and practical, I should have said rigorously and practically here by Costa, Mascot, Sijsling, and Voight I should maybe also mention for other varieties that are not, I believe Costa and Sertoz have some ongoing work on doing this in cases that are not abelian varieties There’s also a method by Zywina that does not depend on a classification. David is going to talk about this later And so it potentially applies to higher dimensional abelian varieties and perhaps even to things that are not abelian varieties So I will let him go into the scope of that but it can do with higher dimensional abelian varieties than the ones for which we have a classification and it can also give you rigorous results in cases where the previous methods would have struggled to do so This is very exciting piece of work that is going to give us a lot more insight as it starts to be deployed But I’ll let David say more about that That concludes everything I said, I mentioned that I have references but I won’t go through the detail I’ll just leave the the page here,stop and then we can go to the questions R. Pries: Thank you Kiran for the wonderful talk K.

Kedlaya: I owe you a question, I have already one question from earlier as I’m doing this 59:33 - Feel free to ask more questions in the chat or you can wait a turn and unmute and ask a question out loud So the question that I postponed from earlier The question is could you please say a bit more about how this motivic Galois group is constructed, what category of motives are you using? So I should start by saying that Serre when he writes his paper, he basically puts himself in a fantasy world where all conjectures about motives are known So Serre writes his paper from the point of view of: let’s assume everything that is conjectured to be true about motives is true then all the different categories of motives collapse into one And then you can really formulate everything precisely, but all very conditionally So the approach in Serre’s paper does not actually pick a choice of motivic Galois group because he just assumes that they’re all equal If you actually want to work in the world we live in then you have to be a little bit more careful For example, for abelian varieties, what Banaszak and I do, is we work with the so called AHC motives, the motives of absolute Hodge cycles, For abelian varieties, Deligne’s work shows that this kind of does everything you want This category of motives kind of satisfies all the things you would want In other cases you have to make a choice, the choice we like is Andre’s category of motivated cycles I’m not going to say much more about that in part because Grzegorz is much more of an expert on this than me I am not a gearhead about motives, so I don’t want to try to say too much more But I will just say that Andre’s motivated cycles is a good kind of intermediate choice that balances some of the issues that come up with different categories and motives It’s less explicit than things like Chow motives, but you know, you can do more things unconditionally But for abelian varieties you can use AHC motives and then everything works without any issues Yes, the comment I should add is that the category of Andre motives is abelian and semisimple So that makes a lot of what I’m saying easier. Okay, so I have question R. Pries: Nathan, could you please read your question out loud? Nathan: Sure. Could be say a little more about the higher dimensional versions of vertical Sato-Tate problems like the one study by Birch that you mentioned Do the Katz, Sarnak and Deligne equidistribution results that you talked about explain what say like Weil polynomials of abelian varieties of dimension g over a fixed finite field F_q look like as q goes to infinity? K. Kedlaya: Yes, so that’s a great question. I should emphasize. So let’s fix g in this discussion So we’re fixing abelian varieties of a given dimension. And then yes, the Katz, Sarnak and Deligne results tell you If you take for example all possible abelian varieties of dimension g over a fixed finite field you do averaging and then you take a further limit as q goes to infinity then what these equidistribution results tell you is that you can Well, there’s a sort of meta result that Katz and Sarnak described that tells you that instead of considering arbitrary abelian varieties, if you consider some geometric family of abelian varieties, you can read off the answers that you’re going to get from the geometric monodromy group of that family In the case of the full family of abelian varieties, that monodromy group is as big as possible It’s the symplectic similitudes.

03:44 - And so that gives you the analog of Sato-Tate, which is that the characteristic polynomials look like they’re coming out of the unitary symplectic group of matrices of size 2g And that persists for various other types of families For example, Katz, Sarnak explained in their book that this also works for the Jacobians of curves which is a much smaller moduli space than abelian varieties or the moduli space of hyperelliptic curves So you can calculate many cases and in fact Rachel is an expert on this - Calculating monodromy groups in families of curves. Actually Rachel and Jeff are both experts on this You can use that kind of calculation plug it in and that gives you more or less automatically out of the Katz, Sarnak and Deligne machinery, equidistribution statements in geometric families Now if you start trying to vary g that gets into more delicate issues That gets more into the spirit of the the previous lecture series, maybe I won’t go there right now But if you have a fixed geometric family, you can do something like this. You can also talk about There are other vertical directions that I don’t want to get into right now. For example, considering modular forms of a given type and so on And so there are lots of vertical distribution results as well Serre has some results Shin and Templier have results I don’t want to go too far in this because I’m gonna start missing names and omitting people inadvertently There’s quite a lot of work in multiple different aspects of vertical Sato-Tate that I did not cover and we probably won’t see in this series, but I do want to give them a mention R. Pries: Kiran, there’s another question K. Kedlaya: Yes, Jack do you want to unmute and ask your question out loud? Jack: Sure.

I guess the question is 05:53 - What is the so called effective Sato-Tate that I’ve seen in a few papers on the arXiv K. Kedlaya: Right. So this goes back to… I can show you the slide that relates to this That’s relevant to this slide. Remember I told you the Prime Number Theorem under the Riemann hypothesis The Prime Number Theorem comes with an error bound of the form N^(¹⁄₂ + epsilon) I sold that error bound a little bit short. It’s not just N^(¹⁄₂ + epsilon) It’s N to the ¹⁄₂ times some particular factors of log(n) and loglog(n) which I won’t remember off the top of my head And you can even make the constant term explicit. It really gives you an effective bound that you can actually, for a given N, you can compute a number out of that bound and the theorem is, under RH, the number of primes deviates from, I should compare it to the Li function, to the logarithmic integral But it differentiates from Li by a bound of the form N^(¹⁄₂ + epsilon) with a specific function and specific constants These effective statements that I’m mentioning on this slide are versions of that same statement using the analogue of the Riemann hypothesis for other L functions For example, Lagarias and Odlyzko gives you an effective bound on the error term in the Prime Number Theorem and an arithmetic progression or in a Chebotarev situation which you can then use to, for example, give an upper bound on the least prime in any given Chebotarev class I don’t want to say too much more about this because Alina’s going to talk about this in her talk But then the point is you can do something similar in the Sato-Tate conjecture so you can, for example If you fix an interval inside the range [-2,2], you can ask what is the first prime for which the normalized trace of Frobenius lands in that sub interval And you can give a theorem with an effective error term for that and therefore give a prediction on how that arises The short answer is this will be it will be discussed in in one of the later talks in the series But that’s sort of a preview Okay, I see a question from Jen Johnson-Leung. Jen you want to ask your question J. Johnson-Leung: Sure.

So if I knew that all geometric Galois representations are actually automorphic 08:40 - would that sort of give you a framework for proving Sato-Tate in general? I mean I feel like those things are sort of related K. Kedlaya: Yes. So the answer is basically on this slide If you knew all geometric Galois representations are automorphic or even you know, potentially automorphic, that would imply analytic continuation for all of the associated L-functions, You have to then do a tiny bit of work to make sure that you have nonvanishing to the right of the critical strip and on the boundary of the critical strip But basically once you know those things are automorphic that is enough to tell you Sato-Tate It does not give you any sort of power saving error term because that would depend on a zero free region But if you knew automorphicity then that would, in all cases that come from geometry, that would cover all of the that would somehow cover everything… everything that can show up when you apply Serre’s logic to the Sato- Tate conjecture The short answer is yes. Knowing that everything geometric is automorphic would then imply enough analytic continuation to get you Sato-Tate Jen asks about the converse now, so that’s an even better question I may not be the best person on the line to answer this question, but I’ll do my best For example, for elliptic curves, there’s a famous result of Weil that shows that analytic continuation for an elliptic curve’s L-function and all of its twists would imply modularity Now, of course the Sato-Tate conjecture is weaker than holomorphicity - it doesn’t Imply that The Sato-Tate conjecture by itself is not probably enough to imply anything useful But holomorphicity, if you knew it in all cases, Well, sorry, I should say holomorphicity and functional equation in all cases Then you have a converse theorem, analogous to Weil’s theorem, by Cogdell and Piatetski-Shapiro that says that in some sense, for GL_n at least, I’m not an automorphic person so I can only tell you about GL_n For GLn, Cogdell and Piatetski-Shapiro showed that if you know some sort of grand statement about analytic continuation for all sorts of L-functions, for a given L-function, you need to know analyticity and maybe the functional equation for its twists by GL(n-1) representations I think So for GL_2 you get GL_1, but for GL_n you need not just character twists but nonabelian twists If you have some grand collection of holomorphicity statements like that, then you can infer an automorphicity statement So these statements exist. I don’t know what the state of the art is, and I don’t know how practical they are because it seems that in general most of the things we know about holomorphic continuation at least on the arithmetic side actually come from knowing modularity statements, but what do I know There was a non math question.

Let me just answer this 12:31 - That was a question about when this would be posted to YouTube and where we could find it and when it will be posted And so the answer is that those will be linked from the from the VaNTAGe site And it will be announced through the Google Group as well Are there any other questions about the talk? R. Pries: Kiran, I want to thank you for starting off this series so well I don’t know if you’ve noticed but I’ve gotten several comments that this was a fabulous talk through the chat window The next talk will be April 7th by Francesc Fite And maybe everybody could just unmute for a second and give Kiran a big round of applause before we stop here .