 Thanks for the invitation and the introduction. So the title is quite a mouthful, and the talk that I will give will be about joint work. And let me tell you who my co-authors are. So it's a joint with Jonas Bergström and Adrian Diakony and Craig Westerland. And there is a second paper, which is also a joint with Jeremy Miller and Peter Pappst and Oscar Randall Williams. And both papers are on the archive, and I will try to make some sort of distinction of which results I'll talk about are from first and which are from the second paper. So the subject of these two papers is moments of the Emily of Craddick L functions over a function field. So let me describe the problem. So we'll fix Q, sorry, an odd prime power, and we consider the following sum, with sum over polynomials of degree 2g plus 1, and we want these to be monic and we want them to be square free. And for each such polynomial, one can consider a hyperliptic curve with equation y squared equals d of x. And the condition that it's square free says that this is a smooth algebraic curve over this finite field. And as such, it has an attached L function. So let me call this curve cd. And what you can do is you can look at the zeta function of this variety and you look at the numerator of the zeta function and that's some polynomial evaluated at Q to the minus s. And this is of degree 2g. And the roots of this polynomial are exactly the eigenvalues of Frobenius on the first comodic group of this curve etal comodic group. And this numerator of the zeta function defines the L function. So we define this to be L of s chi d. And what we're going to be summing is the central value of this L function. So there's a functional equation that relates the value at s and at 1 minus s. And 1 half is the central value and we raise this quantity to the rth power. And finally, we want to normalize by dividing by the number of terms in this sum. Actually, it's going to be more convenient to just normalize this by Q to the minus 2g minus 1 which is not quite the number of square free monic polynomials but they differ by an easily understood factor. So we want to understand this quantity as the degree of the polynomial goes to infinity meaning the genus of the curve. Let's scroll down a bit. And the conjecture is... Let me give this quantity a name. I'm going to call this mr of g. And the conjecture that comes from Conry Farmer Keating Rubenstein-Snaith is that this quantity as g goes to infinity is given by an explicit polynomial plus an error term that goes to zero where u r is an explicit polynomial degree r times r plus 1, 2. I'm not going to tell you the formula for this explicit polynomial. It's not very enlightening when you see it. And it's a massive formula. But the content of the conjecture is not a specific formula but it's rather the recipe that they give for how to predict what are the moments of various families of L functions. So this is an instance of a family of L functions that we're interested in. And we are looking here at the arth moment of the central value of this L function. So this is just one special case of a very general method for giving predictions for what are these moments. And it's very refined information because it's not just a dominant term but all of the lower order terms. And before I go any further I should make some comments about this conjecture. The first comment is that you can derive the same conjecture by different means. And this was done by Diakonio Goldfeld and Hofstein using multiple Dirichlet series. So they gave a different way of obtaining conjectural asymptotics using conjectured analytic properties of these multiple Dirichlet series. And it gives the same conjecture. And I should say that in the original paper of Conry, Farmer, Keating, Rubenstein, Snaith they do look at the quadratic family but not the quadratic family over function fields but over number fields which would be quadratic Dirichlet L functions instead of these L functions attached to hyperliptic curves. I didn't say the word hyperliptic curve but hyperliptic curve means a curve that can be written in this form y squared is d of x. And the general recipe from Conry, Farmer, Keating, Rubenstein, Snaith was carried out in a different paper of untroud and Keating. So they worked out what should be the prediction in the function field case. And this philosophy of moments goes back to study of the Riemann's data functions. So in the case of the Riemann's data you could be interested in there's only one Riemann's data function so it doesn't really make sense to fit it in a family but what does make sense is you can study the distribution of the value along the critical line as opposed to the value at one half. So in a sense that's a continuous family of L functions. And if you're interested in the value distribution of zeta along the critical line you can study the moments like this. And a conjecture which is wide open is that it should be asymptotically equal to C, R, T, log R squared T. Where C, R is an explicit constant which was conjectured by Keating and Snaith using random matrix theory. And this recipe of Conry, Farmer, Keating, Rubenstein and Snaith can be used to give a more precise version of this conjecture which doesn't just give the dominant term but all lower order terms as well when this is given as a polynomial in the logarithm. And this conjecture is known for R equals one and two goes back to Ingham and Hardy and Littlewood but beyond that it's not known. But their original motivation was the study of the Lindelöf hypothesis. The Lindelöf hypothesis can be formulated equivalently in terms of these moments of Riemann's zeta function and that this should be bounded by T to the one plus epsilon for all big epsilon bigger than zero. Now for the quadratic case maybe I should label this conjecture star. Star is known for R up to three and partially R equals four. By partially I mean that one knows the first q coefficients of the polynomial but not the whole polynomial. This is largely due to the work of Alexandra Florian. In the number field case the analogous question for the quadratic family is also known for R up to three and the R equals three case is due to Sander Arrajan. And you might think that the number field case should be harder than the function field case because in the function field case we know the Riemann hypothesis and we know a lot more than the Riemann hypothesis. We know the Riemann hypothesis for we know the Riemann hypothesis so to speak. But previously it had seemed that the number field and the function field case were on the same order of difficulty because we are not much further in the function field case but in fact it turns out that you can go further in the function field case. And the theorem that comes out of the two papers one with Bergström, Diakony and Westerlund and the other with Miller, Pazzt and Randall Williams is that this quantity MR of G you can write this as this explicit polynomial QR of 2G plus 1 plus an error term that looks like big O of 4 to the power G times R plus 1 Q to the power minus G plus 6 divided by 12 and the point is that if Q is sufficiently large this is indeed an error term so 4 to the G times R plus 1 it grows very fast with G but the other term Q to the minus G plus 6 over 12 goes to 0 exponentially fast and if Q is sufficiently large with respect to R then this is indeed an error term fixed R once we choose Q sufficiently big this conjecture star is indeed proof that's a statement of the theorem are there questions at this point I'm going to say something about how we prove this and as I hinted in the title of the talk this is proven using homotopy theory I just feel like I don't have a clock am I sharing this so here we go so the method is to use homotopy theory and the link between topology and arithmetic is via the growth and declifted trace formula so I indicated at the start of the talk that we know a lot more than just the Riemann hypothesis we know the grand Riemann hypothesis in a sense we know even more than that because we have a full formalism of etal homology and etal sheaves and 6 functors formalism and whatnot and this growth and declifted trace formalize part of this thing will be on the Riemann hypothesis let me state the growth and declifted trace formula if you have a smooth scheme over fq of finite type v is an L addict local system on x and there's a formula that looks like this you sum over the fq points of x and you take the trace of the Frobenius on the stock of the local system at x and this is the same as due to the dimension of x times the trace keep writing here Frobenius of the homology of x with coefficients in b this is etal homology and I should make some comments on this formula if you know the v-conjectures that concerns the case when v is the trivial local system and when v is the trivial local system then on the left-hand side you're just taking the trace of the identity on the one-dimensional vector space and that's o1 so you're just counting the number of points of x over fq and the right-hand side is what gives a homological interpretation to the number of points on the variety but in this form you can also allow for a weighted account of the number of points and I assumed that x is smooth here in order to write it in this particular form which will be the way it's used in the argument if it's not smooth you can write it in terms of compact support homology but for smooth schemes compact support homology is isomorphic to homology with a tate twist and that tate twist explains why you take q to the dimension of x because that's the amount that you need to take twist and the trace on the right-hand side is taken in a graded sense so the homology is a graded vector space and what it really means is the sum of the traces and all of the even degrees minus the traces and all of the odd degrees so it's a usual super trace and so far it doesn't really link arithmetic to topology because both sides here are very much arithmetic objects but the further result is the art in comparison theorem which says that et al homology of x with coefficients in v can be identified with a singular homology with a set of complex points of x and the analytification of this local system and there's a canonical such identification here for the art in comparison theorem to be true I need x to be a subfield of the complex numbers so certainly not a finite something that's defined over a finite field so to make sense of what I'm about to say you can assume that x is defined over the integers and then if you have x defined over the integers you can both embed it into the complex numbers you can also reduce it modulo prime to get something defined over a finite field for prime power and then you're forced to ask whether the et al homology of your reduction mod p coincides with the et al homology of the thing that you get by embedding into the complex numbers that's not always true it's true for generic specialization at some prime in our setting it's going to be true at every prime and that's because the schemes we're looking at have normal crossing divisors that are smooth normal crossing divisor compactifications that are smooth over the integers but this is all technicalities the point is that for all the schemes that we care about the et al homology which is the thing that controls the number of fq points and these weighted counts the number of points can be identified with the homology of a corresponding complex algebraic variety that can then be started by topological methods and in our case we're going to take x to be the configuration space of n distinct ordered points on the affine line so this is smooth scheme over the integers and it makes sense to reduce it modular prime and it makes sense to embed it into the complex numbers and if I form this embedding into the complex numbers I get the usual space of n distinct ordered points in the plane which is a k pi 1 and it's k pi 1 for the braid group on n strands beta n is the art in braid group and this is good because if we go back to the original conjecture which is up here we were summing over monic square free polynomials of a given degree and that's exactly the same as summing over fq points of this configuration space of distinct unordered points because monic square free polynomial can be uniquely identified with a configuration of distinct unordered points namely its roots and in the growth and declifted trace formula we were summing over the fq points so it's going to turn out that the right-hand side of the growth and declifted trace formula becomes the right-hand side of the conriform or Keating-Rubinstein-Snave conjecture and the left-hand side is also identified and the particular local system we care about is going to be of the form of the exterior algebra on some other local system w where w is the reduced borough representation of the braid group specialized at t is minus 1 I take this representation of the braid group called the reduced borough representation that depends on the parameter if I specialize it at the root of unity the monodromy of this representation is related to the family of curves y to the p equals d of x so if I take a primitive pth root of unity I would get something related to that particular family of curves and when I take t is minus 1 I get exactly the monodromy of the universal family of hyperliptic curves and I should take a half integer t twist of w to be correct and the description I just gave of w is only valid over the complex numbers but it turns out that there's a natural way to define it in an analytic setting and it's what you should do and the upshot is that we need to estimate the trace of the Frobenius on the homology of the braid group the coefficients in this thing I guess I should do quite this differently what you're taking the trace of depends on r r is the moment and it depends on r in this way and why is that the left-hand side of Groten-Dick-Lefchitz trace formula gives the left-hand side of the CFK or S conjecture the right-hand side of Groten-Dick-Lefchitz is exactly this trace of Frobenius up to this factor q to the dimension of x the dimension of x is n and q to the n is exactly this normalizing factor that I divided by formulating the theorem or the conjecture and the estimate is as n goes to infinity and if you are if you come to this talk from a topology background which maybe not a lot of people are doing this immediately suggests that what you should be doing here is something about homological stability and homological stability is paradigm that's been important in topology since the 70s and applies to many families of discrete groups that are interesting to topologists let me say a few words about homological stability so you can think about topological groups or more other general families of space but let me formulate things for discrete groups suppose I have a sequence of groups like so of the braid groups or the symmetry groups or general linear groups over some ring these are all very good examples to keep in mind and you can ask if these satisfy homological stability and this would mean that these homomorphisms or isomorphisms or k less than say some constant times n then we say that the sequence has homological stability and what's interesting is that this happens for many well studied and natural families of discrete groups such as symmetry groups, braid groups, general linear groups over some rings mapping class groups moreover computing the homology of the groups individually in the sequence rapidly becomes intractable and also maybe the answer is not so structural whereas the stable homology which is the limiting value that you get when you send n to infinity is more computable and in many cases more interesting reveals new structure and I formulated this here just as homology of the groups themselves which I guess implicitly means constant coefficients but it's also interesting to study homological stability with twisted coefficients this has also been done in these classical families of groups like the symmetric groups etc etc and here we're looking at the homology of the braid groups and we're taking the limit of the homology in some sense as n goes to infinity and we're taking a very nice family of coefficient systems for which it was in fact known that homological stability would hold already before our work and it seems that what we should be looking at is what is the stable homology of the braid groups with these particular coefficients already before us it was known that braid groups have homological stability for any polynomial coefficient system I'm not going to tell you precisely what a polynomial coefficient system is and all of these exterior powers of this Bura representation is polynomial this exterior power wedge K of W is polynomial of degree K and W itself is linear polynomial of degree 1 which is good however the stable range is of the form this map is an isomorphism for K less than n over 2 minus the degree of this coefficient system and this is come from a paper of Oscar Randall Williams and they give a very general homological stability theorem that applies to many different families of discrete groups and it always gives for all these families of groups a stability theorem for polynomial coefficient systems and it always has this shape where it depends on the degree of polynomiality and now there is a problem that arises which is that here we were looking at the full exterior algorithm on W which is the direct sum of all exterior powers and these include summands of arbitrarily high degree but that's bad in this theorem because then the degree is unbounded and the effective range of homological stability is zero when you're taking homology with coefficients in the full exterior algorithm which is the thing that we want to understand so something different must be done somehow in our case where V is the full exterior algorithm on W the degree is unbounded and moreover even if we did know homological stability for these coefficients meaning the full exterior algorithm W that would give us the wrong answer because if we had stability for this full exterior algorithm to answer with itself r times then we would end up proving that this quantity of r of G converges as G goes to infinity to the trace of Frobenius on the stable homology with these particular coefficients but this is not what we wanted to prove we wanted to prove that this asymptotically grows like a polynomial we don't want to show that it converges to some number so we don't Randall Williams-Valls result we don't want to prove stability and we also don't expect stability to be true because it contradicts the CFKRS conductor so something different must be done and what is better to do is to decompose this full exterior algebra into irreducible representations for the symplectic group and where did the symplectic group come from well this Bura representation is actually a homomorphism from beta 2G plus 1 to SP 2G Z so this representation W goes from the braid group to the symplectic group and then if I decompose the exterior algebra on the fundamental representation of the symplectic group into irreducibles and the tensor algebra you can tensor with itself four times and still decompose into irreducible so you can ask about what are the multiplicities of individual algebraic representations of the symplectic group inside this tensor product and now the striking fact is that the multiplicity of a given irreducible representation V lambda in this tensor product is given by polynomial in G of degree R times R plus 1 over 2 which is exactly the degree of this polynomial in the CFK or S conjecture which is not a coincidence and I should say something about how to interpret this fact that I just stated if I fix a partition of lambda I can think of that as indexing not just a single representation of some symplectic group but actually a consistent sequence of representations of all symplectic groups simultaneously so irreducible representations of this symplectic group I will index by partitions lambda 1 bigger than lambda 2 and so on bigger than lambda G bigger than 0 now if I have a partition of some finite length by padding it with zeros I can make it into a sequence of partitions of all possible lengths and that gives me a consistent sequence of irreducible representations of all symplectic groups simultaneously and once I fixed lambda any partition I can ask about its multiplicity when I decompose this guy into irreducibles and I let the genus G go to infinity and this quantity is the thing that is given by the degree r times r plus 1 over 2 polynomial and the polynomial depends on that and you see this using how duality but this is the key point but that's great I should also say by the way that if I have G small then V lambda doesn't really correspond to representation because here I said that irreducible representations of the symplectic group corresponded to partitions of this length and if lambda is of length bigger than G then V lambda is should not be thought of as an irreducible representation so V lambda is formally said to be 0 and then the formulas that I write down later are going to make sense but this fact that is given by polynomial of degree r times r plus 1 over 2 should probably be taken to be true when G is large enough that V lambda is 0 anyway that's all technicalities so to prove more logical stability for braid groups with these irreducible V lambda coefficients and now I think I'm in a position to state the main theorems of the two papers the theorem proven with Jeremy Easter let me write it down well the theorem with Jeremy Peter and Oscar is that this stabilization map is actually an isomorphism for K less than approximately n over 12 and I say less than approximately meaning that up to a constant K less than n over 12 minus 2 or something I don't remember and the point here is that this is independent of lambda so I told you that there's this theorem of Randall Williams and Wald that gives homological stability for families of discrete groups with polynomial coefficient systems and the stability range always depends on the degree of polynomiality and it's sharp for a general polynomial coefficient system you cannot get any bound that is independent of the degree and the point here is that if you specialize to this very specific situation where you have a family of irreducible representations of an arithmetic sequence of arithmetic groups and your coefficients are taken from the sequence of arithmetic groups then you can prove a stable range for this homological stability that has no dependence on the specific representation you choose and this is the key ingredient that allows us to control the error term in the main theorem and the second theorem that I want to say which is from this Bergstrom Diaconic Western paper is the explicit calculation of what is the stable homology with these particular coefficients and I'm going to write down the formula but it's not super enlightening I'm sorry to say it's given by a generating series so I sum over all partitions lambda all homological degrees k I take the dimension of the kth homology with a stable braid group on infinitely many strands with these b-lambda coefficients and then I take a Schur polynomial in infinitely many variables I take the transpose of this partition lambda so I flip the Ferrer's diagram around and I multiply this with a formal parameter minus z to the k so I track of the partition by using these Schur polynomials in infinitely many variables and I keep track of the homological degree using this parameter z and this expression now lives in I take the ring of symmetric polynomials in infinitely many variables I join an indeterminate z and then I have to take a completion because this is an infinite sum and the claim is that this formula is equal to something that I will write as follows x of the inverse log of 1 plus z sum of k bigger than 0 z to the k h2k minus 1 minus e2 and this x and log are certain operations called the pletistic exponential and the pletistic logarithm it's not the usual exponential and the usual logarithm which were introduced by Getzler and Kapranov in their work on modular operas and these are universal operations that act on complete lambda rings and this completed polynomial algebra over the ring of symmetric functions that's an example of a complete lambda ring so the takeaway is that there is this formula that looks a bit mysterious but it's actually quite computable you can give this formula to a computer and it will tell you exactly what are the stable homologies with coefficients in various v lambdas and it will do so quite fast so anything like Sage that can do calculations with symmetric functions will do this for you and you can say structural things about what a stable homology looks like and moreover the homology the braid group with these coefficients is pure k of weight I should write k here minus k plus the size of lambda minus 2k plus lambda but the Frobenius eigenvalues are q to the minus 2k plus size of lambda so I know exactly what all of the eigenvalues of Frobenius are on the stable homology and this is precisely what I need to estimate here I wanted to estimate oh sorry I wanted to estimate this as n goes to infinity and now I know both the dimensions of the stable groups and I know the Frobenius eigenvalues of the stable groups and one can give an infinite product expansion of this x log formula and this is you prove a general identity for the plastic exponential logarithm that gives you an infinite product which matches the Euler product from the CFKRS conjecture the Euler product that comes out of the recipe turns out to match exactly this x log formula that's proved by topological methods and once you have these two theorems it's not so much more work that has to be done to prove this theorem that I stated earlier so it's really these two give you the result the first controls the error term and the second tells you exactly what the polynomial is that it converges to my time is nearing and let me just say something about what goes into the proofs of the two so for the paper with Jeremy, Oscar and Peter we use this thing called cellular EK algebra and this is a recent invention of Sander Cooper's Seren Galatius and Oscar Randall Williams and it's a modern and multiplicative approach to proving stability theorems. I don't know if it would be possible to prove our result using this sorry, prove it without this cellular EK algebra but I do believe that they are essential and the key idea is that we want to prove this uniform stability for these particular coefficients that come from arithmetic groups and what we do is we lift uniform stability from the symplectic groups to the braid groups and the point is that we do know this uniformity for the stable range for the symplectic groups from Borel's work so Borel in the 70s proved exactly what is the stable density of sequences of classical arithmetic groups and in his work he proves this for coefficients in any irreducible representation and he gives a stable range that's independent of the specific representation which is exactly what we need for the braid groups and the goal then is to lift this uniformity from the symplectic groups to the braid groups and this can be done by writing down an appropriate EK algebra that measures the difference between these two groups in a sense and something about what goes on in the other paper is that we are inspired by what was proven by Madsen and Weiss proof of the Mumford conjecture Madsen and Weiss proved the Mumford conjecture which calculates the stable homology of the mapping class groups as opposed to the braid groups and there are now several different proofs of the Madsen Weiss one is via scanning and the scanning proof which is due to Galatius and Randall Williams is the one that we try to adapt to our setting and Madsen and Weiss were interested in constant coefficients but one can adapt what they did to polynomial coefficients and non-trivial coefficients and this was explained in a paper of Randall Williams and we try to adapt it to this hyperliptic family and it turns out that many of the same ideas work and in a sense they are simpler and some other things are more complicated and from this you can eventually derive this generating series and if you do things using logarithmic geometry you can make the constructions Galois be variant enough to compute the Galois action of the stable homology as well this was a very short proof sketch or it was not a proof sketch it was a list of ingredients but I'd be happy to say more about it I think I will stop here, thanks for listening