 uh today's edition whether it's morning or afternoon for you if our number three web seminar um as usual we do request uh you keep yourself muted effectively at all times if you have questions for Jordan uh please either virtually raise your hand or ask them in chat and one of our our many of my co-hosts will will convey those questions to Jordan otherwise you'll have lots of time for questions at the end so without further ado it gives me great pleasure to introduce Jordan Ellenberg who's going to talk about what's up in arithmetic statistics Jordan what's up in arithmetic statistics that's how it's supposed to be I get that right everybody um gonna be a super casual talk and as I said in the uh in the back um it's kind of my consolation for not like being in Jim Simon's castle in Germany to like chat about arithmetic statistics with many of the people whose results I'm going to mention here this is just going to be kind of like a lightly curated discussion of some recent results I found striking and I'll try to interleave it with some open questions I know there's a lot of people who come to these talks and uh without further ado let's start with a question that has motivated like lots of interest in this subject which is how many number fields are there okay that's a bad question right because there's infinitely many so we have to refine this question a little bit well first of all let's pin down a degree how many degree n extensions of q are there too many right not a great question um for instance even just even well not equals one then it's okay uh but for any n bigger than one there's certainly infinitely many degree n extensions that's an elementary exercise and maybe just because it's good to have this in mind as an example to think about um eg uh q adjoin root d uh for any integer d that gives you an infinite list of distinct quadratic fields and in fact it's kind of easy to see um how many there are of any bound discriminant but a little bit of care because you might double count so but no it's not counting number fields is not as simple as counting integers because q adjoin root d is the same field as q adjoin root d m squared so you would have to find one representative of each square class of integers we're just typically done by letting d run over square three integers okay um so to make an actual question we have to cut down this infinite set uh to something finite and the most natural way to do that is to bound by the basic invariant of the number field which is its discriminant uh which measures the prime differentification so now we have a question we can actually answer uh how many number fields are there of some fixed degree over q uh and discriminant at most acts and by the way we could in fact if we wanted to um even eliminate this because as it happens you can show that if i fix the discriminant that in fact bounds the degree but it turns out it's much more natural to sort of treat the degree and number field sort of as their own thing and fix the degree for all time and count that actually that's for the first part of this talk um and we're going to pivot about halfway through uh and i'm going to take the other point of view where i fix the discriminant and let the degree of the field grow but that's later i'll just hold the thought um okay so now there's an actual question uh and i'm going to tell you about what we know about it because that's changed actually in very recent months um so first of all how do you even know this is finite i sort of i said now it's a question now it's a question that has an answer it's certainly not obvious that there's not infinitely many fields of discriminant at most acts um even for what i feel you got to do a little something um but it's not hard to check that um the the discriminant of q a join root d is well i don't know it's like about d and maybe there's some factors of two which i'm not going to worry about everything we say is going to be so casual that we're certainly not certainly not going to worry about constants maybe a more important thing to keep in mind is the point that made at the beginning that q a join root d is the same field as q a join root dm squared so maybe better would be to say the square free part of d uh and that's really what the discriminant is but in any event uh certainly there's only finally many integers uh certainly there's yeah there's certainly the only finally many square free integers uh less than x and we could even get an asymptotic so how many number fields are there a degree n with discriminant at most acts is based this is what we mean when we say counting number fields well um as i said the final this year is an actual theorem that follows from work of Hermit and the standard uh quantitative bound um i mean this is essentially something that Hermit knew but this sort of the most general version of it is due to schmidt um so i say 1995 but this is really quite old uh is that you can get an upper bound of x to the n plus two over four so if you think about the quadratic case the case n equals two um you're essentially counting integers up to x so that's x to the one and two plus two over four is one so that's good okay so it gives a sharp bound but probably never again probably there are many fewer number fields than this i'm going to explain to you how this classical bound works in a second and you're going to see that it's incredibly wasteful um and you know i i got into this uh in my first paper with actually Venkatesh about gosh now probably about 15 years ago um where after this bound had been essentially unchanged for quite a long time uh i lost the parentheses in my time sorry uh we were able to show a much improved upper bound uh where that exponent of x uh the dependence on n was much much better instead of being linear in n that exponent is uh e to the root log n um and after that again uh things sat in stasis for a long time and were not changed um until just in the last year uh there was real improvement and this came from first from a paper of genre covenia last year um which shows that you can get an upper bound of x to the log cube then and then just uh two months ago Robert Lemke Oliver and Frank Thorn uh posted a paper on the archive which gets that down to log squared n so we're improving that exponent um more and more and by the way maybe i didn't put a slide of this but i think i'm going to say what's expected but maybe it's a good time to say it actually right now um because it's a rather startling conjecture um is that it should literally be x to the one and you know if um if you're used to this there's lots of reasons why this is a natural projection but if you're not used to it it's very weird because you know if i'm like let me look at all of the degree 1000 number fields discriminant less than x 3 1000 number fields seem a lot more complicated than quadratic fields so you might think there would be more of them on the other hand a degree 1000 number field you compute its discriminant maybe by some like the determinant of some thousand by thousand matrix which maybe you think is going to be quite large then maybe you think it should be hard for a high degree number field to have small discriminant and somehow those two tendencies exactly balance each other out or are supposed to and you get literally the same exponent of n each time would maybe i maybe i should say a different constant um and actually this is a um i should say that i mean there's so much very arithmetic statistics i'm not going to talk about but i should at least say in my mouth even though there's no slide that for extensions with gallo group sn there's actually an explicit prediction for what this constant is uh due to bar gava and we know that it's true for n equals three four and five um for six it would require some monumental uh conceptual advance that we don't understand and people have thought about the case of degree six fields a lot and we just have no clue um jordan let me talk a little bit about how this classical method works how you found number fields and sorry jordan could i could i ask a question or could i ask invite uh thomas bloom to ask his question sure thomas has um yeah okay great yes yes sorry i want to ask this some c sub n factor how bad is the dependence on n here and is it roughly the same in all these different results or these different results differ a lot in the cn behavior as well it's it's gonna be it's supposed to be um a local it's supposed to be like a an oiler product so it's not so it's not so bad there's some i mean okay let's see if i can get this on fly it should go down a bit because there's sort of some business coming from the archimedean places that i think is going to give you sort of a somewhat negative dependence and in any event it's not blowing up but most of it most of its content is going to be um some kind of oiler product and so in particular there's lots of stuff we expect um in terms of oh what if i only look at those who's discriminant as prime to p okay then the numbers to change by just uh throwing out that factor in the oiler product etc lots of expectations thanks to varga va's work and the low degree cases we can prove everything you go from five to six and you can prove nothing not even this not even this exponent so i'm just gonna say like question mark question mark question mark emphasizing this this is very far from being something we know when and is bigger than five okay so having said that everything is pulling hard let me say a little bit about how we can at least get upper bound oh and maybe i should say okay i didn't put this on slide so what how do we even know what do we lower i think you can get um actually kiron the con i think he has a paper about this but i think you can get x to the one half or maybe a maybe actually there's even been a recent improvement about this i didn't write it down um so let's just say i believe that certainly you get lower bounds that are of the order of a power of x and i think you get about x to the one half or maybe one half plus something that's like some modest function of n or something like that but you can't get that close to x to the one as far i mean that we know how to do um okay pretty slide i put up i think of interesting talks i've seen by other people that i should have mentioned in this uh in this one so okay so what's the strategy how do you even start to approach uh this problem the best way to question sorry kiron put in uh please unmute and ask away oh i'm just wondering if is it possible that you get an upper bound of all of x even if it's just a number if you should be in overall degrees um actually once wrote this in an appendix to a little patsky um i guess wrote what would our method give for an upper bound in that case um so in terms of what the truth should be my instinct is yes i think is that they decay pretty fast as n gets big but i don't actually know if that's formally conjectured and it's possible that's not true for some reason that's not in my mind right now but my guess would be yeah even without the bound on degree that you would probably get an x to the one but there wouldn't be like a nice constant it would just be nasty or it'd be a set of some sum of like different oiler products from different situations actually that's true by the way jordan that's true even if you even if you um modules prediction for the constant just pertains to those uh degree and extensions whose gallery group whose gallery closure is full s n so even if you mix the different gallery groups you get kind of a huge in terms of what the content is okay so what's the strategy for getting some kind of upper bound uh and as you have short integers and all the papers i mentioned use this strategy so basically the idea is this um i've got some field i want to try some number fields i want to somehow find it and as i said at the very beginning um one way to describe a number field is to describe an element of that field and one way to describe an algebraic number maybe the only way uh is to give its characteristic polynomial and that's sort of what i did when i described quadratic fields as q adjoint root d i was saying adjoint a root of x squared minus d by just writing root d is my shorthand for that um so let's try to formalize that approach uh i have some field is discriminative is bounded um and i look at the trace zero integers of that field why trace zero because i don't care about like numbers like one and two i just care about authentically interesting elements of this number field so as a stratagem for doing that i can require that has trace zero and that certainly rules out the copy of z something in there um this is a lattice in an n minus one dimensional whose co volume is the square root of the discriminant uh and then the glory of geometry of numbers in the theorem of minkowski is that if i have and i have some upper bounded its value which is exactly what i have when i founded the discriminant then i know that i can that there's some lattice point that's in a small ball i mean any any sort of nice symmetric region but in particular i just want to sort of bound all the arc median valuations and so um oh i sort of said here you know for all this is for all embeddings of the field into r um oh sorry i didn't see that that somebody asks do i need a constant multiplier yes totally and like every constant that depends only on n i'm just utterly suppressing and pretending as one uh in the interest of saying things fast so yes imagine a big oh i can use the highlighter tool imagine a floating z sub n floating over this entire slide and every slide as long as i'm in the n fixed part of this top okay so i have this integer all of its valuations are short um so in particular like we're trying to prioritize like q adjoining root 17 or something maybe i would use root 17 and i wouldn't use like 1000 root 17 plus 5000 i would use sort of in some sense the uh representative and polynomial is some polynomial whose constant whose coefficients i can bound because the i coefficient symmetric polynomial in these arc median valuations whose size is the real numbers i know um and k is not totally real because i mean i should say sorry in arc median valuations thanks um okay uh so from there now all now all i need to use is that once i once i that shows that every field of small discriminant is represented by some polynomial of whose coefficient is small let's go efficiency in some box and now i just have to make sure uh that this map is essentially injected that um i'm not that once i have a polynomial that actually determines the field because then the upper uh then the number of polynomials is an upper bound for the number of fields and this is why i don't want to con this is why i wanted to take care that my um that my short in like five that it was an authentically something which generates uh okay and then uh and then where you get this x to the n plus two over four is just like literally if you account how many options there are for a two a three dot dot dot through a n given these bounds on their on their side the classical bound comes from okay that's not seems like a good method but the question is how does this get improved so um so more generally um and this is in some sense going to be a theme of various things i say today um let's let g be a group and this is going to be the gawa group of the number of fields we're studying so we might for instance think of g as the symmetric group on n letters if we're thinking of generic degree and extensions and let's some representation of g um which i'm going to think of you know where my algebraic geometry had as an affine space on which g acts okay um all right so now i sort of i mischeck this let's write this and what i mean to say is that pi is the map from v to the quotient v mod g which i'm calling w okay so it's this vector space modded out by the action of the group and so this sort of classical example this is this is sort of 19th century invariant theory right the classical example is when i find n space and the symmetric group acts on it in the usual way my permutation coordinates the permutation representation um and when i mod affine space out by the symmetric group i get an affine space again but it's a different affine space it's not the affine space i started with um the affine space whose coordinates are sigma one through sigma n the symmetric in the original coordinate so if you like i mean i'm just sort of doing git here w is going to be an affine variety uh whose coordinate ring is the g invariant functions on the original on the original space so this is kind of my algebraic geometry rephrasing of the thing said before which is that we can um uh well i'm getting a little bit ahead of myself but the the symmetric functions are the coefficients of the polynomial uh we saw the last time and now here's the thing there's a relationship between this kind of crazy quotient i made and the problem we're trying to solve because if i have a point here some rational point on w um i can lift it and look at its free image in the affine space so again if we think of the affine space the space of uh n tuples of algebraic numbers and w as the space of polynomials that's like saying take a polynomial look at its roots i guess actually the way i set this up i mean maybe i should say the space of orderings of its roots but in any event um the inverse image of this point is probably not going to be defined over q right it is going to be defined over some uh g extension of q and that gives me to go from rational points on so this gives me a way to go from rational point to g extensions and in some sense the question before us is i'd like to know that i can um is every low discriminant extension maybe i should say does appear from a low height point of w height is a little bit not the right word because i mean this is really a situation where i'm studying integral points on affine varieties not some protected varieties so i mean one might literally say we're going to talk about points of w of z and ask about small coordinates okay so the idea of getting upper bounds then splits into two parts so this is the one i just kind of scribbled on the last slide the first part for every field of discriminant x some point of w whose coordinates are not too big some power of x um which gives that view that's exactly what we did a couple of slides ago and we said every using minkowski we said every field of discriminant k has a short integer um whose characteristic polynomial the thing whose coordinates the symmetric functions has coefficients that are not too big bounded by powers of x and we even wrote down explicitly with those powers as well um so that's part one and then part two is you actually have to show that there are give some upper bound for how many points there are which in the case of counting polynomials was like really easy right if i wanted to know how many polynomials there are such that the second the first coefficient is this big the second coefficient is this big the third coefficient is this big i'm just counting integers in a box that's the kind of analytic number theory i know how to do and i multiply by multiplying numbers and and you're able to do that um but of course the reason i set it up in this level of generality is that the standard permutation representation is not the only representation of s n so in some sense the question is can we do better by choosing different representations um and and this is sort of the idea that animates paper i wrote a long time ago with akshay um and in some sense it's also what's going on uh in this new paper of robert and frank uh link yalber and uh and and and frank thorn um instead of taking if you thought i was going to have sort of some super fancy representation theory to take on this no instead of taking the permutation representation i'm just going to take our computation representation direct sums together and let s n act back okay so that is a different representation um and in some sense if i look at what are the points on v that what this amounts to instead of looking for a single short integer in k i'm going to look for an r tuple of short integers um it's not immediately obvious why that would help you so let me say a little bit um for part two now we have a different kind of variety like we have an to the r mod s n and this is not just affine space anymore it's something more complicated so part two immediately becomes harder part one is sort of pretty similar in terms of you just use minkowski to show that you can get a point whose arc median valuations are not too large um so all right my tech got messed up this should be i'll get this c i um so again if you thought i was going to do something really really smart don't worry i'm going to do something really dumb i have some gigantic affine variety which i know nothing about and i want to count points on it a bounded size and the means of doing this is just going to be to think of functions on this variety and map the variety to affine space so um before we use the symmetric functions as coordinates on w and i took a short integer and i said okay look at its uh symmetric function so maybe i'll write this down here um when r equals one my functions of um my functions of alpha are like well actually what i used was the symmetric functions um i could also have used that at amounts to the same thing the power functions right the sum of the powers of the coordinates so i could have used trace alpha this one i actually chose to be zero trace of alpha squared trace of alpha cube up to trace of alpha n um and if i know those i know what alpha is um so if i have more integers so if i have a pair alpha beta the function now i have a lot more functions i can use i can use uh trace alpha and trace beta i might have said those would be equal to zero but i can also use trace of alpha squared trace of alpha beta trace of beta squared all of these are going to be integers right and all of them are things whose size i can bound if i've bounded all the arc median valuations of alpha and beta etc etc so um and so each of these functions i just wrote down is a function on this quotient space um i just want to say i like how bjorn has this static picture of bjorn kind of making a nice smile as his zoom thing because then i just feel like wow bjorn's really liking this like everything i say i get i get a big happy smile from bjorn on my screen um the uh yeah all these functions are bounded in size and in fact um something really good happens which is that um you know what's the size of this thing if the absolute value of if every absolute value of alpha is less than some constant y and these get pretty big right this is going to be less y to the n um but here i have a lot more functions of low degree right because if i look at polynomials of degree up to uh up to m in two variables i have m squared then instead of m so i actually have more low degree functions when r is large than i do when r is small um so again so i wrote a lot on this slide but um all i'm saying is if i if i have a point uh on w i can i have capital m different functions i can that's the same thing as saying i have a map from capital w to affine space of dimension m and then i can go from my g extension hopefully by minkowski i can show that that comes from an integral point of w whose coordinates are a lot too large and then by applying these functions to a point on affine space whose coordinates are not too large and then i can count in a box know how to do um but we're not quite out of the woods yet because now how do i know i mean this map could be horrible i mean if i chose capital m too small for instance this map is just going to down and then i'm going to lose the injectivity and certainly if i wanted to arrive an upper bound for the size of this set from an upper bound from the size of this set i'm going to need some kind of injectivity and in some sense all the technical difficulty is here uh you need to control uh the positive dimensional fibers of this map uh and make sure that you're not crushing down lots of number fields uh onto a system of am so what that amounts to is you have to study the positive dimensional fibers of this map and um be able to say that not only can i find uh an r-tuple of integers whose heights are pretty small i can find one that doesn't fly on one of these bad loci so it's kind of a fun 19th century style thing to do maybe i'll just sort of say in passing we're not we're working on this long ago our goal was actually this sort of it was when bombieri pila and heat brown were coming on there was all these new exciting techniques and we were excited to use all this stuff to bound number fields and we actually put the paper that way and then realized that in fact i'm sort of sadly none of it was necessary and the techniques for doing this will be classical in 19th century so that's what we do oh no this one i well don't worry i i'm going to tell you um so we are able to do it but we need kind of a lot of functions we need about two to the two r times n functions um but uh robert and frank are able to do much better going to sort of complete the sentence above it make it work with capital n on order of rn you know the fewer functions you have the harder it is to control the positive dimensional fibers right so having a lot of functions makes it easier but it also makes your bounds worse because you're mapping to a bigger space and going to functions of higher degree um and i just want to recall that of course remember the dimension of w is our end because it's a quotient of a to the end of the r by a finite group so better than this like we can't go any lower than the actual dimension of the space so their result is in some sense like best possible uh by this method and it gets you down to a much better dependence in and an exponent uh than we were able to get and to get past that i think requires doing something truly different so i promised an open question so let me sort of say uh just a few thoughts about where one can go from here one obvious question is what about other groups so i mean this is sort of set up to count degree and extensions but of course there's a huge interesting which hopefully i'll get towards the end of this talk about counting number fields with specified gala groups uh so evan dummett has a nice paper from 2018 uh in um about using methods sort of akin to those in my or actually uh to count number fields with specified gala group um and i think it would be very interesting to now return to that problem in the light of these frankly much easier and less techniques of of lenke alliver and and thorn uh to see if you can get improved upper bounds um you know the x to the some function of of g i think it would be really interesting to know now what's best possible for this problem in terms of upper bounds um and question two of course is i said you know the only thing finding points on this affine variety w particular points in this incredibly mindless way of just trying to map it to some affine space and then counting points in the box but of course there's also a huge rich literature about counting points of you know rational points of bound integral points of bounded size uh on varieties that really takes into account the actual geometry of the variety and these varieties are very special i mean they're very natural things quotients of affine spaces by finite groups um so can one even conjecturally even if you give yourself every possible conjecture of sort of bacteria of mining type which is the sort of governing family characteristics for questions of this kind what would follow uh about uh counting number fields of bound discriminant so there is work um by Yasuda about this from 2015 but i think this is an area where more could certainly be done um and finally i'll just say as i said the bound for this is supposed to be x to the one but we're very far from knowing this or even x to the one million so we don't have any bound where the exponent doesn't depend on n and i will just comment that in the function field case um we do get something like this in um by very different methods in the function field case so this is a paper that i wrote with Pritank Pritank Tran and and Craig Westerland um but it's on archive and which everybody including us finds very hard to read it's like complicated algebra and typology so we're sort of in the process of sort of trying to boil this down uh and write a more readable version of it just from the title right can you tell it's like pretty different from what i usually work on um but so i think something like this is function field case by methods that are in some sense in spirit uh similar to the work i did with Akshay and with Craig uh on the Cohen-Lensberg Conjecture um which made me lend some creep if this is true but i would say it still remains uh very far out of reach okay i'm gonna make a total effort Gordon yeah we have another question Peter Sarnak please unmute and uh ask away hi Peter all right uh actually maybe i must understand early on but if you can't all number fields up to height of discriminant less than x without any restrictions whatsoever not fixed degree is that conjecture to be a constant times x um i have formally conjectured i think it's probably true and i would say it's i mean you made this agree i would say it's slightly less natural question no it's a question when faultings proved these uh model conjecture in 83 i was an instructor and system professor NYU and Harold Shapiro who was a student of MLRT and actually he said you know this theorem that he's proved which the Chevrolet which state conjecture is a generalization of the fact that they only find out too many number fields of a given discriminant so he said to me you do weird things he says to me why don't you count how many number fields have discriminant less than x and then he said for former series and does that have an analytic continuation but i misunderstood the beginning what you seem to be discussing the case where you fix n that's what i've been discussing but i'm literally about to pivot and not do that but you believe it's it's believed that even unrestricted with it i'm not sure people have formally i would think so that would be my guess but i'm a little hesitant to but you know i mean even with that bounce that you gave or so poor say again the upper the upper bounds that we give you can i i would pay i depend on n those depend on n but those can be made to you can get an upper bound without the defend well it's the word best upper bound you have independent of in yeah that that's my exact question oh it's uh the number of number fields whose discriminant is less than x period okay that was my question i look in the appendix action i wrote to this paper of that's where you wrote it up i do not remember it's all right because i wasn't gonna that's like what was up in it that's all so i don't remember sorry um okay so uh this was this looks like the question i started with but i'm changing in the spirit of people were asking about what happened the degree changes what if instead of fixing the degree and letting the discriminant grow i fixed the discriminant and let the degree grow well i've written this is not a great question sort of for the reasons that i said that there is actually some bound on x in terms of x so let me mess with this a little bit uh instead of that let's let let's say i'm gonna number fields of discriminant x to the n of degree n but uh fixed and n is growing and the reason i write it this way uh this sort of a more natural uh first because this actually can happen with n growing and one way to get such extensions is that you fix some number field k of discriminant x and then take unramified extensions of that so now you might ask a different question i fixed some extent no if i could start with k equals q but q doesn't have any unramified extensions if it did i would ask that question so i got a pass to some auxiliary number so if k is some number field of discriminant x tell me about all of its unramified extensions um but this is another another way to sort of capture that is to say what is the i can just composite them all together and say what is the maximum extension of that number field now interesting questions here but uh and one problem is that this i'm not going to answer this question for one thing it depends quite keenly on the specific arithmetic of k so having set up the question in this way and letting n grow and thinking about all kinds of higher and higher degree i actually do want to let x vary a little bit and ask the following question okay what if k is a randomly chosen let's say g extension of q what does randomly mean it means that i choose it from among the ones of discriminant of most x so for instance so e g q would join root d with d uniformly random in one through x maybe chose choosing a square free or something like that um what can i say about the gala group of the maximum unramified extension of k and maybe if you haven't thought about this question before i will emphasize that this is a crazy group might be trivial like some fields like you don't have any unramified extensions it might be infinite it might be an infinite pro group and so this is kind of a a very exotic kind of question in some sense i'm asking for a probability distribution on isomorphism classes of profanite groups but even more than that because my k carries an action of g the gallo group of maximum unramified extension carries an action of g too so i'm looking for a probability distribution on this very weird family of things and it's hard to even imagine how you would describe a distribution on morphism classes of profanite groups um what's the president here just make it seem a little less weird is that if instead of looking at all tensions i only look at the abelian ones now we're in a much more studied realm of number theory because the maximal unramified extension of k is by class field theory just the same uh as its class group and so now i'm saying k and g extension what does its class group look like and that's the subject of the cohen and lendstra and oh i meant i should put in and martin a um and by bin tamala my student dana carton etc etc etc um there's a huge long literature starting in 1980 with this paper of cohen lendstra about uh what does the class group of a random number field look like and there's a great story i'm not gonna tell this okay i'm not gonna tell by the way sorry i know that was the abstract but i ran out of space um there is some developing sense in which we now understand what to mean by a random finite abelian group be a random finite abelian p group and that cohen lendstra uh conjecture is saying that the class random number field is a abelian group and if that number field happens to have an action of g then maybe it's uh instead of being a random abelian group it's a random abelian g action which is to say a random item of this category anyway the problem that we're talking about seems much harder and so the development i want to point to is a very beautiful new paper uh by yuan liu melanie matchett wood and david zirak crown it's on the archive last year um which is one of those papers that does write what it says in the front of the can a predictive distribution for gallagrups of maximal and row high extensions um they produce a probability distribution on this set of isomorphism classes on such group in other words they give like a sort of good uh motivated definition of what we should mean by a random profini group um and they conjecture that this is indeed uh the limiting distribution for the problem that we talked about and so basically if that's true that would allow you that would allow you to answer any question about often a certain kind of extension of q to making those discriminant up to x um how often it has an unremovified extension with a certain gallagrups so you could think of this as a non-abelian coenland stroke indexer and it builds on like a large body of work leading up to this a lot of it uh involving uh nitha boston who's probably good wisconsin um jordan yeah question joel rozenberg um please ask away so maybe his mic is not working uh i'll just ask this question or read his comment so in between abelian and non-abelian would be solvable extensions would that make uh the problem easier to handle i mean i think so what's true is that a lot of the earlier work so nijl's work on this i think largely concerns pro-p extensions so the no-potent case so i would say the no-potent case is a lot harder than a billion but a lot easier than this so that's what i would say is i'm not sure solvable is that much easier but no-potent is definitely uh technically easier and that's some of the that melanie and david and juan are are building on here um so so one thing i'll say i didn't know hi but one thing they do that's quite um cool is that they observe that there's certain kinds of groups that cannot occur which i think had not really been noticed before you have to sort of set them to zero in the probability distribution um the fabulous creates a certain level of things i in a sense okay these three folks noticed which no one had noticed before that certain kinds of groups are ruled out okay so how do you know there's not some other clever person who next year things are some other class of groups that uh that is ruled out and can't occur like why should we have confidence that this particular prediction is correct and in some sense i mean i will warn you for those of you who have not worked in this area the original conjectures of cohen lester and martin hey we're like not quite correct and they're basically based on things should be completely random unless i can think of a reason for them not to be but sometimes there's a reason you didn't think of and so there were modifications and mala and garnton of the names i put on the last slide were people who sort of slightly had to tweak those conjectures to make them more correct so i will say that um one thing i find really impressive about what you and wood and zero ground have done is they not only formalize these this conjecture and say these are the kinds of groups that can arise and subject to that we make some principle prediction on with what probability they'll arise um they prove a kind of a geometric version of their conjecture and it you know given the time um let me not say too much what have i got on the next slide um so of the following nature um and actually i wrote this and then actually they don't need this so let me i'm going to process that because it's not actually i put this slide but i have to repent of this um so ie um if you ask about the limit as n goes to infinity i'm going to write a very informal version of this of the limit as q goes to infinity of and now i'm going to just commit a thousand sins against probability theory by writing something that's not really true i'm really taking limits of moments of distributions not limited distributions okay um so distribution of um unremifed gala groups of g extensions of fq of t with discriminant less than q to the n okay so i'm replacing the number field with my favorite field the function field fq of t which is supposed to give us a lot of clues to what's really happening over a number of fields um and the really important thing is that this gadget can be expressed in terms of algebraic geometry in terms of the algebraic geometry of of well a g extension of fq parentheses geometrically is a g cover of p1 over fq those are parameterized by a certain modular space called a hermit space and you can express these questions in terms of the geometry of those spaces um the real question at hand would be to ask about how this thing behaves in the limit as n goes to infinity for a fixed q that's mostly out of reach but as every algebraic geometry in the room knows you can do a lot if you let q go to infinity there's a lot of geometry that becomes simpler if you like let q go to infinity and take limits by vacant structures a lot of stuff drops out and you're left with geometric questions that are much more handle so i guess what i'm saying is that um this kind of reasoning you should think of it as like it's like kind of like an error correcting code for number theorists it's like it doesn't prove things about number field but if you have two competing hypotheses over number fields typically one of them is going to pass this test and be consistent with with this kind of geometric reasoning and one of them is not and so what they show is that their conjecture while you're sort of in the middle of a sentence here i say if you ask about and the answer is yes the predictions of their conjecture gives are in fact true in this large q limit setting uh so in particular any group that they predictor actually does occur in this setting um okay so 10 50 all right let me just say a tiny bit about uh the sort of venn diagram intersection uh of the two cases that i gave and then we'll quit and take questions um because i've said okay what if we i started this we fixed a degree and look at fields of that degree of growing discriminant and try to count um and then i said what if i fix the discriminant and look at fields of growing degree with the same discriminant like we're now applied to that same set of primes what can we say now it's not counting problem but it's still like a problem what happens uh as we came to that variable okay what if we fix both what if we say what can i look at number fields of a fixed discriminant and a fixed degree and i'll just comment that there's a conjecture there's a conjecture i don't think anybody really knows exactly whose conjecture this originally was but it's kind of full um which says uh that if i look at the fine at the g extensions with discriminant x you don't have too many that hit exactly the same discriminant but there's an upper bound of um on order x to the epsilon so maybe i'll just say um so one way of thinking of this is that um the first part of the talk i'm asking if i average over all the discriminants up to x so that's kind of like an l one question right what's the average of the number of degree on extensions up to x and this would be a question about like l infinity like what's for any x what's the maximum number of number fields that can pile up and all have the same discriminant um and you know for quadratic fields for instance you don't have two different quadratic fields that have the same discriminant but for larger degrees that actually can happen but one feels it should be quite rare um evidence for this is pretty sparse um but it's what people believe i think in the interest of time let me just i'm going to elide the connection with bounds on class groups um well i sort of have to at least mention it because of the theorems i want to mention and beautiful theorem of a long list of authors um from 2017 uh that give upper bounds um for the two part of the class group of an arbitrary number field and the point is that this can be thought of as saying if i fix a number field and say it's too class too big it's saying i give some upper bound for how many for how some particular gala group uh can can pile up on the same discriminant and uh i have last month by Jia uh Jia Wang who's a postdoc at Duke um and as student of melies uh gives similar non-trivial upper bounds for the l parts of class groups for any no-voting gala extension with this of an interesting set of exceptions so um people are working hard to get in this case getting upper bounds is um even beating the trivial bound is quite hard so um all right let me maybe i'll skip this um i'll just circle this to be like this slide and i'm happy to talk about it after as about what you might guess for what that x to the epsilon might in one case happy to go back to this people want to talk about it uh let me just close by saying something that was promised in the abstract this is what i've mostly been giving seminars about lately but i've decided i'm not allowed to give whole seminars about it anymore until we release the pre-print which i promised we're going to do soon it's been floating for a while so this is uh work of myself and uh the aforementioned david zara crown and matzapriano um so i'll just say that this whole set of problems can be conceived in a different way the extension of q which we think theory can also be thought of as part of arithmetic geometry if we think of a g extension as a rational point an entity called bg which is the classifier of the finite group um and so it's long been seen that somehow there's some kind of formal similarity between these problems of counting number fields about the discriminant on the one hand and problem of counting rational points at bound to height on the other hand um and in the work that's forthcoming uh we've been able to unify these questions by defining a notion of height rational point on a stack a definition that didn't exist before so this is a paper with like the subject is really a definition in terms it's like a definition in a lot of questions um so just to have one short slide to sort of show you what uh properties of some of this definition is it does indeed capture the notion of discriminant of a number field um it captures the sort of usual naive height of an elliptic curve so we always call this the height of an elliptic curve but this kind of ratifies that choice of notation by saying it really is what we would on a stack of elliptic curves um but let me just close with one very concrete example and a question so a very nice stack which i want to talk about here is and if you ever going to talk about stacks you're like oh i just take p one well then i take three of the points and i kind of fold them in half okay that's my picture of a stack um and and by the way i just have to bring this up because mike is here and mike invited me to give us talk of course mike and i know each other because we worked on generalized sparrow i equations and in some sense the modern way to think about equations like a to the p plus b to the q equal c to the r is as certain kinds of integral points on this stack where instead of a half point a half point and a half point i have a one over p point a one over q point and a one over r point so this is sort of a point of you introduced by by demont and and grandville so so i'm going to just close with one more question uh the height of a point on this stack is given by the which is represented by the stack is by rational to p one so it's rational points of the same as the points on p one is pairs of co-prime integers it's this quantity i take this free part of a square free part of b the square free part of a plus b times the max of a b and i think the square root of all that and then the question is very simple if i'm going to count points of bound in our sense in that stack i've got to know how many pairs of integers there are um so that's bounded so this seems like a very tame question of analytic number theory but i gotta admit i don't know how to do it so i'm going to present it as a question how many pairs of co-prime integers are there uh satisfying this uh inequality and it's pretty easy i mean if you just let the integers be anything you want you can easily get b to the one half just by um letting little a and little b be at most capital b to the one fourth um and our conjecture would predict the actual answer is of is on order b to the one half plus epsilon and some numerical evidence where what does some numerical evidence means it means that like when i talk about this in carbon number theory seminar by the end of the talk no one already computed this like up to like something and gnome told me that it looks like it's a root b log square b which is a conformity we don't know how to predict the power of log by the way for questions like this so um so this kind of new conjecture that we make creates lots of questions about arithmetic statistics of which this is sort of a sample one which we don't know how to do so hopefully we'll put that paper out soon and be able to ask lots more questions so i'm gonna stop there thank you for asking questions during and after thanks so much all right thank you jordan everybody would like to jordan a round of applause