 Okay, great. All right, so yesterday we had a kind of whirlwind tour about how you might have various G look at G number fields and maybe count them asymptotically. And somehow in spirit behind counting them asymptotically is like producing some list of knowing what they are. Now today we're going to imagine you already have those number fields and then you want to ask among those number fields not just how many are there but what kind of behaviors do you see? And the behavior we're going to focus on today is the class important invariant of number fields. And so the question is for today is on the distribution of class groups of number fields so if you have some number fields or care of a cure number fields and they're varying the question is what's the distribution of the class group of K? And in this lecture series for simplicity we are mainly going to focus on the C L O P subgroups. So I'll write it like this for the C L O P subgroup for a prime P. So the class group is a finite abelian group. It's a product of its C O P subgroups and we'll mostly be asking for one P at a time. What distribution of abelian P groups do you see as you look among your number fields? So if we take this question here of extensions of discriminant say bounded by X that have some particular class group divided by the total number of extensions you are considering. And if we want to sort of think about all the number fields we'll let X go to infinity and this is asking how often do you see which C L O P subgroup among class groups? Yes. Oh, sorry. That's just a poor writing. Thank you. I didn't mean to say that. I meant to say as K where K is K, you know, for K some number field. That was just an abbreviation for writing K as a number field. As K number field varies. Thank you. What's the distribution of the class group? And more generally we should ask over other functions of the class group that you might want to average what are their averages? So this is saying as I sum over K and some, I wrote F here, some family. So I'm also trying to open an idea that maybe we won't just take these families we were counting last time G extensions but maybe we'll put some other conditions on our number fields and we can say count them up to discriminant X and then on the other hand we can sum any function we want of their class groups up to X and then we could take that ratio to get and this is supposed to be the average, you know, this is the average of F over class groups of K, 4K in this family. And really if I want to be general there's one more thing I should generalize. What I wrote here, you know, we would say is ordered by discriminant the way I take this limit is the discriminant up to X. And so a more, more general thing to do here would be to consider maybe some more general invariant of the number fields and which could be the discriminant, it could be some other invariant and then this thing here is, you know, the average of some function F over class groups 4K in some family of number fields by some invariant. So that's the sort of most general question you could ask about how the class groups of number fields in a family behave. So for this lecture and the lecture tomorrow we will mainly be focusing on quadratic fields. After all, one thing to say is implicit in either of these questions, this original question or the more general question is the denominator of these questions is about counting the extensions in the family. So that's what we were talking about. So it's like, oh now we have a problem that has a numerator and a denominator. The denominator is the kind of question we had last time. Now of course in the case of quadratic fields we saw that you can really count quadratic fields so at least the denominator is not scary in the case of quadratic fields. And so one of the first things you might notice when you're thinking about the class groups of quadratic fields is that you have any experience with them either theoretical or empirical that you will notice that the class groups are quite different for imaginary quadratic fields versus real quadratic fields. Well what do I mean by that? One of the many things is that the imaginary quadratic fields have much larger class groups if you look at them empirically and we know for example it's a theorem that there are only finitely many imaginary quadratic fields with trivial class groups. On the other hand for real quadratic fields it's a quite open conjecture of Gauss that there should be infinitely many real quadratic fields with class number one. So since that behavior of class groups from the imaginary quadratic case to the real quadratic case is quite different that suggests that one might want to consider these families not just to say be all quadratic fields but perhaps divided by some other conditions like which ones are imaginary and which ones are real. Okay they're finite abelian groups and we know what all the finite abelian groups look like and it turns out for the class groups of quadratic fields there is one further thing that we know which is what genus theory tells us and so I'm going to explain now what genus theory tells us since this is a lecture series often you'll hear okay and genus theory tells us something so we'll ignore that I'm going to actually get into and tell us what genus theory tells us and I'm going to again kind of like yesterday I'm going to give it through a very general lens through the lens of class field theory of course genus theory for quadratic fields was understood by Gauss in terms of binary quadratic forms and one can understand it quite concretely however I'm going to explain it today through the more modern lens of class field theory because that same lens can be used to understand what this kind of phenomenon is in other situations where the same concrete approaches don't necessarily apply so okay alright so class field theory first of all tells us about the class group that it's the Gauss group of the Hilbert class field the maximal unramified abelian extension of K so we have this is for any number field K that the class group is isomorphic to the Gauss group of this specific canonical extension of K so if you haven't seen that that's like a miraculous thing like why should such a thing be true this was about ideals and principle ideals okay that's a beautiful beautiful miracle of class field theory but we're going to just take that and use it okay and what genus theory tells us about is the following thing so if I have this is my extension K if I have an abelian extension E over Q of course this this extension E K I take the composite extension this E K over K will also be an abelian extension and if that extension happened to be unramified then it would be contained in this maximal unramified abelian extension and genus theory is about when that happens okay so I'm going to define a genus field to be the maximal extension of K that is unramified and abelian so that it will be you know part of this part of this Hilbert class field this maximal unramified abelian extension and it should be a composite it should be of the form E K for some E over Q abelian now you know not every abelian unramified extension of K has to come about this way and maybe not you know maybe you could think maybe none do but alright there are some in that you can see that there is a maximal one because you have to take their composite and it would still be be such a thing okay and so whatever this this genus field is the maximal unramified abelian extension of K that we can get by pushing up here an abelian extension of Q it's since it is a subfield of this maximal unramified abelian extension by Gao Law theory we have a quotient from the class group of K to the Gao group of this this genus field over K and so that is the genus group so that's you can think of as some piece of the class group yes question no no I did not yeah no I didn't say that the maximal unramified abelian extension is of the form E K I mean in some case it could be but mostly it is not just that that kind of tall logically almost that there there is some maximal abelian unramified extension that is of the form E K where E over Q is a billion and whatever it is because by definition it's unramified at a billion it is a subgroup so E K subgroup sorry subfield it's a subfield of the maximal unramified abelian because by definition it was unramified in a billion so so because of that by Gao Law theory it's Gao group over K will be a quotient of the class group of K and and what so that provides us some piece of the class group piece by which I just mean quotient all right and so what can what can E K be what kind of what kind of unramified abelian extensions of K can we get this way so class field theory and a lot of these the claims that I make many of them are spelled out in more details and with some exercises so you can really work through them in the notes but I'll just tell you that the class field theory tells us that this Gao group of this genus field is the semi-direct product of so this is the Gao group of the genus field over Q so going all the way down to Q and first of all you might think oh it's not even clear that it's Gao law but it is Gao law by the usual kind of argument that it's defined by this this universal property and the Gao group is the semi-direct product of the Gao group of E K over K with the Gao group of K over Q now of course it has to be some extension and so so there are two facts in this in this claim one is that the extension splits but the other and the more important one is that the action so this is a semi-direct product and I wrote this minus one here because this Gao group of K over Q I should say here now I am talking about the example that that I said we would focus on which is K is a quadratic field okay so this Gao a group of K over Q this is just the group of order two and so to say how it acts on something we just need to say how the generator acts and this minus one says it acts by minus one because this is an abelian group and so you always have a action by multiplication by minus one of the group of order two on an abelian group and it it's that that action now on the other hand so that that sort of class field theory tells us in some abstract way on the other hand it's a composite extension it's a composite extension so we actually have a much more basic act that as a composite extension this Gao group is a subgroup of the the product of the Gao group of E over Q and the Gao group of K over Q and in this this here this action is trivial of you know I mean you could say this is a semi-direct product as well but with a trivial action alright so on the one hand we have that the action is by minus one of the Gao group of K over Q on this this piece here and on the other hand we have that the action is trivial so the conclusion of that is that the Gao group of E K over K must be two torsion because of those of the abelian groups the only one on which the action by minus one is the same as the trivial action is if the group itself is entirely two torsion yes is it obvious why E can't be equal to K ah I count that as two torsion so if E was equal to K then it would be the trivial group and I will count that as two torsion I mean it's it's it's killed by two so absolutely it can be the trivial group so the two torsion includes like the trivial two torsion group the trivial group yeah other questions okay so now not only does classical theory tell us that this group has to be two torsion it can help us find what what it can be so what are we looking for if it's two torsion that means it's generated by by quadratic extensions so I'm looking for for quadratic extensions here quadratic extensions here that when you when you push them up they become unramified so what does that look like alright so here's here's my picture so K over Q K over Q is a quadratic extension and I'm looking for quadratic extensions F over Q such that when I take their composite um that this extension L over K is unramified so that's that's what I'm looking for now I you know of course it'd be great if we could take F over Q unramified then this would definitely be unramified that doesn't work because we know there aren't any any such F over Q but maybe just maybe um it is possible to have ramification that that when you after you take a composite disappears and we're trying to look to see if that would happen so this is my field diagram here and next to it so I've drawn the gawa group so so of course if this is a composite of two quadratic extensions the gawa group is going to be Z mod two cross Z mod two so this is the gawa group of L over Q so it's it corresponds to Q in the the gawa diagram and so I'm going to say that K is the field fixed by by the subgroup generated by one comma zero alright so that means that the gawa group of K over Q is like the second coordinate here it's the quotient by this and the gawa group of L over K is this subgroup and so what does it mean if we want L over K to be unramified given this gawa diagram L over K is going to be unramified if and only if the inertia in this extension intersects trivially with this subgroup since that's the gawa group of L over K right so I've just I mean I've just said like for this to be unramified that means the all the inertia should in you know in L over Q should not see or not touch this gawa group of L over K which is this this order to subgroup here right okay so if we think about this we have the gawa group of Q bar over Q all the possible extensions and we already have a K we're starting with a K we're imagining we have this K its gawa group is is Z mod 2 Z and we're looking for lifts of that we're looking for L's where the gawa group is Z mod 2 Z cross Z mod 2 Z so we're looking for lifts of this to some Z mod 2 Z cross Z mod 2 Z such that projection here so this this is I drew the arrow over on the right because this map is going to be projection to the second coordinate so we're looking for maps here such a projection to the second coordinate agrees with already the map we have here that is is K over Q and I wrote this this way in terms of the absolute gawa group and maps because class field theory tells us what what all such maps maps could be right so if we have any abelian group these homomorphisms or continuous homomorphisms from the absolute oops there shouldn't have been two bars there the homomorphisms from the absolute gawa group to A so A extensions here are given exactly by homomorphisms of the Adele class group and as we talked about briefly last time those homomorphisms can be specified by giving a homomorphism from Zp star the periodic units to A for each P and this restricted product here means that only finally many of those homomorphisms are allowed to be non-trivial but so class field is telling us in this in this way what all the maps from gao q bar over q to any abelian group including say Z mod 2 Z cross Z mod 2 Z R it is a finite abelian group oh yeah yeah let's finite yeah thank you good yeah so so that means we can replace in this sort of picture where we're looking to lift maps from the absolute gao group from Z mod 2 Z to Z mod 2 Z cross Z mod 2 Z we can replace that picture with now we're trying to do that with not the you know the absolute gao group but the product of Zp stars okay the product of of of Zp stars and and and for these it's pretty pretty easy to say you know for each of them what the maps to Z mod 2 Z and Z mod 2 Z cross Z mod 2 Z R but also also in this picture the Zp star themselves are inertia groups they are the inertia groups in you know the image of the inertia groups in this in this gao group so remember we said now we're not just looking for any any Z mod 2 Z cross Z mod 2 Z extensions of Q we were looking for ones I'll just go up here remember where the inertia needs to intersect trivially with this this subgroup here generated by an element non-trivial first coordinate okay so the image of these Zp stars it well I said it can't intersect the subgroup of course it can I should say it can only intersect it trivially it can't intersect it non-trivially and then that doesn't give a lot of a lot of options you already know you already know how your Zp star maps to Z mod 2 Z and you want to think about lifts here but you are not allowed to to intersect this and so in particular if Zp star if P was unrammified in K so remember this is the map that corresponds to original quadratic field K so if P was unrammified in K that means that the Zp star here maps trivially to Z mod 2 Z so the second coordinate has to be trivial but yet you're not allowed to intersect the subgroup where the second coordinate is trivial and so there's nothing you can do except to map the Zp star trivially in the orange map up above on the other hand if P is rammified in K then that means your second you're you're mapping to the second coordinate such that the second coordinate is one and then you have sort of two ways to lift that you could lift it to one one or zero comma one all right and so what this allows us to see is that you know from this from this point of view the number of ways the number of maps that you could create like this the number of L's is twice the number of ramified primes because for each prime that's unrammified you have no choices about the map Zp star to Z mod 2 Z cross Z mod 2 Z and if P is ramified you have two two choices and now these aren't don't quite all give us what we want first of all two of these two to the number of ramified prime maps are not surjections on to Z mod 2 Z cross Z mod 2 Z if you always lift only to the the second coordinate or only to to to one comma one and each field if we care about fields each field gives two maps so because of automorphisms of Z mod 2 Z cross Z mod 2 Z that fix this you know the fields correspond to the kernels of the the the homomorphisms not the homomorphisms themselves and some of these things that we constructed even though we made sure that they were unrammified at each finite place some of them may actually be ramified at infinity so this is why genus theory is usually a statement about the narrow class group so the narrow class group by class field theory gives you the maximal abelian extension that's unrammified at finite places and this gives a perfect story if you just ask about being unrammified at finite places but any case those are sort of some easy things to fix and when you put it all together that tells you that this this genus group so which we know to be of the form right so we said before it's a two torsion group so it's mod 2 Z to the K or maybe I'll call it T you know for some for some T right this genus group its size is at least two to the the number of ramified primes of K minus two one of those those minus twos is for the second point and one is for first in fact maybe I should have said here I mean we actually I can put in the put in the upper bound as well to the number of ramified minus one here and the reason for this this question mark about is it minus one or minus two has to do with whether this question of being ramified at infinity comes into play and cuts things down by a factor of two or not so that's how we can see here all of the all of the unrammified abelian extensions of a quadratic field that come from the composite from Q with an abelian extension and they give us some two torsion and we can say basically how much two torsion in terms of the number of ramified primes of K and of course we know sometimes we have class groups that aren't just two torsion so we know that we don't get all the unrammified abelian extensions this way but the moral the takeaway from this is that we feel like mostly we know the two torsion in the class group for K quadratic because it the number of ramified primes is we think of as a pretty basic invariant of the quadratic field and in contrast for the whole rest of the class group we have nothing like this at all the whole rest of the class group is vastly vastly more more mysterious alright so that brings me to the Cohen-Linstra heuristics so we're going to fix an odd prime and the reason that we're going to fix an odd prime is because of this moral here that well we know about the the two torsion and it follows this specific behavior but for the odd you know CLP subgroups we don't know know anything and they seem quite mysterious and so Cohen-Linstra made the the following conjecture for an odd prime which is that and I wrote it this way so on the left hand side we so we have some function f and on the left hand side we have this average of f over the p torsion in our class groups here and we're taking imaginary quadratic fields and well and so the sort of two things that we pointed out so far was one we should probably separate real and quadratic fields that's happening here and then that we should perhaps ignore two because you know we already know something about genus theory and what's going on it too is why we have an odd prime here and so this conjecture predicts that for some reasonable functions f that this average should be an average over a very concrete distribution that has nothing to do with number theory so what is on the right hand side on the right hand side I'm summing f over all finite abelian p groups and I'm weighting each group by one over the number of automorphisms of a all right now the since those weights don't sum up to one for this to be an average you know I need to divide by the sum of the weights so this on the right hand side does not you know doesn't have any number theory content it's like a it comes from the the theory of finite abelian groups and just says with these with these weights on each finite abelian group what is that the average of the function and then we'll see here as well the similar the similar conjecture now oh oh except I forgot to change the okay the similar conjecture for K real quadratic and it looks very similar we have just this class group average on the left hand side but now just over real quadratic fields and on the right hand side we have a similar average except you'll notice that the weights the weights have changed just a little bit they have picked up this additional factor of a and so you you know the sort of first natural functions to consider in these averages would be the characteristics functions which is like how often do you see a particular group ie f could say like be the function that's one when you're a particular group and zero otherwise so then on this side you'd get the proportion of fields whose you know p torsion p silo subgroup oh wait I wrote p then should be an infinity in these I mean the kind of you know of course yeah silo p so if f is a characteristic function on this side you have the proportion of fields whose silo p subgroup is a particular group and on then on this side you would get you know one over the size of that group number of automorphisms of that group divided by this constant so the moral of that is that among the various finite a billion p groups when we're looking at silo p subgroups of class groups we should have quadratic fields some group a should appear roughly you know c over automorphisms of a of the time where c is some constant that doesn't depend on a if we're looking at you know so this is among imaginary quadratic class groups and this is among real quadratic class groups okay so I want to say that sort of tables of class groups of quadratic fields both helped motivate these these conjectures that in the date in the tables so presented some mysteries that need an explaining like why are you seeing you know this class group more than this other class group especially if they're the same size like if you want to compare how often you see as a co3 subgroup Z mod 3z cross Z mod 3z and Z mod 9z you might at first think well like they're both groups of size 9 maybe they appear roughly the same time but that's not what it looked like empirically and these conjectures provide a very precise explanation for what is going on there and of course the fact that we have very good tables help provide evidence for these conjectures which look very very true so we have good class groups of good tables of class groups of quadratic fields and and these these conjectures look look spot on with those tables and we'll talk a little more about what what we might be able to do with with such tables later but now I want to also present another perspective on on these these conjectures which call a matrix model right and so this this perspective comes from a paper of Venkatesh and Ellenberg and it says well let's let's think of the class group the following way so I have a quadratic field k and I take s some set of primes of k that are sufficient to generate the class group so I mean it's a finite group so certainly there's some I maybe I should set s I'm thinking of s is a finite set of primes so there's certainly some finite set but indeed you know even very naively like we know by the minkowski bound that you can have some concrete bound in terms of the discriminant that okay if I go up to all the primes of norm at least this I'll have a prime in every class of the class group you know so certainly enough to generate it so it's not hard to imagine that you can get can can can think about about such a set and being confident that it generated the class group and once we have such a set of primes we're gonna take the s units so these are the elements of the number field that have trivial valuation outside s so if s were empty then these would be the units but once we throw primes at s we're we're saying okay we're allowed to have some some of those primes in the numerator and denominators kind of nothing but nothing else so those are the s units and then I'll write is for the s ideals so these are just the fractional ideals generated by the primes primes in s and then I'll write mu sub k for the roots of unity and k and there's a map from the s units say mod the roots of unity to the s ideals that says well if I take an s unit and I take the principle ideal generated by it well that that principle ideal you know has since I have a s unit it it doesn't have any valuation outside of s so it can be written as some product of primes in s with various exponents so I get a map there and because I took s large enough to generate the class group the co-colonel of this map the quotient of the s ideals mod the image of the s units here is the class group because I mean what is this it's ideals mod principle ideals except I'm only I am only only looking at the stuff that involves s but if s is enough to generate the class group I'll get the whole class group okay so that's a way to get the class group all right now we were interested in this co-p subgroup of the class group and so one way to to pick out just co-p subgroup is to not take this map but just tensor everything involved with zp so we're using tensoring with zp mainly as a convenient way to kill all of the parts of our abelian groups that aren't the the co-p subgroup all right and so if we actually so what are these these groups that we're mapping between well they actually as z modules os star mod mu k so this is just a free abelian group on s generators so it's z to the size of s sorry the size of s generators if k is imaginary or z to the s plus one if k is real so you might have just proved that when you prove Dirichlet's unit theorem or if not you can use Dirichlet's unit theorem to prove that that's the size of the the s units and of course the ideals the s ideals are just the free abelian group on those primes and s so they're also isomorphic abstractly as a z module so these are just as as groups these are just z to the size of s or z to the size of s plus one and so we can imagine if we picked a z module basis of these things it's a little easier to imagine doing that for this group than than this one but in any case certainly there are some z module bases then the reason for doing that is then this map becomes in particular this map that we care about becomes a very concrete thing it becomes a matrix because it's mapping zp to the s size of s to zp to the size of s or maybe in the real quadratic case it's mapping zp to the size of s plus one to zp to the size of s so here so it's a matrix an n by n plus u matrix where u equals zero or one according to which case we're in right so if you have these matrices so for every k okay we made some choices but then we get this matrix we get this piatic matrix and you might wonder how might these these matrices be distributed like among zp matrices how are they distributed and very concretely there's zp matrices you could take the mod p right and then mod p you have an n by n plus u matrix mod p what might you get and maybe if you have an idea you think maybe they're uniformly distributed why not they could be anything maybe there are every possible matrix mod p with equal probability and then well what what would you guess how might they be distributed if you have no idea you might think well maybe they're equidistributed as matrices mod p squared as well and if you sort of continue that then you would be imagining that the matrices were distributed via harm measure over over over zp so that is just so if if you think that they somehow come from a uniform measure mod p on the finally many matrices and a uniform measure mod p squared etc that leads you to say that oh they're distributed over over zp from the harm measure on on this group and if you haven't thought about the harm measure on this group then it just says the same thing is basically there sort of should be maybe uniform mod p and mod p squared mod p cubed and etc and maybe they're not exactly but maybe just approximately distributed that way that's certainly the most natural measure on this base of matrices and so that leads to a random matrix question if I take say a random matrix from harm measure so this ncp I'm thinking of as a random matrix that means a random variable valued in these these periodic matrices from from harm measure what is the distribution of its co kernels and why why did I care of the co kernel because remember that this periodic matrix that we built its co kernel what so this is I'm thinking of as a matrix that actually came from some number field and its co kernel is the class group of the number field and now I'm now I'm asking for some matrices that didn't come from a number field these matrices I'm just saying by fiat I'm defining them to be the matrices that come from harm measure on this base of p adding matrices and what is the distribution of their co kernels and so I'm going to briefly briefly sketch how you could answer this random matrix question this question about patic matrices from from harm measure and this is worked out again in in much more detail in the notes but here's a sketch okay so we want to understand the distribution of the co kernel so the co kernel right is zp to the n mod the image of the p to the n plus u so this is you know N sub p remembers a map from zp to the n plus u to zp to the n because it's a matrix and this is the what I mean by the co kernel and I want to think about when is that isomorphic to some fixed group B so B is just some is some fixed fixed group and if I want this to happen for such an isomorphism there have a map from zp to the n to in fact I know exactly how many such maps there are there be to the n and I should maybe say this is sort of a fixed a billion p group since these are the sort of possibilities that I'm interested in there so there are there be to the n in such maps I can just send any generator you know any of the n standard basis elements here to any element of B and this question from another point of view given maps what is the probability that that map gives gives such an isomorphism and so for such a map to give isomorphism first of all it has to give a map from the co kernel to be so this image of our matrix NP has to actually be in the kernel of f so we need need this to be in the kernel of f well that just means that the image of each of the n plus you generators should should be in that that kernel and since it's a map to be each of the those things happens with probability one over the size of B and so that all of them being in the kernel is this probability the size of B to the n minus you and then once we have this in the kernel we need to make sure that there's nothing else in the kernel for this to actually be an isomorphism and you can compute the probability that this generates the kernel in a pretty straightforward way because it's something that you can check mod p by not not yama's lemma if I have this map of zp modules and I want to know does or I say have some elements of a zp module and I want to know did they generate the zp module I just need to check if they generate things mod mod p so that's that's a sort of finite computation about whether certain you know what the probability of elements mod p you know vectors mod p will generate a space and because we're taking everything from harm measure mod p becomes the uniform measure and this is a computation that one can do and that leads to the conclusion that the limit as n goes to infinity of the probability that this random matrix from harm measure is B is this is a constant again over B to the u times the number of automorphisms of B and so here that you can sort of see the B to the n and the B to the n minus u they combine to give this B to the u factor in the denominator this probability of generating the kernel actually in the limit as n goes to infinity turns into the constant and this odd B factor comes from the fact that I was thinking about actually counting isomorphisms but if you are isomorphic to B you'll actually have odd B isomorphisms to B so by counting isomorphisms I've sort of over counted by a factor of odd B so that's where that that comes from in that argument and these are indeed in the cases of u equals 0 and 1 the Cohen-Linstra distribution conjectured for the class group of imaginary and real quadratic fields and so then then you might you might might wonder or hope so maybe these if we go back to these matrices that actually had to do with number fields these matrices that we got from looking at the map of s units into s ideals whose co kernels were the CLP subgroups of our class groups maybe those matrices are distributed from harm measure on matrices now there are a lot of things to you know that's not quite a precise statement what I mean by that I should say you know I sort of sneakily took this limit here as n goes to infinity of course what was in it was it was analogous to the size of the set we needed to generate the class group but we can always use a bigger set to generate the class group so it seems okay to think of n as being very large but it isn't fixed in is in is is going to infinity and so to even say this this is there approximately distributed from harm measure on this you need to have some notion of what that means when n is not is not fixed but you can make such a notion and I point out in the notes Friedman in Washington who were thinking about these matrices in relationship to the Cohen-Linster heuristics for function field did did even make such such a notion of what this just you know approximately distributed from harm measure on these n by n matrices where n is changing might mean and you might hope that if if you knew such a thing that that would then imply the Cohen-Linster distribution for quadratic fields because we have these these matrices whose co kernels are literally the class groups and then we know that a certain distribution on matrices gives the Cohen-Linster distribution and so if at least asymptotically or approximately the the matrices from class groups look like they come from this harm measure then then you could hope that you know with with enough precision in what these statements mean that that would imply the the Cohen-Linster distribution for quadratic fields all right and I just want to say and to even make sense of this to make these matrices we need a basis for the s units and that's a that's that's maybe the thorniest piece of the whole thing I mentioned above to choose the matrix we also needed a basis for the s ideals but that's that's there's a very natural basis you just take the primes at s so yeah so that is a thorny thing to think about on the other hand if you are using a computer algebra system that will compute for you s units it gives the basis so it's not it's not like there's not a sort of deterministic way to make make a basis and in particular some computations that a student working with me a few years ago did suggest that that this thing that you might have hoped for here that these matrices look like they're coming from harm measure on matrices over zp really doesn't doesn't doesn't appear to be to be true and even if you look at aspects that are are agnostic as to the your choice of basis here that that doesn't look to be the case and so okay so that then you might wonder so so what's going on what does that mean for this for this whole story and so now I want to tell you about another random matrix phenomenon which falls under the the broad umbrella of universality and probability theory so it turns out actually many more distributions of random matrices have their co kernels giving the Cohen-Linster distribution besides you know in the key here is not just from harm measure so let's say we take any distribution for example and this is really kind of the tip of the iceberg but it will give you the sense so if we take any distribution on the Piattox that is not is not completely concentrated on one value mod p all right so it needs this it can't just be a deterministic value that puts the entire distribution on some value mod p now I just want to say that is a contrast I'm not saying you have to be uniform mod p you don't have to see everything mod p with probability 1 over p nice and even I'm just saying you can't be completely a hundred percent of the time you know three mod seven you have to at least like point zero zero one percent of the time also be two mod seven but you don't even have to see every value mod p the distribution does not have to be supported on on all the values mod p it has to just be to support it on at least two values mod p so this is like quite a this is not just like nearly uniform distributions this is like everything that isn't you know that is not that deserves to be called a random element at all and let's say we took our our matrix not with our random entries but entries so this iid means that the entry should be independent and they're identically distributed from this distribution so maybe that distribution is like is that you know you're ninety seven percent of the time your three mod seven and three percent of the time your two mod seven and that's your your distribution and so you take such a random matrix so this is this is not at all like it's coming from harm measure it's not equally spread around on zp it just prevented from being really the same thing all the time if you do that then one has actually it's a much more difficult result but that as in goes the size of these matrices as in goes to infinity the distribution of the co kernels of these matrices is in fact exactly the same distribution as if you had taken the matrices from harm measure and so so it's not it's not a special thing about matrices from harm measure that their co kernels give you the Cohen-Linster distribution indeed it's a very very general thing so this is analogous like to the central limit theorem if you average a bunch of Gaussians appropriately normalized you get a Gaussian but the central limit theorem tells you that you don't have to average Gaussians to get a Gaussian as your sort of normalized average you can put any kind of garbage in and the central limit theorem tells you appropriately normalized you get a Gaussian out and this is this is in that same same vein you can put kind of lots of garbage distributions in to make your random matrices and you still get the Cohen-Linstra distributions out so that I should say shows how in the sense of universal and important these distributions are so that leads me to to my final question which I think is very interesting question to start pursuing computationally so what is then since I said it doesn't look like it's coming from harm measure what is the distribution say empirically of the matrices that actually define class groups and can you at least conjecturally look at those matrices make some conjecture that appears computationally true for for what kind of distribution those matrices are coming from and then now a theoretical question does the universality hold for for for that distribution meaning will their co kernels have the same Cohen-Linstra distribution we certainly would expect that to be the case but I think it's an interesting opportunity to to try to understand these heuristics through a different angle so that's it for today questions yeah and basically you can see that I mean like for example like if it was always zero mod p right you can easily see that you'd be something and then if you're a matrix and you know like and you're all to mod p then mod p you can't have very much right so you yeah you can easily see that if you took a distribution that was 100% of the time the same thing mod p you would get like in fact a very funky co kernels because the matrices would have have very low rank mod p and the co kernels mod p tell you about the p torsion of the groups so so the question was when I talked about these these the specters that need to be in the kernel were they independent events so the reason that they are independent events is because we were considering matrices say like in by in matrices from harm measure and a way to get an in by in matrix from harm measure is to independently take each column from harm measure on just Zp to the n on the columns and so that basically comes from that comes from this you know sort of symmetries in the niceness of harm measure that that a harm measure and by in matrix is the same as the independent harm measure column from each column all right let's thank Melanie again