 We have a brief advertisement for Central Number Theory, which is a new journal that is starting, and it's under a very different model than traditional journals. In particular, the standard for publication is that articles should be extremely useful to the community, as opposed to, you know, original research breakthroughs. The standard is usefulness. So if you are doing some computations or developing computational tools, in particular, that might be of a lot of use to other people. That might be a kind of paper that you could consider submitting to Central Number Theory and feel free to talk to me or contact any of the editors if you have an idea and wonder if it's an appropriate paper. Okay, and so I will get on. Okay, and now we have the next announcement I wanted to make. Yeah, a little follow-up from yesterday. So I should have mentioned when I was talking about error terms, Bob Hoff has a paper called Equidistribution of Bounded Torture and CM Points where he gives a theoretical heuristic suggesting, I mean, yeah, that the average of the Z-mod KZ moment, say over imaginary quadratic fields, where K is odd, might be one, and then a secondary term that's decaying like X to the minus one half plus one over K. So for K equals three, that would give the minus one sixth exponent that I mentioned was actually a theorem of Bhargava Shankar, Zimmerman, Tegucchi and Thorn. And then for K equals five, it would give this minus three tenths, which would be a faster decay, which was the direction we saw on the graph that the five moment seemed to be approaching the limit faster. In any case, he looks at just a small amount of data for this and says that, in the paper, says that for K equals five and seven that this looks good with the data and for K greater than or equal to nine, it looks not so good. So I think that it's still very much open to understand what these secondary terms would be, but I wanted to make sure to point that out this is one paper that's in the mix that you should know about. You wanna think about the speed of convergence of these moments. All right, so today, I'm gonna talk about conjectures for class group distributions for higher degree extensions. So we're gonna let G be a finite group and we're gonna take K over Q Galois G extension. And we're gonna say ask about the distribution of the class groups of K as K varies over G extensions. But one thing happens here is that these class groups of G extensions, they're not just finite abelian groups, they have an action of G because the Galois group of K over Q acts on the class group and so they are modules for the group ring Z join G. So that's just another way of saying that they have an action of G that's compatible with their abelian group structure. And so since these class groups are ZG modules, we should ask for their distribution as ZG modules, not just abelian groups. That's sort of part of the philosophy. So here's a sort of thought experiment. You know, when we were just safe for imaginary quadratic fields and real quadratic fields when we were talking about the class group, we asked about the distribution of the class group, not the class number. I mean some people, you might think about the class number, a lot of people, especially you're doing analytic number theory, a lot of things work just with the class number. But if you had tried to do that, you would have for say some number in these relative probabilities, say for the case of real quadratic fields, that would be the sum over all the finite abelian groups of order in of this kind of term. Now you can write that all explicitly because we know, you know, in terms of the prime factorization of in, what all the finite abelian groups of order in. But this is in terms of in, is just some really garbage expression, completely impenetrable what it means. And then once we, but once you consider the group structure, then for each group you're just saying it occurs with this relative probability and those terms all have a lot of meaning and are much easier to understand. So similarly to the fact that the conjectures for class groups of quadratic fields look much nicer and more sensible when you think about them as distributions on groups and not on numbers that record the size of the group. We should think about distributions for class groups with the full structure that they have. So in this case for Gaoua, G extensions that is as a ZG module, okay. All right, so let's think about the class group as a ZG module. So there is a particular element I'm going to call in thinking of it like norm in the group ring, which is the sum over all of the group elements. And this in element on the class group acts as zero, sending it all to zero. And why is that? Well, the norm thing was a hint, right. This would take takes an ideal and multiplies it all by all of its conjugates. And maybe we should say, I just realized maybe I need some, some let's say, we're at primes away from dividing G. I'll just say that for now. Cause this is what we're really going to talk about. Remember P is a prime. What? Oh, okay. I was just, we're about to restrict to that. So I was just doing that to make it a simpler statement, but yeah. Okay, all right. Okay, which I was about to write here. All right, in any case. So the, then since we're going to think about just as before, because it simplifies some issues, we're going to just think about the P-torsion or the P-power-torsion, the C-lo-P subgroup of the class group for a particular P, then the ring that is acting more precisely, we can say instead of just the group ring acting, we actually have the action of this following ring R, which is ZPG, so this group ring. And so the ZP here is acting. And it's that is that we're putting that there basically to emphasize that we're only looking at the C-lo-P subgroup that there's only P-power-torsion here in the finite modules. So the ring that acts on the C-lo-P subgroup of the class group is this group ring, ZPG, mod this norm element in, okay? All right, so as before, in the quadratic case, we separated the fields based on their behavior at infinity, and there's not so much in the quadratic case that can go on in terms of behavior at infinity, but it can get a little more complicated for larger G. And so we're going to keep track of the decomposition group at infinity, G infinity. So that's a subgroup of G. So another way to say that here is what that decomposition group is. So that is a group of order two that is where complex conjugation lands. So here what this is, is keeping track of the conjugacy class in G of complex conjugation, okay? So this S, G, Gamma is gonna be all of the G extensions but that have a particular behavior at infinity, which we mean like complex conjugation acts as a certain element or a conjugacy class in G, okay? And so, Cohen and Marnay, building on the conjectures of Cohen and Lindstra, made conjectures for class groups of higher degree extensions, and in particular their conjectures imply that for P that doesn't divide G and F a function of R module, so this is the group ring thing acting here and we're thinking of this as an R module, this following kind of average. And you won't find it written quite this way in their paper, which is why I sort of wrote over here a reference where we show that their conjectures imply this distribution, okay? And so this is the sort of technical kind of statement here that we saw before in the quadratic case, so we're taking a limit as X goes to infinity, we're averaging over fields of discriminant up to X. Here, this is saying what fields are we averaging over? The G extensions with G infinity, their decomposition group at infinity or how complex conjugation behaves and over those functions, we're averaging, sorry, over those fields we're averaging some function of their class groups as R modules, all right? So F is a function that sees the R module structure and what do we get? We get the average of that function over P group R modules with a particular distribution and each R module occurs with this weight and then there's a denominator here that makes, you know, there's the sum of the weights so that you have a proper average here. Can I ask you this? Yes. Does it, do those conjectures imply this for all F or are they applied only for reasonable F? Well, the conjecture says for reasonable F so it only applies it for reasonable F, I suppose. Okay. Yeah, so the conjecture and the conjectures don't give a precise definition of reasonable and the notes I give a few references to a few papers that have kind of tried to grapple with what reasonable might mean, but yes, yeah. So, yes, yes, yes, that what a G extension properly should be is an extension with isomorphism of its gaol group to G so that you can make, otherwise it doesn't make, right, otherwise the class group is not a ZG module. Right, right, yeah, great question. Yeah, so again, I suppose I'll just say these should be, I mean, this is reasonable, where reasonable isn't defined after all, they're just heuristics or conjectures and yes, one, I'll just sort of repeat this question in order for this class group to actually be a G module, one has to have an identification of G with the gaol group so that has to be part of the data of what we mean by gaol G extension and that is handled in the notes with some terminology of when we fix the isomorphism and when we don't. And what is this, so this term here makes a lot of sense, we've got the automorphisms as an R module and what, there's this term here, so this is the elements of A that are fixed by this G infinity that are fixed by complex conjugation, so this is the fixed elements of A, okay. So that's how this term is where the behavior at infinity comes in and so now you can see that, oh, you might have different elements of complex conjugation that give you the same signature, say, of your field, but they're going to give different behaviors here. And summarizing how you might say this more informally is that the conjectures say that the class group is distributed as an R module with these relative probabilities. So, I mean, we write this big equation with the two sides and the big fractions and everything and that's sort of informally what's happening that in this family with this ordering, this is saying that the class group is distributed as R module with relative probabilities and what relative means is that the probability isn't this, it's this divided by the sum of all of these weights so that they're actually numbers that add up to one. And this distribution, we were talking yesterday about how the moments are so important for distribution so this distribution has moments, the beef moment is one over the size of the part of B that's fixed by the G infinity. So that looks a lot like what we had for quadratic fields except now that A to the U is being replaced by this and you can think it's exercise, I think it's exercise in the notes, you can make sure it was that, is that consistent with what we said in the quadratic case and indeed it is. Okay, and just as in the situation we were talking about for these distributions that were predicted for class groups of quadratic fields, these moments determine a unique distribution and I should say a couple of things, the moments here now are surjections in the category of R modules and this everything kind of continues to follow this philosophy which is like you should use all the structure that you have. So before surjections were just in the category of a billion groups, now that we're thinking about distributions of R modules there in the category of R modules and this is a determined unique distribution of R modules. So it's important to know what structure your class group has, what category you should be working in. All right, so now some warnings about these conjectures, these conjectures or maybe we should call them heuristics, the heuristics conjectures need some modifications. So first, Mala did some interesting research in empirical computations of class groups and his work has really been the main work looking systematically at class group distributions of higher degree fields and also over other base fields other than Q and it's now a decade old. So there's really a lot of space for more computations to be helpful but through his computations from the tables he was creating it looked like the conjectures were wrong at two and even though for my talks I've been focusing using Q as a base field for everything that we do, of course there's a theory where you could replace Q by an arbitrary number field say K zero and so more generally his tables, if he was using say Q join the square root of minus three as a base field he was having some problems matching the conjectures at P equals three and so more generally when P divides the order of the group of roots of unity in your base field it looks like the conjectures were often so this is why people are referring to this as roots of unity issues. Now since Q just has two roots of unity over Q this only affects the conjectures at P equals two and there is work trying to understand what one needs to do in this case and there are several references in the notes and this is also an area that I'm actively working on and trying to understand what should be fixed here and then another caveat, another way in which these conjectures are wrong as stated so Bartell and Linser have a paper in which among other things they pointed out that for some groups when you order by discriminant G extensions a positive proportion of those will contain some fixed subfield and that subfield will have some class group and that class group will affect a positive proportion of the class groups of G fields and that can really throw off the conjectures and so they suggest to fix this by replacing the discriminant by another ordering, another invariant that doesn't have this property so perhaps the product of ramified primes which is similar in a lot of ways to discriminate we kind of feel like it's in some ways measuring the same thing but there are a lot of reasons to believe that if we order fields by the product of ramified primes you won't have this positive proportion problem and that makes the conjectures more likely to be true and there are also other invariants depending on the group for which one doesn't expect to have this positive proportion issue including for many groups by the ordering by discriminant and so it's kind of interesting to wonder like well what should work should any ordering work that doesn't have this positive proportion or should we stick with something like the product of ramified primes and there are a couple suggested projects exploring this question of what the appropriate ordering might be in the notes and I think that somehow there is nothing in the spirit of the original heuristics or conjectures are tied to the discriminant ordering it's more of a oh as we look among fields and okay we have to order them somehow so we order them by discriminant so I think this is a really an interesting question to understand how different orderings affect things and which ones might be most appropriate so those are our two known problems with that conjecture that I just stated I will remark that if you avoid roots of unity in the base field and you do order by the product of ramified primes that a paper of mine with you on Lew and David Zergbrown shows that these the conjectures that I just stated in the Galois case hold over the function field at QT with a early Q goes to infinity limit so in some limit where you take Q go to infinity before you take the X going to infinity one does have the conjectures so that suggests that there aren't too many more things to fix but there could certainly be both things that disappear in the Q go to infinity limit or arise in the analogy between number fields and function fields being not perfect that could be going on in this situation not to mention this only addresses this one ordering yes question, no you do not fix the characteristic yes, I don't, wait are you saying usually referred to? I don't know if people use the language large Q limit I mean there's sort of two, I mean let me just say there's sort of two ways that you could take a limit where you put the Q limit here and you put it outside here and roughly this is the easier one and I don't know if people say large Q limit specifically to refer to one of those two places or I would just use it more generally when you know somewhere you've taken a limit Q goes to infinity, other questions and sometimes when people say large Q limit I think they do mean that you fix the characteristic and like Q go to infinity that's a totally different kind of thing yeah but okay so that is what the conjectures are for distributions of class groups of Gaoua fields so what about non-Gaoua extensions? Especially because one might want to start working with low degree extensions you know you start your cubic extensions while some of them are Gaoua but then you have non-Gaoua cubics and obviously like nice accessible case of fields you might want to understand something about but they are not Gaoua all right so here we're gonna take G to be a finite group and H to be a subgroup and we'll start with a Gaoua extension L over Q which will be a Gaoua G extension but then we're gonna take the H fixed field and that will be K and so this is how we're gonna think about our non-Gaoua extensions they all sit inside of some Gaoua extension for example their Gaoua closure and so we can think of them as you know for a certain G and a certain subgroup H the fixed field of some Gaoua G extension by H all right and so for a prime P that doesn't divide the order of the group G it in this setting the PCO subgroup of the class group of K is simply the H invariance of the PCO subgroup of the class group of L just like K is the H invariance of L so the elements fixed by every element of H and you can see this is shown in the notes or maybe there's exercise about it in the notes how this is the case but this map is for example just by the inclusion of ideals of from K and L so because of that in principle the Cohen-Martinet conjectures for the distribution of the class group of L as a G module imply conjectures for the distribution of the class group of K as a P group because we have some distribution on G modules you can have the forgetful functor where we just take every G module to its underline abelian group and we have some distribution and it pushes forward and then you could say well what is the probability of getting some sort of P group okay it must be the sum over all of the all of the G module structures you can put on that P group of one over okay but that's a lot that's maybe not so easy to think about and in fact it kind of reminds us of my thought experiment at the beginning where I said oh you know what if we tried to do the Cohen-Linster heuristics but just asked about the distribution on the sizes of the groups and then your probabilities are some weird funky sum of something. So are you right for a getful? This is the H invariant functor yeah. Oh yes sorry yes yes yes thank you and it's not the yeah yeah that that yes thank you. I guess it does forget something but that wasn't good to write forgetful there yes I mean the functor that takes the H invariant thank you. Okay but so we so in a paper of mine with my former graduate student Waitong Wing we actually we work out what this push forward is what distribution you see in the non-Gawa case implied already by the conjectures in the Gawa case. All right so I'm gonna tell you about that. The easiest case to say what is implied is the following so all right we always we have our G and our subgroup H that work our non-Gawa fields remember are the fixed fields for H so G-AX on the cosets so the of G mod H so this is a have to be a group this is just a set of cosets so since G-AX on this set G-AX G has a representation over say the complex numbers of dimension the number of cosets and this is also I mean this is also the induced representation from the trivial representation from H up to G is another way to write this representation so this is a representation over the complex numbers of the finite group G and then I'm gonna build and inside of that there's always one copy of the trivial representation because say G-AX on the basis elements here by permuting them around so the sum of those basis elements is always gives you one dimensional trivial representation so I'm gonna get rid of that I'm gonna throw that out and then I get this, oops, this representation here so the induced representation minus one copy of the trivial representation and the easiest case is when that is a irreducible representation and I write absolutely, I called it absolutely irreducible in the notes because one also sometimes considers the representations over Q and I'm saying it should be irreducible over C all right and so in this case where that induced representation minus the trivial representation is absolutely irreducible the conjectures of Cohen-Martinet say the following that if you take class group averages so here notice how I set up this average the sum is over Galois fields these were our Galois fields and it's ordered by the discriminant of the Galois field that is just how it works because of the conjectures that they made, the one as I mentioned before might wanna consider other orderings but then this here, this is we're taking the class group here of these, what I was calling K the class groups of the fixed field so we're only asking about the distribution of the class groups of the H fixed fields and we show that it's given by the following you know the conjecture suggests that it should be given by the following oh I forgot to put the F up here the following distributions there should be an F up here where this average with weights A to the U times the size of the automorphism group of A and so here U is the number of cycles of this decomposition group G infinity on the cosets minus one which is also turns out to be the unit rank of our non Galois field here, the fixed field all right so U is the unit rank and so that is maybe easier to match to the quadratic case where we had U was zero in the imaginary case and U was one in the real case okay so again just to sort of we have the big formula just to summarize of what that is saying is it saying when we take these H fixed fields their class groups should be distributed with relative probability one over size of A to the U automorphisms of A where U here is the unit rank of the field K and this distribution here you know we already talked about it a lot in the cases U equals zero and one before and it's just it is like those determined by its B moments which are the size of B to the minus U and I should just point out you know I said that there were some copy of some warning some roots of unity issues and which counting invariant you use issues with the original conjectures and they all apply equally well here because this is literally just telling you what the Galois conjectures imply in the non Galois case however this gives one you know many many more examples you know so for example this includes this this case where this this particular representation is irreducible includes the degree D S D field so what we sort of think of as the generic fields the degree D fields whose Galois closure has group S D and so in particular the non Galois cubics alright okay and so one also take away from this is that when this representation this induced representation you know minus one copy of the trivial representation when that is irreducible the class groups of the fixed fields here these non Galois fields they have no additional structure and why is that the takeaway well in this philosophical context the fact that the probabilities mainly had to do with the size of the automorphism group just as a group of the you know of the different options sort of indicates that that we you know found the structure on these objects which is that they are groups alright however so when this representation is reducible then actually these class groups of you know non Galois not necessarily Galois fields always actually have extra structure there are not just groups so here's maybe a first example which is easy to think about so we were talking about quartic D4 fields on the first day so if we let GBD4 say the symmetries of a square then there's index for subgroup H say generated by the transposition to four so this is a index for subgroup and so when we're taking a Galois G field and then taking its H invariance we're getting quartic D4 extensions and as we mentioned before these are quadratic extensions of quadratic extensions and that top quadratic extension we know it is Galois so there's there is some automorphism of that that top extension fixing the intermediate extension and so quartic D4 extensions actually have two automorphisms so they're not Galois they don't have four automorphisms but they have two automorphisms and of course if your field K has automorphisms those automorphisms are going to act on the class group and so the class group is not just here then a Z module it's a you know module for the group ring of of the automorphisms of the field if at least however that is not the only way in which the class groups acquire extra structure so for example if we take G we could take G to be H5 sorry A5 alternate group and here I wrote down a particular index 10 subgroup so in this case when L is a G extension and we take this fixed field so this is a degree 10 A5 field you might call it it's a degree 10 field whose Galois closure is A5 that won't have any automorphisms but this induced representation is not irreducible and I promised you that in that case there would be some extra structure on the the class group and that occurs here and so let me let me tell you now about what that what that additional structure is okay so alright so I'm going to write down E for a particular element in my group ring here and I'm thinking it was my group ring but there are these decorations we took like the ZP version because we're only thinking about the CLP subgroup and we quotient it out by this norm element because we knew it acted trivially and so but in that in that group ring which I was calling R I'm going to take an I'm potent element here the average you know of the of the elements H for each H in our in our subgroup H so one over the size of H times this sum and this it's I'm potent in the group ring that isn't isn't hard to check since H is a subgroup but it's not the kind of you know I'm bones often in this kind of semi simple situation you think a lot about the central I'm potent this is not necessarily a central I'm potent because H there doesn't have to be a normal subgroup okay but I can build that I'm potent and then from that I'm potent I'm going to build another ring a T and T is the the elements sort of you can think of as a sub as a subset of R of the form E R E so that subset of R E R E is additively closed you can check that and it's multiplicatively closed because if I have like something any something E I still have ease on the outside but probably most people would not want to call this a subring because while this ER E has a multiplicative identity which is E that is not the multiplicative identity of R but in any case it does have its own multiplicative identity so so this is it this is this is a ring here okay and another way to think about this ring is that it's an order indeed maybe I should have said like a maximal order it's a maximal order in the hecke algebra of finite groups now that we write like this so that's only useful if you've already thought about the hecke algebra of finite groups I'm just saying this is another way of writing that same sort of thing you know except I'm taking an integral integral version of it and the thing about this ring is that if I have a R module so then then this this new ring T the R with the sort of these impotence from H on either side of it T naturally acts on the H invariance of of B and so there are some exercises about that in the notes but sort of briefly you know you you you just you know literally act by how this element you know tells you to act and because you have an e here on the left you'll end up with with an H invariant so you know when you multiply something by e it becomes it becomes H invariant that's exactly what the averaging over over e does so and I should say I have been using all along continually the assumption that P doesn't divide the order of G which makes this all much simpler however one can develop this this theory in the case when P divides G but it's much more complicated and where is I using that I'm using that so that I can invert the order of H and not have to to worry about that and it's much more subtle to build the appropriate maximal order in that in the hecke algebra when P divides 0G but yeah so I'm taking advantage of that so that when I oops when I act by 1 over H I'm I can just do that easily because I'm acting on a on a P group where it where P doesn't divide the order of H alright and so you know we've just said so if B is an R module then then the H invariants are a T module and so remember that the case of interest here was we had some class group of L which was an R module and then when we took the H invariants we got the class group of our non-Gawa field and so then that tells us that those are a T module okay so maybe I'll just say that I said axon that's kind of how you would say it if you're talking about a group I should say B sub H is a T module alright okay and so now that actually works in any case even when you know the the whatever representation was irreducible but of course sometimes T is just ZP so if I say okay you're finding to be in P group and you're also a ZP module that's not extra structure that's what you that's the structure you already had and so that's the case that this T is giving no additional structure if and only if this this representation this induced minus trivial representation is irreducible and so when it's not irreducible like in that degree 10 a 5 fields example that I gave then there is additional structure as a module for some you know non-trivial ring on the class groups even when it's not say coming directly from from automorphisms of the field any questions yes yes yes yes so you can yeah you can see in particular that that the the group ring for the automorphisms of your non-gowl field will sit as a subring of T but he can be bigger than that I mean as we saw in the trivial case in the D4 case it's actually not so I gave that D4 example T is just the group ring of Z mod 2 Z in the D4 case but you know you can have other cases where there are two automorphisms but the ring T is bigger than that yeah that's a good exercise I should have put that in the notes to show that the group ring of the automorphisms of the non-gowl field sits as a as a subring of this T other question yes yeah so this is a great question so we what if say like H was was a normal normal subgroup so I mean I kept saying oh we're talking about non-gowl extensions non-gowl extensions but I never said oh yeah H isn't allowed to be normal so actually this applies completely generally including the case when H is a normal a normal subgroup and so part of actually what we show in this paper with Wei Tong Wang is that there's a sort of self-consistency of these of these conjectures because in fact even in the gowl case it can also be always embedded in bigger gowl field and then it's the H invariance for some bigger gowl field so you could always ask you know do the conjectures for some bigger group and then we take the invariance to get down to the group that we want you know are those are those consistent with the direct heuristics for G fields and they are and we show that and yes so so you could you know if if you and then the upshot to answer your question directly is like yes if H is a normal subgroup what you get here is just the group ring of the quotient group so that it's consistent with the original gowl conjectures other question yes yeah I guess I would just like do do a computation in the you know with the characters you know for the for the representation so right you can you can take the pairing of the representation with itself and that'll be one if and only if it's irreducible so yeah in some particular case that that's like a not so bad computation to do with the characters of the finite group G good question so but I should say that that leads perfectly to what I'm gonna say next but sometimes VGH is not not irreducible and in those cases then I hope like I've repeated the philosophy enough that you all could sort of know what is going to happen next we should ask about the distribution of these class groups of these non-gawa fields so secretly they're just like any fixed fields which they could could be gawa but we should ask about these as T modules since indeed they are T modules and then in that case the Cohen-Martinay conjectures imply that these class groups of the fixed fields are distributed as the following so I'll just sort of scroll up so we can see the the the informal version so we have you know the function with these weights same thing same thing so the informal version is that a group a T module B appears with this probability so we have the automorphisms of B as a T module certainly that we expect and then this this factor here which we knew somehow has to involve the behavior at infinity to be consistent with everything else it's a little more complicated and before we took the invariance of the decomposition group at infinity or the invariance of complex conjugation but that doesn't act on B because B is not a mod it was the you know it's a T module it's not our module it was you see you're thinking of it is like it's the H invariance of something so it doesn't have a G action but if we tensor product with T over T with R e so then this this R e is a is a right T module that's where you put the E there so that it would be a right T module and it's a left our module which is the the group ring then this is precisely the thing that does have an action of G and so you can take G infinity invariance there all right so that's that becomes a little more complicated of an expression but that's that is that is precisely what the conjectures predict that a particular T module occurs you know in this family with this behavior at infinity with this relative probability and so it would so these are a lot of predictions and there you know there are a lot of more specific pres predictions by not just considering the gaol fields case but the implications for the the fixed fields inside those gaol fields a lot of predictions for a lot of low degree fields besides just quadratic ones and it would be great to have computational evidence for or if it happens to be the case against these these predictions and so in the notes I make many as specific suggestions for for projects of cases where I think computations would be particularly enlightening or use useful so especially around what I said with these this warning these caveats and their their their corrections so around like oh well maybe we shouldn't order by discriminate it should be ordered by product of ramified primes what about some other ordering that people have that's well studied and maybe has good properties so there are many many suggestions around those and then also there you know our suggestions in cases where no prediction is made so in particular I just gave a fairly exhaustive descriptions of predictions for the primes that don't divide the order of g and when primes divide the order of g it is more subtle so I should say when p divides the order of g so sometimes sometimes the paper of Cohen martin a makes a prediction and sometimes not so it depends on the p in the g and it's not so complicated for which to to say for which p and what that sometimes means but maybe a little bit beyond the scope of this talk and so so also in the notes I give several examples of cases in which there is no P and G for which there are a pretty low degree fields for which there's no prediction made at all by anyone so I think there are yeah there it would be very exciting to see more more computational evidence for the the distribution of class groups and so yeah one other thing that I wanted to say was yeah so when when p divides g you know there's there's more to say for example like when there is a heuristic event and when there's not and what what the different behaviors could be and even though that's beyond the scope of the the talk I feel like I'd be really remiss to not point out one of the the major developments in this area of recently which was Alex Smith has determined the distribution of the LCO subgroups of class groups for CL extension so this is the cyclic group of order L I'm talking about here and so this for example this includes quadratic extensions and their two CO subgroup and so this is in that case for example precisely the part that we in my second lecture I said oh we're just going to ignore because we're genus theory tells us about the two torsion but it only told us about the two torsion it didn't tell us about the whole CO2 subgroup and so there is certainly more that you could say and he finds that that distribution of the rest of the CO2 subgroup beyond the part that we already knew about from genus theory and then also the analogous thing in the CL extension and his very long-awaited papers are now on the archive and so you can read those and actually if you're interested he is going to be running an online seminar going through those papers that are available this fall and so if you're interested to learn turn more about that I encourage you to look into that seminar all right and that is it for my next