1/4 Analytic number theory around torsion homology

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1/4 Analytic number theory around torsion homology
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Point (geometry) Filter <Stochastik> Group action Randomization Presentation of a group Homologie State of matter 1 (number) Student's t-test Element (mathematics) Different (Kate Ryan album) Square number Modulform Energy level Number theory Integer Addition Theory of relativity Generating set of a group Gamma function Subgroup Arithmetic mean Vector space Quotient Lipschitz-Stetigkeit Quadratic field Family Abelsche Gruppe Near-ring
Axiom of choice Point (geometry) Complex (psychology) Group action Divisor State of matter Dimensional analysis Element (mathematics) Prime ideal Frequency Centralizer and normalizer Complex number Moving average Modulform Series (mathematics) Endliche Gruppe Gamma function Weight Physical law Variance Basis <Mathematik> Sign (mathematics) Vector space Energy level Spacetime
Presentation of a group Group action Homologie Hecke operator Multiplication sign Water vapor Inverse element Mereology Sign (mathematics) Many-sorted logic Different (Kate Ryan album) Analogy Forest Matrix (mathematics) Square number Körper <Algebra> Fiber (mathematics) Process (computing) Moment (mathematics) Homologiegruppe Radical (chemistry) Quadrilateral Order (biology) Right angle Abelsche Gruppe Resultant Fundamental theorem of algebra Spacetime Directed graph Point (geometry) Geometry Divisor Real number Maxima and minima Rule of inference Theory Element (mathematics) Goodness of fit Degree (graph theory) Hyperbolischer Raum Average Term (mathematics) Operator (mathematics) Reduction of order Modulform Energy level Integer Maß <Mathematik> Module (mathematics) Time zone Multiplication Focus (optics) Dependent and independent variables Weight Prisoner's dilemma Model theory Limit (category theory) Einbettung <Mathematik> Coefficient Local ring Maß <Mathematik>
Point (geometry) Group action Functional (mathematics) Divisor State of matter Multiplication sign Water vapor Mereology Prime ideal Pi Many-sorted logic Different (Kate Ryan album) Natural number Square number Number theory Ranking Absolute value Social class Potenz <Mathematik> Matching (graph theory) Endliche Gruppe Regulator gene Thermal expansion Linear algebra Mortality rate Measurement Klassengruppe Network topology Order (biology) Heuristic Object (grammar) Pressure Directed graph
Point (geometry) Geometry Presentation of a group Multiplication sign Direction (geometry) Water vapor Mereology Quadratic equation Analogy Number theory Analytic number theory Theorem Fundamentalbereich Theory of relativity Regulator gene Gamma function Generating set of a group Special unitary group Elliptic curve Proof theory Klassengruppe Heuristic Right angle Hydraulic jump Fundamental theorem of algebra Resultant Imaginary number Spacetime
Group action Presentation of a group State of matter Multiplication sign Algebraic number Set (mathematics) Water vapor Solid geometry Inverse element Mereology Proper map Food energy Many-sorted logic Matrix (mathematics) Square number Number theory Ranking Körper <Algebra> Determinant Social class Compact space Potenz <Mathematik> Fundamentalbereich Endliche Gruppe Rational number Theory of relativity Regulator gene Generating set of a group Gamma function Price index Measurement Flow separation Connected space Arithmetic mean Vector space Klassengruppe Normal (geometry) Heuristic Right angle Fundamental theorem of algebra Imaginary number Point (geometry) Geometry Computer programming Functional (mathematics) Free group Divisor Real number Event horizon Theory Product (business) Element (mathematics) Prime ideal Frequency Well-formed formula Natural number Antiderivative Absolute value Dependent and independent variables Focus (optics) Scaling (geometry) Model theory Equals sign Physical law Mathematical analysis Volume (thermodynamics) Limit (category theory) Subgroup Torsion (mechanics) Factory (trading post) Quotient Universe (mathematics) Homomorphismus Musical ensemble
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Anwar and Wong in the war and located just on the say-so in every working here come if you if you can so I tried to make this the the level of this and other people of different levels and different backgrounds of its size furor it occurred a a graduate students at the songs with some knowledge of the whole Moffat forms and Moss forms of this ominous start going to look quite slowly and all the other thing is I don't know how many people half a million people are with some homology so uncharted by and large stick to the simplest case where I don't have to make any use of it at Lima's stock for a while it is still to the and and and then will suffer the 1st half an hour Saunders get the question which wouldn't discuss the rest of the year this was just some simple addition GAM as a group we get our whole gamut have means for us to be unionization of gamma OK in other words that's the largest abelian quotient Indiana so it's the quotient of gamma by Wallasey take gamma quotient and by the subgroup generated by the subgroup generated by all elements X Y X inverse widened West extra wiring again so if what we're going be talking about this for what is wall what happens when we do this to groups like cell 2 over an imaginary quadratic field right so if it is Janet comes lips his gamma comes the presentations is very easy to compute so for example if gamma if it if you happen to be given a presentation would say something like has generated by 2 elements subject to the I'm just making this example at random bits of the 5 is day-to-day then so this Debbie isolation will be so you will have to generators so squared any quotient by relations which come from this so in this case he cautioned by the row vectors 2 3 and 4 or 5 K and I guess that's the Group C 1 2 alright so you know what our start by doing is what it will do this for subgroups vessel to how explained why you might be interested in doing that from the point of view hold more effect on forms and then so former about his will discuss this gather and 1st 4 subgroups of Tuesday and then and then after we do that then I'll make this easy into Aziz Square D and we'll see how the situation changes OK so right so so I'll stick to the you it it doesn't really matter which family of some procedural also up all stick to the standard ones that we talk about a number theories of gunmen ordered an old just as usual be the elements in the CD In consultancy that is determined 1 of tremendous 1 and and divides appeared and the 2nd the same definition also makes sense Sosa right now ABC near integers but this also makes sense that if we did the square it OK OK so what happens for subgroups a vessel to the 1st OK so far are just 1st some filters the This is it a what happens when you believe that this is very closely related to wait to modular forms so some garment if you take this and you think it's appealing ization what you get some comprises roles in Sequoia comes from is you get us yeah actually I guess that the state is the amount not to make it is literally right only 1 and is
primal so even at 2 major plus 1 plus a small town in the central state small finite group and where H is going to be the dimension of the space S S 2 and so this means wait itself we too level and homomorphic more cost for us to get some innocent people familiar with that by Wednesday a small finite group in this case what it means is that every element has ordered at most 6 cases in all it's a really is very small but in in particular but the most extortion for the arts 1 explain where this comes from because this may be due to some motivation about why you would be doing this billing ization thing in the 1st place those of the point is that in fact this is very closely related to the space in which to forms and if you and if you didn't know definition away to form you would notice many of the same things if you're studying the soul of all kind explain where this relationship comes from so in fact other questions this child and if there are questions of because they're so this this relationship between these comes from there is appearing OK bi-linear appearing on the same by law period are this plus 1 and he added that this is a it's right event prime and this has to do with Eisenstein's series also it's a little bit of offenders not prime spots on something a bit bigger than any Minnesota not worry too much about it that is a binding appearance so if you had from this to the complex numbers and all the fight for use of this so given so give an element that call at gamma far are into civilian ization OK so this is an Nabiullina vision so it comes from some element gamma so that image of of some gather in the actual group to redefine the variances integrate performed so so gamma you know form path in here could be defined as to be the integral From W 0 0 mostly W. gamma W methods CPC we hear W's any element in the upper half plant itself because this interval independent Of the choice of W OK so you could so this means you take any path from W gamma W. doesn't matter which team was in very taking WA pattern governing gamma W. integrated and this gives you a complex number OK and so this gives you a kind so the uninfected except for this annoying plus 1 this is actually a perfect caring in the so that that means in fact cell in fact there is we can take a basis but gamble 1 of the underwriters very explicitly again I want to get that there is a basis for this labeled the bars for this is either so is such that these Gamal what it's just that the functional you get from them priorities the 1st weekend and gives a basis 4 the deal In this case the cost forms of that as something that strange about this of cost form this is age dimensional to each conventional complex vector space so this and do this deal space is also a two-dimensional is an H dimension complex factors but this actually gives a real basis OK gives the basis so this is H dimension will oversee but it's too H dimension overall so that's what the that's where the 2 countries OK so
this is the real is that the reason that you see this to a chair and that the tuition the judicial sources of sorts to the side I said Did you can choose a basis in such a way that history so and Morial aside from the past 1 this is it this hearing itself I cancel with the wheels of person OK so and then and then even more it's true so there's also so 1 can also defined the action of the Hecke operators defined and action of taxi operators so for every school in the southern end where n is relatively primed and say on this group In a way which is compatible with here in such a way that chief you if you apply Hecke operator on 1 side is the same as applying it on the other side OK so what this means is that in other words we could so can I recover all of the you want everything learned a lot this week to modules together with the action of Henke operators on them you can completely recover if you know the same things about this these no look at this is this the start of on the algebraic theory of modular forms that all wise this is useful for example this is a very efficient way it actually computing OK so if you take it in is a few hundred anyone write down 1 hour the week to modular forms of that level it's actually this is actually rather computable because all mentioning admitted that the very easy to write down things like a presentation for this which is not help people is exactly do it but so this is this is computationally useful and also for for example it tells you things like this is the result effective fairly basic importance for example accounting guiding values of the heady operators algebraic integers OK because it feared that you think solely solely about whole market things there is no reason why this should be the case but he over side of the secular operators are represented by integer matrices of that the but if I considered weighed about what he may still have room move to receive it and the whom I was out so if it were not to this will literally speaking this would no longer be it is no I would have been a sort of so there's no problem making an analog of this theory but instead of the billing aviation and I would have take homology with some twisted coefficient said it signed a ceasefire sure so think we need to let me know so this all no not also all the statements that we make about it'll parallel statements and higher weight which would be the only difference is there more annoying to formulate because instead of this I would have the right the homology this group with coefficients in something I was hoping to avoid that this was 1 of privatization would be more work Beyond that that that's true but so for example even for is 1 I a so so you lose 1 and yeah yeah we don't really know that we really want to go there the average value make good player at some point I'll put a prominent disclaimer that will make many statements exactly correct in the Compaq case just because it's quite OK so this is this is source speaking introduction of why you might want to do this process of appealing nicely in the 1st place a place I say that it has the same information as the space of which forms and now I would do the same thing but for others so "quotation mark gone are yeah sell the same thing but I want to replace SO 2 z of rule by something bigger appears now let's now we study the same question 4 OK so now I so very replacing z vizier squared the or by any running water and replacing the dollar as the square OK so in other words that let me call 0 z Square D and now this definition as I said still makes sense Limoges right out so work so we're clear some downward and is still it's now at Terminal 1 matrices in all now the zone where the German 1 and now this element and is allowed to belong to both in fact and could be an ideal but just to keep it as a similar to the previous day's close economic and
and divides quite so what the let's 1st take the case of where Wendy is blue with greater than 0 by the way it everything I say In these talks has an analog fought for any group that it's safer for bromide Geel firing any balanced out tries forest possible to stick to a cell to over-reported feels just because statements are clear that I don't have to talk about higher homology groups OK at the minute this is not the Avalos there are now distinctive the rights of 1st debated and 0 so again I'll this somewhat imprecise and explain what you will find if you do exactly the same things would be better than 0 OK so for example 0 0 should be set up Oscar for for example may be perceived square to OK so what you will find if you do this again I will be entirely precise but is that this is always a small finite abelian group a or say exactly what small is but I can get you some explosive bound but in fact I can cook almost computed advanced but it isn't very explosive belt of what the size of the order of elements are in terms of an abusive 1st of all it never happens in this case you have these factors of our of OK it's always a small fine doing good to let them just explain why this case behaves so differently that disease no cases of the main reason this behave so differently that their unit appeared so it is forced to be small because old has many units parts of the largest city hall so we need forms commentator group due caution by elements X Y X inverse wine verse so if you take if you as a unit prison anyone password to and I the UK X to be the elements you you inverse 0 0 and why to be element 1 1 0 1 my only prisoners dropped short on their only see 2 pieces and I dropped them both OK so so if we take this plan X Y X inverse Weinberg is on 1 once you squared -minus 1 0 OK so it's like the man immediately clear white useful usually when you take this commentator to matrices you get a giant mess OK but if you have units you can use it to produce a very simple classic elements of this type and then you can try to check the danger that went out with the definition that they generate a lot of focus of basically you can use this fact to show at a sort Faustus appealing education to be small now from somebody it's a little bit disappointing but I'm you can recover an analog of the theory about so 1 can recover analog of theory for if we replace this sounds abelian ization so rather than work with the abelian ization there's another way to talk about a billion ization which is as the 1st homology groups as I'll briefly talk about homology but with a it's just it's not necessary it is just a part of a passing comment about what the how we generalize this so instead of this we work with the 2nd homology also called the shore multiplier OK if you were to work with this you would see something very parallel to this where I S 2 then pullout be replaced by it's right to 2 men if you work over real realpolitik field you can talk about whole waterfalls with 2 different weights 1 for each of the real embeddings so they'll be a very powerful story was appearing and so forth but with this kind of what is sometimes called parallel late and here again as before there is a corresponding story for any waits as long as the boat tour larger cable we 1 is a completely different story and British anyway so so if 1 wants stood generalize the things we're talking about it at once needs to make this kind of thing but OK we don't have to do it for it turns out we will have to do it for the and groups so personality the Western 0 peso for example the abide by and here we find actually do so all the reasons yet so so here it is these are analogous groups part that local Bianchi groups and not so desire Colby and you all are at least maybe well you maybe this time is usually used for just O 2 all but is colder it's so the vessels to call because they were studied by Bianchi and he's a very pretty they act on so just as they act on fiber hyperbolic respect and I'll describe this action later when we need to say more about it but the delivery very parallel to that SL to see which acts hyperbolic Tuesday's within and beyond the actually used the geometry of this these actions is so this is more than 100 years ago the already figured out fundamental demand reductions and 1st service certain small values and the heat essentially computer presentations for these visibly and he did quite a lot already here what we find a phenomenon that's quite different so as another no units and it turns out the dam ordered then have can be quite large parts of limitation should write some data related to this and then will so the other questions OK so this indeed unwillingness of data that was computed by Hollioake's saying good and it was computed in response to a conjecture that version I mean all say the conjecture moment limit jury the data 1st and that's the hook I this is 4 0 is III of III and I believe that the stated Xia for S O 2 before follow-up up PSL too but OK so here we go 4 in his 9 calls for 5 this could be lionization but is the 1 5 4 3 1 3 pursuant to the 6 Inter for and is obvious signs have been stopped and for instance only so here it the model
for those 7 8 7 9 3 5 1 3 6 7 1 3 question :colon 2 9 2 3 0 6 3 2 1 2 2 0 0 0 people will cost of stopped there goes on actually having to Proberta once see another version of that but to let some of the oldest just to abbreviate you go up a little bit higher the same thing because this is the so it can the angioplasty where he is a finite group he is finite of water that order is bigger than the candidate 310 OK so unless you really wanna see it OK by the way that the primes and so for the chop and these are essentially random appeared and that is that there is no naive there are no size these but more or less at random numbers from the factorization point of view OK so this was actually in this this kind of thing had been computer before but somehow you only kind of see how this blows up when you go moderately deep like it did today this computation is not so trivial but it is doesn't look like such a large number but what's relevant is that square and then you have to do linear algebra into Germany algebra things of that size not and so this was his so computers because Nicolas Bergeron I made the following conjecture and high conjectured arise again the general conjectural specialize in this case specialized in here that there that this grows at a very specific rates so we take the size of this now ever saw as you see this was could be also have an infinite part so when I see the size I just mean define partly portion heart so what we conjectured is and I guess that maybe I should say that the amid media the constant all writers who looked in his various to Prime's for this to be a part this should converge to land over 18 pie were Landers also somewhat alternating on just .period squares so that or is it this is this is the so insured it grows exponentially with the absolute value of this number market but violated the agreement with this conjecture is very good if you mean numerically speaking so I know what I want talk about in the is more or less discuss some of the things that are around this conjecture OK so in a way that maybe the could the conjecture itself is not as interesting as some of the things which show up when you start to try to understand what's going on so the focuses the ball literacy is trees to give some credit to some understanding of the world "quotation mark what is happening here in 1 it's related to the they should get out of this but made In this yeah the size of the yes sorry I said that pressure Britain it yes it is the size of portions of group so it absolutely let the gasoline can be infinite and and by far the most interesting phenomena here happen in this case when is it hasn't it's part so I did that match while they also had to work on the request of the policy those are on squares here yes to this number is likely to be hit by the health of the know the rank three-part bids in extremely irregular way it's also it's so out of all so so certainly at the rank of the group it is in a lot of the time is 0 the list for that way it is hard to even make any sensible statement about it because it behaves so regularly this is because I feel we would set the example we have to know the 1st 1 of them have 0 0 free part is the last Oh decide this and this is at the house and these 2 have 0 free from the front of the yet yeah so so most the time what you'll see is a large simply a large fire group in summertime when you see those typical pictures a small three-part accompanied by very large portions all of them the question answer is so here's what I'll talk about and you get so question so that was the 1st thing I wanna talk about is she up of described to heuristic so it was kind of plan heuristics has to why this should be true a sewer but will wear thin descending this kind of expansion this means exponential growth of poor exponential growth this fits all described to heuristics Fred and quite different and so they both have some advantages and some problems but so I guess probably was already taken there as attended the seconding I'll talk about is the analytic portion isolating just tell you roughly now what that's going to be the statement I make here not all state in the context of discussed it's literally true only in the in the Co Compaq version of it so this so let me say that the the following the statement about to make his His really do the work of singer and lead singer she and Mueller so this comes from a statement that hasn't no number theory it is not a member of the erratic nature but here it is it says that the size of this group multiplied by the number of ah so how clueless regulator is equal to ways saying it for for now all state exactly when we get to it for often Alameda states that you can think of it as the value of the Solberg zeta function capello also I don't really like thinking about it this way so I'll say something different but this is in a getaway all make precise is the sort of billion class that number formed so this is then again have something more precise and my when that class from before OK you want this book this all are measures the size as so often are as 1 if this is really a finite group so it would measures in some sense of the is finite so are measures the size of the free part so far measures the size of the free part there there's several ways you can think of measuring the size of the free part that they're all
full and a constant battle so I think is a good way to think about what walk no explain why more about why later but want to think of this object it measures it's kind of an automobile in class number again and again that there's something precise to be set up so that you should do it in some way it has certain features which are like usual class numbers and in particular has the source
of analog of a class number before but the downward talk about is also the mother's regulator are this regular is itself extremely interesting today at 4 4 I think for a long time I somehow ignored it because I was focused too much on this but it is something very interesting in its own right and I'll talk in particular by its relations it's its relations with El values and with heights of elliptic curves Alameda sale values and Heights presidents parts next at the plans of the lectures and for the rest of would talk about heuristics as to why this little why we might expect this kind of phenomenon "quotation mark on top of the say 2 different heuristic so as to why you expect this thing to me behave this way so that the 1st case of 1st heuristic but it's also the jurisdiction which is pretty simple minded and then I'll discuss Sun then I'll discuss it for a little while including some ways in which it's not right so this but we can give up so I haven't yet described How but when we when we come to this analytic portions Toro also about ascended discussed this with Michael I think the main point delegate to look or maybe I should say yeah well when I don't
like to and I will say some 2 and 3 I will talk about some of like we have several theorems in the direction in which I will discuss and me later on but I'm not going to go into the proof of anything I'm just gonna try to give some sense of the story prior to the 1st year stick is so we can a write-down of presentation for a few gamma North and so From now on what I write down order and we're always dealing with Gavin on an inside so so here this is inside cell to all 0 where 0 is now an imaginary quadratic for water Price always work from now on and I never mean inside a cell the anymore was inside the US requires thing so we can go right down the presentation for its using the geometry of the fundamental Jim demand all of the fundamental domain so it in other words socialist gamma note this guide now some hyperbolic 3 space and I will say more about that and use the geometry of the cautioned you can write down a presentation for it the result I won't go into that but will what this this this presentation looks like so this gives a presentation of the following types so it gives you a presentation with some generators 1 X G OK and the generators have to deal with it take the fundamental demand and you look at the fundamental demand adjacent to it and this and then you impose some relations cake and if you do this the more than 1 way to this but is carefully see you can get the same number of relations is generated OK so this number
in this that this group has a presentation some generators a number relations and she's some measure of this size all the fundamental domain OK now to get the sizes fundamental demand is going to be this sense it's a measure of how small this group is so it's more was proportional to the index of this group Gammon inside gallon or 1 and so very roughly that's going to be on the scale and square OK I'm going is this approximate sign the means really approximate as mail that know of and squared and square login souls was the the rooms being here is that this is meaning that if we did it matches because this is the 1st that an issue that has long compact and that naturally give you fewer relations but that yes we we we can do this explicitly say that you have something written as explicit explores the complex you can always write a presentation "quotation mark what we do not say algorithmically you might write a program produced by the principle it is not known OK so now bill now of how I'll say something more about the nature of these relations in a 2nd but I'm pockets sold so if we had really nice this disguise what we get then is we get a G copies of CDs and this just like the computation errors at the start you have a group with an explicit presentation To make it billion just have 1 copy of z for each generator and the new quotient by so some vectors which uphold B 1 2 the subgroup generated by some vectors the after the G where the eye is if you look at this example it's kind the export director "quotation mark this relation are OK so for example if RI worth far I was something like recalled X X 2 it's 5 inverse X 2 X 3 than the corresponding Victor I would be on it's 8 0 in the 1 0 2 1 0 minus and a case of these numbers here the 2 means X 2 shows up twice the matter where the water was a matter minus 1 ex-fighters of experiments on so OK now it's given that this presentation came from geometry so these relations come from something like ages of the fundamental demand greater that generators scope confluent faces that relations come from edges and you can see these the eyes hold most of the entries of the the eyes on the I will be in use minus 1 0 World appears so that's not it's something about them how we can expose we constructed present OK so now we have to address the problem of 100 bring down that bought and sold right so distant so left for a moment forget about our exact setting and we can consider the following problems so if we take all of which use the same total around them of the eye skin disease she with entries date OOD just use all the entries at random we can so that some sort of naive model of this week and ask how largest group is going to be and the size of this the proceedings Aegean take the quotient by the group Jemaah by these factors the size of this is going to be the determinant all the Matrix you get whose Rosa given by these factors prepared absolute value that determinant and if these are if you take a matrix Friday start filling the entries in with 2 0 slums and response at random the I'm that the day the size determined you expect so depending on how exactly what would you expect this to be a size exponential Ng maybe G factorial that saw the state 1 expects this randomly speaking to be exponential Ng again you because you know you want just 1 of those yourself it you'll see me doing that it has at least of the scale the Virginia individuality so this is this very naive thing is Of these compacts remember I said this quantity G is of size roughly and squared so this is at least it means such as heuristic is certainly not makes no precise conjectured this nature but at least it suggests to you that I so in other words you can in some way to make a random model for this group ,comma North and maturity point of view and you just think about the geometric modeler does suggest that it does suggest that the immunization should be locked OK this model is so it is that the 1st heuristic is not bad but there's there's several problematic things about adult discussed a couple of them but it's it's odd sort of gives you the right idea of what's going on many questions yeah was lottery what sold to the doesn't have such a hassle to and some groups are basically free groups that I didn't save any specific wouldn't say where this factor presentation is balanced in that proposal to the EU and its subgroups are up to the top of this small phenomena their frequent there's no relation to talk this is another 1 of the most heuristic is that it's all very robust like if you if you're analyzing S L 3 it's much much harder to figure out from this naive .period you would have yet the and only yes and I did that that's the conjecture which is there so this number comes from thinking about all of the people this In the case of the and that's essentially where this comes from yes yeah yet and the problem in this story is not the analysis of the solvency of function all I have to say exactly what that is but that's not the issue that added the issue well there are 2 issues but the most interesting 1 has to do with this he was totally useless here about the law that is the university's accused of so at a given little on talk about that right now now how the three-part affected with another question you later of wireless this literally yacht that the EC items comes from the geometry in their different ways to construct a geometrically but you know that these relations something like this you have a bunch of fundamental demands that meet around edge and the relation is a product of elements is equal to 1 so not too many things need to be adjusted OK so forgot 1 of the only outkicked so who never lets I will talk now yet in this whole story at the more the most interesting thing is what happens when there's a free particles so-so OK so a statement of this heuristic now now let me say remarks clinical heuristic yes heuristic a baby also be a heuristic be remark upon heuristic OK I guess I'll we only have time to make 1 of these so there to comment on it but it's so personal what happens here so what is this what does say about the case where where Gannett ordered by and ad the following year it has it has probability 0 but nonetheless it happens it's certainly have to be right initiatives it certainly have prodded expeditiously small energy for 1 thing authority a point is that it happens much more frequently than this OK so it has been if I just buried his number and it happens far more than you would expect from the flutist rights of some of right but then also let's look at a case so if this supposed this group were actually infinite they were infinite then right then this matrix is happens when this matrix is of whose entries are lazy eyes this happens when there's a singular OK has ranked rank at most while has ranked last in G so let's suppose just put sake of discussion Allen has Ranji minus 1 but Frank as she minus 1 the sulfur this we're now nowhere the qualities this write to this summer with the modification of so that's a modified version of Star operate they can so the following a modified version of Star still holds looking people in what I'm about and what I'm about to write but I know let me assume that asymmetric OK I'm just because they're too there is a reason why this is a sensible thing to assume that it's not mainly just makes a formula but simpler so what you still have is the size of the the portion part of this group OK so so now we just take a finite part and now we multiplied by something which call or because it's going to be similar to the regulator will occur later and this is all too dead prime of a and this and that and that prior they is the product of the non-zero lighting that OK so there's still some of the determine 0 but what 1 still has something like this and you excuse to expect the proper non-zero Eigen values to be pretty large Castillo 1 expects a still to be exponential size and say what art so here is ah is given by our rights are so skewed by some of the some of excise squared where where X 1 of the X G is a generator for his is interval generators the limit their primitive meaning role the Hornets relatively prime primitive interval generator for the colonel of a acting on the cheap but it has ranked it's a major matrix as wrenching months once it has a colonel and so you can do it you can pick a generous for the Colonel and this summer's are measured the size of their generous OK so another way to think of this so another way to think of is that it measures the size of the new measures stood the size of a homomorphism from the scrutiny and from the home office and from the the G quotient by this the 1 the G so this is no longer find a group that has a homomorphism disease and not explicitly this homomorphism sends and while that state it sends them In 1 of the the said to son of excise I pulled 1 key so this solid excise square countermeasures decided you actually try to write down in expose according to homomorphism disease there are measures how large centers so now you have this problem which is why should so for our gamma Norton have food for this to be large part should be small it's all about so many things this and is naive thing above will have a precise a configured as a kind of crude version of this Alan torsion formula which is literally right but anyway that the fact remains as you have a competition between 2 things 1 is the size of the final part of this group and 1 is this are OK and this for example as happens in the context of class groups if you take the class group of imaginary Quarterdeck feel the size of the class group time decides the regulator is something you can compute explicitly tell functions but what happened is in the real ordered case they are the are always winds almost for typical real parade field regulators very large in the class group is very small OK so this does not happen In the Senate OK so for some of the 6 from an experimental point of view Moss the time that you see at the ah does not appear to be that large and so this was something that mean trying to understand this was 1 of my that motivated me for a long time understanding why it should always stood up so to speak in favor of this despite the finite group OK so I think I'll stop there and next time I will start after the 2nd remark I have this heuristic well explain away in which it's definitely wrong just like this heuristic is clearly crude dolls .period 1 way that gives you wrong answers there was a no lot questions of the of on the other side of the border we always are you saying with what part of this is heuristic the other partners heuristic is a low is that you have very like this you have no actual idea that this determinant is there's lots of really good right on the presentation this distraught size but that determine might be 1 all this is the the only because but there was no answer at the start of next time now in particular explain what I mean by Norman billion class for so so that it is there is a connection so the next time will talk about more what this has to do with algebraic number theory was and the number of courses across the numerous but this measure something this and at the end of this nite I think this is an ink OK so I might have missed remembers by things as wasn't this in exactly quality and adjusted to the fact about the quotient of the world this is our you can think of it as the volume of the colonel want yet yet when the square of it was this confusion about squares that I would like to work with him the event the guy not learned that it makes a good point so sorry I should explain that a little bit more for 4 and sell to Zeidan's index would be essentially of size and but because it's over the magic word field it's the norm of the ideal generated by which and there is his recently that nearly the 40 years he worked as a nightmare assume he's had here is that many of the and that all the the conjectures that we all talk about the conjectures are completely general if you look at my paper Bertrand their formulated for any group isn't yet yet what is not that the methods focus later on it's also point things are more specific but that injectors conjecture there for example is completely general so what you told that I know that the civilian uses the amount of music you go there there was 1 of the things that you believe there were was no longer the computers in the number of I'll explain exactly what I mean is it didn't have a precise meaning that which is not but also the next soldiers the value of the With all that would decide this over and over the phone the 1 with the failed values at the center of the storm so it is that when I say that that comes in knowing this is something like more like it's off to a rational number no value which the Russian number can be 0 but that maybe it's better to it's hype also at


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