3/4 Analytic number theory around torsion homology
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3/4 Analytic number theory around torsion homology

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36

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2014

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English

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08:12
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20:25
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28:55
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37:18
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49:36
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52:05
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00:02
so if a In and we can win in the
00:12
world and orders for the
00:14
sale and then explain little more I want explain the remaining terms and this little torsion formula and then after us and then also fired discussed what we expect and some heuristics for announcing that I will discuss a bit what or what can be done what we can prove that I can make a little room and little remarked before that which was in in response to a question and Paul Nelson at last time which is that I've been talking might imagine requiring fields all the time there is some kind of parallel phenomena which you can see for just classical model for a case of the to the kind of thing we've been discussing so there is a so it will be another the sort of thing we've been doing is you take this grouping you take a 2 billion isolation and what you find is it has typically a small it's a small repartee can have very large portion part and there is a similar phenomena for for SL 2 over for just classical model forms but it occurs somewhere which may be 1 doesn't think about so much which is for weight 1 4 OK so if you look at the best 1 and there's space of weight 1 forms now the space in the wee 1 forms are in some ways the most interesting weight because many things 1 think of don't work here from friends that the simplest example is there's no dimension formula became the dimension is typically very small and very easy regularly with em go where we have so much some on the underside of an image it's very regular but it turns out the this that this has a very similar you can interpret this as part of the classical space await 1 forms you can interpret as I see you can make an integral conversion of coal S 1 and the broader space so that the classical space is just what happens when you take the free part of creating another way to say it is it's possible for a weight 1 form it's possible to have a weight 1 for Model T which doesn't lift the characteristics 0 OK except for very relatively trivial thing instances does that phenomena and doesn't happen in Highway Highway there of any once you've worked out what that right notion of Vermont 7 homeoffice form is always lifts to characteristic 0 OK and then I met here this can really fail and this was so that this story is much higher too much less so the 1st air that the 1st evidence that this was given unlike some might 1 to classes that don't delivers were found by master and then some more refined by bothered but the Swiss Bill relatively sporadic and then of on the basis of the commonality of what I was saying I had a Ph.D. student was asked to computers integrally for many and that so that that George Schaefer and he found that in its impact is really does behave like this that is to say it has a very large typically has a very large portions components this is sort of hide the classical space await 1 forms like this tip of a small tip of an iceberg but underneath this is vast number for a further of any very large Prime's where you have a more Peewee once formed a doesn't have the characteristics of this story is less well understood I don't have time so it in principle that this an analog of everything I'm saying that data so it will be nice to have a conjecture here that this is growing at some specific exponential rate and so on there some mysteries in this data which I don't understand but it anyway this Salemme say media say announces is less understood and it would be worthwhile to develop it yes usually don't think await 1 forms but somehow as time passes as I sort of see them as the most they have many boundary behaviors which are very interested in rights saw his moments and I'm going write what was what was also are city this is so rights abdominal and so now pose owes symmetrical erratic water something its eastward minus the URA and we take this group inside sold 0 what citing questions about the story so there you go above the Kevin brother has a nice they provided most of my students teachers she ceases all lies it's interesting also from just a computational point of view there little review even computer of we go here and remember we have this subgroup and again I'm going to say that although I I am stating the state's setting of made many of their the results a R Our proved only the Co Compaq setting all told should certainly extend here in here including the 2000 yet so for example I give you like some and like 500 and I ask you to give me a basis for the space of a week 1 forms over the complex numbers or more said and even over the complex numbers it's not it's not trivial question of course clearly computable but at the it's it's it's it's been better than mask forms but it's so need to say 1 word right you can embed the space and a higher which space by multiplication but then you have to figure out what images that is it's not clear that when you divide your Interpol OK so Surbakti writes that for what is the candidate conjecture we've been discussing is that the size of the billion as they can this Standard and the correct scaling is the volume those to 106 so let let me say state again this summer time are so as far as far as I spent quite a while giving you heuristic motivations for that now as far as analysis goes the main where we have to understand this this for love story of analytic torsional and I right the names against lists of discovered by rinsing their hand crews given by triggering dealer and what they say is well we can we can access at least in part this thing analytically so the size of this against the rights of this statement is valid in the cool Compaq if I have time all
08:13
say some newly know we now know fairminded and in in this case will assure governing complete formula this multiplied by regulator divided by the ball hyperbolic Volume 2 is equal to sell some some the ratio of but clash so and this righthand side solvency to function well and you right now while would discuss the Saxon hyperbolic space so you know what this is you know this is ultimately what this is and then I'll spend some time saying what we can prove words and they will come back to the extent there's time left will discuss this right hand side but the say so this picture this for example those Delta 0 is just the usual pop clashing operator on functions built the the 1 is means on forms on different weight and indeed so it has some eigenvalues so 0 but is in the Co Compaq cases has no continuous spectrum and this determined is some way of making sense of the product of the item value is willing to talk about that despite sample after it at the end of lecture comes off as I said that this summer I also think of isn't the sovereign state of functional actually prefer thinking about it this way because this way it's a clearer but what the effect of the eigenvalues spectrum is on its so that slices of the waters are This is what I call the regulator and what is meant to measure so far as 1 as I said earlier 20 billion isolationist finite sold general what's meant to measure isn't it measures the size of in some way although the Free Parking quite so just as in the case number fields this and general it's it's some expression would determine OK so they had to keep keep let simple others talk about the case where it has ranked 1 so I have to write down determinants so let's say that suppose that this gamma North and was actually a copy of the integers a finite To this end generals story will be similar to what I say but with determinants so what Osmond to measure more precisely so in this setting here right you have so you can make there's going to be we have set up there's a unique up to sign surjective homomorphism like this OK so it's just projects such as the and that's only half the size of its ejection from Camelot and the Osman to do it's meant to measure that in some ways make it measures the size of this homomorphism quite so so I will give you the the precise Sudanese a precise definition but I wanted just described so this quantity is interesting has also let me describe a couple of different ways that you could try to measure the size and it'll it'll turn out all these ways are from the analytic point of view the same Hockaday 0 essentially distributed the same order of magnitude from algebra point of view of the specific definition this is interesting because it's it's meaning its exact definition is related to other arithmetically interesting quantities but that analytically Ozawa and so some ways we can measure the size of a most naive ways you could just say Well this is a homomorphism from this group is the eldest let's look at a bunch of small elements here and all the think the largest value takes results Sunday as a group theory way but you just take say that the largest RIC pick some bound for how big a entry should be out so maybe you go up to so a reasonable thing to do would be to go up to some power I need the largest see how they can but only because you multiplying matrices right accountants who a you made this into the 20 innocent normally expect get that much bigger because we knew multiplied many matrices increase of what multiplied the matrices that what larger without feeding roughly you expect us to just grew logarithmic so does exactly what you put here is not supported by the way what we are concerned about for our applications all we're concerned about is will be does this or grow exponentially with an on off a case of SOEs were for system for distributing mothers problem we need to understand only the very core scale it is a topological way of thinking about it so hard so our people so I'll say it that again a fact from a topology which is true exactly as been known in the complex case otherwise you'd have to be modified slightly but it's a hot at the
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Hospice of homomorphisms from this group to the integers is isomorphic too the 2nd homology so I I I set out on China could put to much about homology but this is dyslexic be quite explicit fall explained so this or homomorphism the you can represented by something the 2nd home manifold says all say what the map is so suppose that we have Esq which is the surface inside are this threedimensional manifold How do we make I would say from it's how we get a homomorphism like this involvement is gonna make this explicit so this s went overboard here so what you do is you intersect you or do you send S 2 so the homomorphism so for sane gather endowment then the number of intersection .period itself all areas with the past From wt gallon W so for any OK so in other words down on the dispatch of W gamma W it's really a of closedloop here and you see how many times as it interceptors s countered with a suitable was OK now given that what you can do so over here I Mobil One reason I mentioned it and notes a little bit of field always but is is that it's actually very useful it in estimating are himself it's something but at the time right so soon are the question emitting the size of this translates sequester mission size bare but over here there's of the famous more muted grow love Thurston which is that you can just measure minimal complexity of the minimal genus of such a surface appears you look at all services as that induces an easier howl I know the words this right then the simplest possible surface I can hear that induces new measure how complicated OK now that is not that the amount of finally company definition which is the the most relevant which is that the correct definition for this which is analytic on Saturday questions so far this is the was also I have worked for the world although officers who were Yasser so that that like it's cleared arrived just this way so if you the fit fact none of which there is a nice amorphous that doctor OK so analysis and for announces Limoges quick 1st remind you of something that we said at the very start of lecture series for SL to see his whatsoever very start something I said is for a subgroup of S O 2 zany prices we had this pairing between Gammon an abelian iced and the space of a week to forms of level and and so was how the scaring went was something like behind some damage hearing as here you descend the interval of Athens From W together that was OK so a similar appearing here and all the others say where there's a mulligan basically transferred his feet over 10 something here and we can measure told him it's all right out of riders out there so the solidarity of the is represented by the French harmonic differential form and we can take it all to itself right so there is a similar story this select I'll use the same location can be because suggested so let s to and this so over here you had way too we do homomorphic forms which you could think of was homomorphic differential forms in our sitting there's no homomorphic but we can still see harmonic so harmonic differential forms on the our gamble on 1 it's Tracy we can't even think of these as being masked forms assistance specimens Eigen value and weight and an exact that this the story goes over word for word taken care the use by sending it into US exactly the same as the same formula OK so appearing from this in both cases appearing from this to the complex numbers are right now having said that although I know what happens is that but they'll be a unique there's a unique Omega such that the of gamma is given by appearing with that unique honor OK so what we've done in other words begin this discrete object which is a homomorphism and kind of spread it out over the manifold to get this represented on guy and we define our regulator can be that L 2 in a product so so outside this symbol used in 2 different ways but that this is L to long on the manifold so it's like a Peterson this not
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this object is very delicate and would be saying more about its algebraic properties in next lecture it's not so hard to try to see what it is up to rational numbers OK but you that you notice the address and the size of something it's not so great if you know it only up to to a rational number and if that's very subtle OK so I don't know also next I'm also trying to say a lot of our mentioned some problems the next lecture also but the point out the US Sprint sometimes in the 3 things because they're they're actually all essentially From the size .period view they're all the same that the their role essentially the same OK that is so these all all have fun asunder say essentially the same size so that the the most useful of these is that this is not a hard it the a hard fact and maybe it's nice to have always the most useful as the relationship between B and C. OK and if you want exact statement you can see my my paper all refer to it again say my people with Nicholas version on and hold single and there but only a hard fact we just might not exactly what I mean when I say that these have and we we don't write the relationship with age but I don't think it's very hard to check unfortunately is almost useless but in and somehow in the sense that in new computationally it's easy to compute but theoretically it's very difficult to get a handle OK theoretically what easy to get a hand on his desk on a gun because it comes from automorphic forms or physically constructing a surface but you could also do geometry farright so that is what art and next time on talk more about more about are about and it's especially well what exactly know how is it related to health functions and so forth very quite so so now let me I have I still have to say something about this but let's let's not cause I'll tell you so what we know how to prove in the direction is conjecture the case of the tool that only real tool we have to prove things like this is this formula but even here there are still many obstruction biscuits and any questions the definition of the relationship even as I was sorry I I might embarrass any question that I'm assuming I'm in a setting where this happens this is this is a unique such thing up sign all get on 1 of different parts and saying well here different ways to measure the size of the new directly or you can translate feeding a surface they can translate feeding in differential form and this 1 is kind of that may be the most canonical 1 but sighs Weisel by the way I should that this this thing you can also use studies was somewhat just for a sultan's even ask how big the Home Office and and there's a nice people gold fell when he discusses his quest it but in that case you have a much more so than concrete way of understanding it because you have these modular uniform ization the VOC problem here a basic problem is that we have removed ourselves from the world where you have at the end of we were where you have to have real objects like algebraic varieties to play is everything is purely analytic OK so so now what about so now limits they spend some time telling you what can we prove to words to conjecture but any Israeli presence alright so so let's let's look at this formula for a moment political formula star and let me who also list that the difficulties you have and so 1 can try to apply Star Wars so 1 can analyze within analytical behavior all the right hand side so this question I haven't precisely defined this for you but as the level goes to infinity is related to something Thornton was saying you understand what the distribution of eigenvalues converges to very precisely and correspondingly it's easy to fairly easy to understand the asymptotic said this except for the following painful issue maybe you have an extremely small widened value OK maybe do this this this this operator has an idea about a very very close to 0 and then but then this becomes small knows nothing can do about it so so as it is there are no small biden values or maybe more but they're restated his few small values for dealt lot OK so the exact conditions looked coldest conditions no fuse f a few small eigenvalues so if you go through this you find everything is fine you understand exactly what's going on except for it to drag out the small eigenvalues by hand and and and when I think about this and now I still know absolutely 0 about they're absolutely nothing so the but the nature of the statement that you need is the product so we need we would need to know something like this this summer over all eigenvalues Landauer cities are hiding values of this operator the slope plush an operator here your life is going to basically we need to realize that there are a whole lot the exponentially small volume which is absurd cases like the sort of clearly will never happen but still it's a heck of a minute called the the volume of fear here things were looking at so so Lou let's just say that we were always assuming call compactness and I will comment about what we know about the the noncall Compaq case but grew from an initial set up let's not worry about what you need is for for any positive Delta you take all the eigenvalues lest in volume the Delta is some them logarithms appeared to worry the and the skillet by the this should go to 0 OK so if you think about what the limiting distribution is it assert that this is very very unlikely to fail but think seems extremely difficult approval so as a as a .period case of the kind of issue here you can see and ask for a sultan's and do you have any understanding of how close Of the died invaluable must form can be to a quarter without actually being at a quarter of it so that there is it's so the Mom Romans this clearly should be true but I see no way anyway so this is that's 1 thing so the 2nd thing that you need to know that are articles are for regulator you need to know that the regulator is also growing and some exponential rate but would only probably at a it's very easy to buy hour from below its
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on top problem anyway flowers small and increased its that OK so the point is that given the size of this about which is the focus is on so I know Andreas Stromberg said I talked to him about this many years ago was probably that I like certainly looking at that the spectrum near zeroed in showing peculiar behavior and that is a behaves as you will get from the air limiting point of view and I certainly would I would love to see anything in the evening much weaker and like additional 0 about this this is low the situation here is a lot better about a 2nd but so what do what do we know given S and are done so so here's a fact given ethanol are now we get the conjecture note this is and so I am going local Arizona nothing like this we know quite a bit about this but still the that actually what got Bergeron I started on this was the observation that in fact it is closely related contexts we can completely sidestep both of these the same time and then we get unconditional results so let me say that about this I have this change the problem slightly and then will really get the analog this conjecture and they will come back to the lower what more we can stay here so Bergeron it this isn't a paper I saw that is another paper of just Nicholas version I found a few years ago but not what sink OK so you see can avoid both problems by changing the way so now that he's here I pretzels in this in this portal say sense the most of the really unconditional things we know so when I say changing the weight when they go back for 1 moment again to hassle to see it so I'm not going to out Of the resolve without defining all the terms because I get I wanna going to homology but suppose for S O 2 the ways as I said is appearing between this and I and which informs now support to want to know as Henrik enough month for such awards the analog for higher which is normal for weight was OK we won visually nothing but this works perfectly well for you just have to be willing to suffer foot for higher wages that an understatement is rather than this you have to take the homology of this the same group with coefficients in some Montiel so this is Montiel depending on uncanny so are and the same thing is true here that the analog of changing the way it is looking at homology with corporations and different modules and so what we show is that so proved the analog of the conjecture so that before in this context so in the context of cool contact group insult to see for what we call for what we call strongly a cyclic weights and I want to see what this is but the point is that in fact it goes all the way here and in this case the waiter Prempro by appearing to jeers and strongly basically means that the 2 images of different so generic condition not in another special 1 OK when I the analog of conjecture there's a there's a different constant but otherwise I the the conjectures exactly the same thing so this is very satisfactory in the sense that from the point of view sailboat those levels program there's no 1 way this is other so it certainly gives you a pretty strong evidence for the underlying phenomenon but in fact so we did this for all groups and we need whole group something very pretty patterns which I doubt I'll say OK so for the exact definition of this he created the exact date when you can see our capability that statement with a different constant and with appealing as it replaced not 1 thing I want to say that almost the same time as our paper Mueller Marie Miller approved a similar statement but at possesses same kind but as the week goes to infinity rather than the levels OK so it's not sort of strange that we were yet so we gave the vote of the yes so I OK so strongly basically case the point is that it's a little bit surprising is that the strongly it's a good thing exists in the 1st place but given that they exist what happens died values are all forced to be away from 0 and Horace 1 yet so it's not that it's a complete surprise and we did exist because they don't exist version were writing so it's not anyway so this is a it's interesting as far as I know no 1 is really thinking about this for a before and that Mueller some hosting wanted exactly the same time as we work Li say something about it and add different general groups because it you there is something to be learned from there so we prove a similar statement so we need nickel her alive but with 2 see replaced by any group a similar statement means the exponential growth of some similar objects by any group g satisfying OK some conditions with Richelle at the rank of G minus the rank of a maximal compact subgroup is 1 and gamma have was replaced by some higher homology groups OK so you for some higher you so I'm logic you now this condition here is a strange condition meant she might Frankie is once again so for for example it includes 2 groups S L 3 R R itself are maybe SO look at USO Peachey with P and Q 1 but not a cell and 4 and bigger than 5 2 the debate equal to 5 so the user groups outside funerals it does matter but this is almost there's only maybe 1 of the examples from that
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sort of thing so you know you go through this method the point is that all sorts of complications to be done here depending on the group and you find your disappointment that you get a result of this nature only in this case but here's something wonderful happens is that according the heuristic be so you can work through this this heuristic with the on corners sticks which I mentioned earlier and you find this wonderful thing that needs to be the only groups only G With the exponential growth Of course ,comma abusive here's where this heuristic B by itself is certainly something should be somewhat skeptical about but it is a very nice thing that matches exactly with what you can prove so the proof for reason that I won't go into that you know you put something here than otherwise you get nothing you do you're left wondering what happened before so heuristic says is an is that what the cases you can prove I really I think the only cases where you have to say so in other words we conjecture but we can't prove that there is no the exponential growth in any other case the cases where square Ranchi she was ranking is not 1 I think I know we can't cruise maybe struck again the the sort of thing about which I know 0 that is I have no way besides completely trivial ways of giving you opera bounds for a non interesting upper bound for the size of the sort of things were discussed so it's OK so so all this is under this method was under this switch that you switch to a different weight will you can avoid these issues and now come back to to do what we what kind of things we can we can say about Spector's 2nd success and I will go back to the original question trivial weight corresponding to be renouncing all our about that in the noncall Compaq a soso now alike so at least for S O 2 c so I and then and then noncall Compaq case there's quite a bit of work in and out of the the most of them women of this work of Jonathan Safran John Rambo which gives now similar results and not come back by the noncombat case here is is really painful OK it's not like it's much worse than the than that that just a trace the horror the trees from when you try to regularize this and you have a continuous spectrum it's really really awful but that that this this strongly a cyclical thing makes life easier but anyway I just wanna say it's not I put it like a small this so I think that that the number of pages involved in the small 1 line is probably greater than that the number of all the other papers mention together so it's not it's it's a substantial problem it is even more substantial problem to do the same thing in again this is like really not just a technical problem I have thought about this sultry 3 and I really in fact I mean ranked 1 of these you can work hard and sort of the idea of a accidents even see her news for selfish a right to know OK so let's talk a little bit about what we can we let's go back to the issue from what we can say about this and any questions yeah the fastest man in the world is there anything like result is just I Presley noted I O trivial results 1 could give abound for the help of I'm not sure how the made showed goes to 0 but you can give some not that the players in this case and is for example in some cases you don't even know how to give any manner have 1990 could be a 0 he some priory bond you can give over how close adding value could be 2 0 because if fear because somehow you can switch the value problem for a discrete problem you know with the integer matrices and you at a lawabiding how close citing the image matrix could be 0 that would give you some amount but it is totally useless docket social Yanyan questions White said nothing about death but is better and and so said today I let me say that about the analysis of holding lot of something so much the size of our and then back to moral talk about that that the kind of or whatever the next lecture about that I will more about that it's an algebraic be interesting subject but it also what we want shell so we'd like to show the service ride over here involved are over so are the regulator should be bounded by the exponential of epsilon times right if this fails right if you look at what am I saying I'm saying if this were
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to fail means you have this homomorphism disease which take exponentially larger value on small matrices which seems absurd to it might seem that if this is something which again this this is survey this might seem like something which was so to speak you should expect but that's not it does Oracle clearly so so up I don't know the exact the exact current status but also the broken field have these producers have produced baby archive examples to this exactly but of closely related counterexamples for nontariff many gamma OK so that they they don't produce a counterexample exact setting where Sabres rhyming conjecture but they they produce something very closely related so it's important I'm completely possible but this is sums that is always trees really has some special features in some special feature of arithmetic and not as a general the group theoretical or a geometric properties OK 1 case so let let's study ministry a bit more about what this so made of the same words hold 1 case where we understand are well what 1 relatively easy case so easy from the point of view of 4 bad for understand having a good up about an hour is when all the knowledge is when all of this space which I called older Oracle well let me just say what all gamma or in ads comes From base change comes from Q by base change most allow the violence in the sense of politicians talk about when I say this I just mean that the characteristics 0 parts cancer would cease right so I am looking at it so without going into details suppose you start off with and which is in integer and then you can also think of em as being in all so so you can sort of former subgroup Gammon on em In a to see and you can form the corresponding subgroup of Dow German or em in the big In insult to Nestle to all get it turned out that there is a lift that goes from the sort of free part of this the free part of lifted coalbased itself so this is 1 way of producing examples where I yet that it's a fairly common it's part 1 of these local uncommon that Walter the free part of this comes from base change in this way OK because it sort of not not here like Ethel to the the this threepart somehow usually so small except if explained that somewhere so in that case we can understand we can give the court the correct upper bound on the regulator appeared and that uses this topological interpretation Part B so here what you do is you you produce enough enough collegiate Isaac surfaces and use part B as you spot the now that there is a reason why we want to study the specific case is that it's in for numeric summit portion of this case actually look pretty bad so it will be a little bit worried that this would be this was a case that had some reason to believe 0 might have some reason to believe this would fail but in fact that we can give us rather good balance that actual bond we get in this case is something like so I we never wrote down that we disproved some exponential but I think what we actually prove something that it's exponential square root of all something the in which question all I and I know I can but it is certainly no I certainly expect so this case in the other case I mention I expect them to be valid fully many in the numerics we did their book Ballard saying you know around half of the time he opted out of course some of the things change from a very large eyes that but when I say it happens a lot I mean I don't remember the numbers we were the paper but maybe half time instantly OK and that all out OK so more interesting case is is went wild really say next faces III died Wednesday at least a few words about this whole plush indeterminate so that old next time will be devoted to studying these are in the case where you have a single form the Komsomol of occur despite based OK so so related to an elliptic curve overall I will see that this conjecture is then becomes related to questions of of diophantine geometry in some very surprising way
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boxes of and the last American is going to go back to the analytic portion formula and talk and just give you a quick very brief flavor of what these plotting determinants it looks like any questions right so I I was I was planted and I would like to spend more time on this lush indeterminate thing but it's probably the least number theory part of sociologist so hard we define defined this sort of determinant of Delta again and added that I I prefer this way around Solberg said a function because this way it's easier to understand this effective small light and values of the EU can really see if there's a small valued cause a problem but that the definition actually makes sense for any company makes sense for the plush operator any for any contact Riemannian manifold again this definition is due to reinstate the introduced it for the purpose of stating that formula and so what do you do so let you have the eigenvalues beadyeyed values and basic fact true in this context
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is that the function some socalled there's some over I'd be given equal to 1 planned the miners Esq so mobiles law tells you that this is convergent when the real part X large and it actually has an analytic continuation to the complex plan but while it no more fixer can you understand exactly where polls are and sort of this is this result is obtained by sword very hard assistant is obtained by taking the Mellin transform things we know about the heat but it is true for
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any Riemannian manifold now formally let so ignoring issues in love of itself in the region of convergence the region of convergence if you differentiate this with respect but that's what you get is negative wild 1 guy I overland IT S so sort of formerly if you put S equals 0 so the bulk of the sin "quotation mark slog of land eyes equal to In quotes the negative of the derivatives as a 0 sum Justice formerly presses 0 and this is what we take as our definition of this so we defined the determinable willful passion to be the exponential called minus the prime minister that's the definition and it works Compaq remind him right so let me just do a lot of to give you a flavorless to a single example animal stopped and any questions about the definition you have to that yeah this is the says no and that there there's no solution none of the other outbreaks such as has been the continuation it's very soft provides the example in our mode is a flock of flat forest so far too 1 of of the 2 would be the usual metrics that looks like this OK with periodic boundary conditions and portion operator is our is minus accepts minus the y y and so this example kind of a special example but it alone
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will learn a little bit by the guiding functions are used to pile higher next year 2 cry and why and Eigen value is is so as to 5 squared and square possum square so sold other words this function NMR introduces the EMS is quiet and then forget other lose to keep I decide I don't know almost certainly screw something up of incubated so that this function is EMS looks like this so in other words it stayed on the value Eisenstein's series and said this is side seriously for S 2 the evaluated at the point 5 OK so at the point 5 right so to do this computation will we actually computers absolutely explosively and also very briefly do it some more generally we can look at latitude for the so what we want right is we want a computer the exponential or mine the prime 0 let's do something more general so we will computers bewildered by computer by computing it for all lattices at once so we look at PS which is the sum of YDS this adviser standardizes assigned series for to the right so we can recover this look at on the upper have played wonderful to see was a standardizes sensors and so Our function we want is the value of the supply now all alright so what we want is to differentiated 0 get it turned out you can do this and basically by pure thought so this is on side sensory satisfies 2 properties it tonight and function and it's constant term is YDS plus the IFC U.S. minus 1 over perceived to ask why the onemonth compared with this the completed zeta function so now if we want to work out the derivative DVDs CDs at a sequel 0 was calls function capital you learn something about this from but by from both of these facts the 1st thing you learn is that the law unit that is equal to 1 outside this is not actually quality of it it's this class the king get so we just sort of this function differentiated 0 get something oldfashioned 1 and the 2nd thing says that F is equal to why no French Lord y minus pious wife over 3 possibly tainted stock research you if you differentiate this "quotation mark you evaluate the state of the pie 3 consumers 8 of 2 so what this says that alright so I claim that in fact this this function that is but the logarithm although why times Delta to the 1 Over something 6 OK we this and my money or for and the reason I noticed this point is that these 2 function the difference of these functions is now a harmonic function which is going to 0 infected so it has to be easier so in other words what we've done so in the end if you go back to this expo mind is the primary 0 something like this wide times Delta to the 6 million I might have missed in person and I'm not sure if the university there should be an inverse or not but but when put minus 1 so there OK so this is certainly a very striking calculation because it shows this invariant in this simple case and gives you a number theoretically interesting thing I think that was a rinsing a discount competition I think it's certainly suggested to the HMOs a good definition but actually the important thing to learn from this competition is that this like what I do this LipoScience thing this series itself I evaluated this at a specific point by looking into the function of the whole space again the use harmony city and that's very that there's sort of moral of the story is that this will clash in determines are very difficult to understand themselves but you can understand how they vary but they can differentiate them when you when you are change parameters I'm just a bit to finish it's also 1 of the motivate 1 of the reasons that rinsing realized they were onto a good thing is that the right hand side of that did their formula the righthand side of the street singer formula the the which involves a determinants they confuse what they did is they said Well here's something which depends on a metric and then they change the metric and they they watch what happens and they find it to be metric independent prices very hard to understand those directly but they could check the doesn't changes you change the metric so if you have such an In variant of a man metric which is then independently metric it's reasonable to think that it's something purely topological OK so that's the 1 didn't take waving is not a specific example but that you can understand how the before plotting determined varies will you make when you change things so sorry I went a little bit Overtime thank you for this work the to we believe that this will have a way so where are they going to do it as long as it's 0 1 holy I'm sorry I didn't try should put the items that specializes in the point by that's right so I think you the should be 1 Europe question the as