1/4 Automorphic forms in higher rank
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Title 
1/4 Automorphic forms in higher rank

Title of Series  
Number of Parts 
36

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CC Attribution 3.0 Unported:
You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. 
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Institut des Hautes Études Scientifiques (IHÉS)

Release Date 
2014

Language 
English

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00:00
Polymorphism (materials science)
Classical physics
Multiplication sign
Parameter (computer programming)
Weight
Maxima and minima
Sign (mathematics)
Hyperbolischer Raum
Matrix (mathematics)
Lecture/Conference
Moving average
Ranking
Gamma function
Analytic continuation
Determinant
Social class
Series (mathematics)
Product (category theory)
Quadratic form
Bilinear form
Local Group
Subgroup
Connected space
Modulfunktion
Boom barrier
Computer animation
Hausdorff dimension
Ranking
Identical particles
08:05
Complex (psychology)
Group action
Free group
Logarithm
Real number
Multiplication sign
Plane (geometry)
Matrix (mathematics)
Hyperbolischer Raum
Positional notation
Lecture/Conference
Term (mathematics)
Square number
Subtraction
Descriptive statistics
Multiplication
Product (category theory)
Spacetime
Theory of relativity
Scaling (geometry)
Element (mathematics)
Basis (linear algebra)
Thermal expansion
Mathematical model
Hyperbolische Gruppe
Cartesian coordinate system
Complex number
Quadratic form
Bilinear form
Connected space
Field extension
Vector space
Hausdorff dimension
Order (biology)
Clifford algebra
Equation
Right angle
Object (grammar)
Multiplication table
Identical particles
16:05
Group action
Multiplication sign
Mereology
Theory
Subset
Quadratic equation
Inclusion map
Sign (mathematics)
Matrix (mathematics)
Degrees of freedom (physics and chemistry)
Lecture/Conference
Determinant
Subtraction
Social class
Spacetime
Product (category theory)
Element (mathematics)
Mathematical model
Set (mathematics)
Complex number
Local Group
Vector space
Hausdorff dimension
Clifford algebra
Order (biology)
Universe (mathematics)
Phase transition
Linear map
Identical particles
22:21
State of matter
Multiplication sign
Correspondence (mathematics)
Parameter (computer programming)
Complete metric space
Mereology
Group representation
Heegaard splitting
Sign (mathematics)
Roundness (object)
Matrix (mathematics)
Conjugacy class
Plane (geometry)
Orthogonale Gruppe
Negative number
Ranking
Spacetime
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Hyperbolische Gruppe
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Functional (mathematics)
Automorphism
Modulfunktion
Hausdorff dimension
Order (biology)
Figurate number
Sinc function
Similarity (geometry)
Event horizon
Theory
Number
Prime ideal
Frequency
Hyperbolischer Raum
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Lecture/Conference
Wellformed formula
Operator (mathematics)
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Series (mathematics)
Eigenvalues and eigenvectors
Chemical equation
PoissonKlammer
Element (mathematics)
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Subgroup
Summation
34:32
Series (mathematics)
Modulfunktion
Plane (geometry)
Lecture/Conference
Energy level
Parameter (computer programming)
Quadratic form
Variable (mathematics)
Subtraction
Food energy
Subgroup
Automorphism
36:37
Point (geometry)
Discrete group
Real number
Tournament (medieval)
Inverse element
Mereology
Food energy
Prime ideal
Group representation
Goodness of fit
Hyperbolischer Raum
Matrix (mathematics)
Plane (geometry)
Causality
Root
Positional notation
Lecture/Conference
Operator (mathematics)
Square number
Energy level
Determinant
Symmetric matrix
Series (mathematics)
Spacetime
Block (periodic table)
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Set (mathematics)
Mathematical model
Complex number
Variable (mathematics)
Bilinear form
Functional (mathematics)
Sequence
Local Group
Commutator
Symmetry (physics)
Hausdorff dimension
Identical particles
Resultant
44:57
Discrete group
Musical ensemble
Multiplication sign
Inverse element
Weight
Mereology
Group representation
Matrix (mathematics)
Binary quadratic form
Analogy
Determinant
Modulo (jargon)
Position operator
Social class
Fundamentalbereich
Spacetime
Product (category theory)
Commutative property
Gamma function
Bilinear form
Measurement
Functional (mathematics)
Category of being
Modulfunktion
Hausdorff dimension
Order (biology)
Slide rule
Observational study
Diagonal
Mass
Event horizon
Theory
Number
Power (physics)
Root
Lecture/Conference
Natural number
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Symmetric matrix
Skalarproduktraum
Series (mathematics)
Element (mathematics)
Quadratic form
Subgroup
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Summation
Finite set
55:32
Classical physics
Diagonal
Multiplication sign
Thermal expansion
Set (mathematics)
Student's ttest
Mereology
Inequality (mathematics)
Time domain
Number
Hypothesis
Modulfunktion
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Position operator
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58:03
Standard error
Multiplication sign
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Mass
Weight
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Power (physics)
Lecture/Conference
Simplex algorithm
Term (mathematics)
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Series (mathematics)
Eigenvalues and eigenvectors
Chemical equation
Basis (linear algebra)
Hecke operator
Bilinear form
Functional (mathematics)
Local Group
Lfunction
Connected space
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Arithmetic mean
Modulfunktion
Voting
Oval
Linearization
Right angle
Object (grammar)
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Invariant (mathematics)
Limit of a function
00:07
In you can all women on stage so actually the titles of dimorphic forms on hiring groups at least the official title but the but the world that there will obviously be obviously be some overlap with other lectures from but that's probably not a bad thing so against the leap in his lectures has focused on the group's vessel to are the leading to the 1st half OK well never mind him perhaps in any case they sell to hours the underlying group and for classical out of Norfolk forms and there are many ways to to generalize this due to certain special isomorphic 1 can view is so too are as the connected component of S O 2 1 and then the natural thing to generalize this is To look at it so 1 Of course you you can generalize this even more and look at O P Q let's not be too general another way to look at itself to our it is to viewed as this P to R or depending on your taste is P 1 hour of need and then the natural generalization is S P 2 and R 4 what Emanuel suggested you can view this as to Class AA plus means positive determinant and then the natural generalizations is PGO are now this has ranked 1 so strictly speaking this is not a hiring grow but in some other sense is a hiring group it has racked up rank 1 over par but over QP it may have much larger and so perhaps the best qualified as something of higher rank this has rank and and this has rank and minus 1 I will mostly focus on the PGA land but it will also mentioned some things about our hyperbolic spaces and some discipline the group C modular forms of so let me start with with hyperbolic spaces OK so we are interested in and out of Norfolk forms on S O N 1 and to get started we need some so we started with the Medellin decomposition of S O N 1 and what abstractly it's a product of 3 groups an alien came I'm not quite sure how to no I'm just trying to find out how to organize but the blackboards appropriately brought soldiers continue but still carries the maximum Compaq subgroup that's mostly to describe this as 1 and then as and so that's obviously isomorphic to SOS and then determines the rank and you can easily see its rank 1 group it depends on 1 parameter and then the rest is the identity matrix the Mets isomorphic to install 1 1 perhaps connected component of the identities I and then there's a bit harder to describe our it's easier to describe the correspondingly algebra and then and then it's just the exponential of of of the Lee algebra and so it's matrices of the form identity plus the matrix and plus a matrix and squared over to that's the beginning of the tennis series expired and turns out that the rest of 0 where members of the following fall in it is well this and it's not just I'm so this is a matrix and take a different font 11 so it's it's a matrix of dimension and plus 1 and it has a victory here of dimension and minus 1 and the same Victor again and it has affected here With a minus sign and the same victory here and the rest is 0 that's it this is the victim an oddity and minus 1 and this is isomorphic has agreed to oddity and minus 1 OK so there are there are other ways to choose coordinates and here I'm assuming that my quadratic form is something like minus X squared and then plus y squared blasted squared plus W's with so it's minus 1 plus plus plus plus plus 1 and so on but sometimes but that's right this time Is this go on forever I said this use the underlying quadratic form minus 1 and that 1 1 1 1 1 I or I don't know but maybe decides a different it's 1 and then minus 1 1 or whatever this is obviously a form of signature 1 and also the quadratic form 1 1 and then this is
08:08
used so this all this has won hyperbolic plane and then I hear it's the identity matrix that is also a form of signature when and is used in everything looks a little bit different OK so hyperbolic space is it's all in 1 22 and that's a natural generalization of the upper half and indeed if if any calls to and perhaps I have to take the connected component of the of the identity matrix if any calls to them this is just the usual half playing and if any equals 3 this is what actually introduced yesterday that hyperbolic free spirits then again there there are several models of of this hyperbolic space there is the hyperbolic model this is the 3rd of all it's not up to extend the positive real times the real so the pollen ends such that makes 0 squared minus X 1 square mile expanse where equals 1 and then you have a natural action of SO and 1 on this hyperbolic many more familiar to you if you're if you have grown up with the with the usual upper half plane the upper half space model their importance in this case he will plus 1 cordons but 1 that equation here we have importance and we just require the last coordinates to be positive so in the case and stewardesses just the usual X Y complex plane and there is a 3rd I description in terms of Clifford algebra which is sometimes quite useful and let me spend a little bit of time defining the relevant objects see there is the cliff order of Gibraltar which you may or may not have heard of so that Clifford algebra is an algebra all dimension to to the end so as a vector space it has dimension to to the Powhatan and a vector space basis is given as follows we take the palace 0 some basis elements I 1 up to iron so I take elements and I take the powers and so I didn't obviously to to elements and I interpret a subset but so any element in this policy is a subset of of this said I went up to iron and I interpreted as the product of a bill that would give examples I so this is about a bit of abusing notation Sosa strictly speaking the basis is so it's that are plus are I 1 plus up to ah I and then all products are I 1 no when I won the right to and are 1 3 and so on so that's a product and not really a substance but it's clear how to how to do this and so this defines the Clifford algebras resurrected space which is not particularly interesting but it's algebra so we need some multiple application relations and I call them because they play a similar role as I of complex numbers so in particular we have IGA square is minus1 fall and we have I a IDB is minus I agree I a this is what you know from Hamilton kwacha and in fact it turns out that the Hamilton quarter engines are a special case of this then there are some more relations and this is not enough to get all the relations but there so the multiplication tables so as an example C 0 it's just the real numbers see once the complex numbers and see to it that Hamilton's so to has it is our plus RIA plus RJ plus I J and J's scale so inside Syrian minus 1 we have an important vector space V. N minus 1 and this is the vector space consisting so generated by all basis elements of degree at most 1 so I A 1 plus plus up to I and minus 1 that's a vector space of dimension and inside the Clifford algebra of dimension to to the months 1 and I can the upper half space age then this up there should be a page and as sitting inside this vector spaces and in in a very natural brave I take the upper half space model then I have and coordinates the last of which is positive and I simply map to X 0 X 1 to extend minus 1 so I don't I don't do anything in this matter OK so you can view would
16:11
be the upper half space as sitting inside a Clifford algebra so in the the familiar case of any equals to it sits inside see 1 and see once the complex numbers that's what you know the upper half playing is part of the complex numbers and as action mentioned yesterday the upper half space in equal to 3 can be viewed as sitting inside the courtyard now the important question is how does the group that has so so hard as the group is in 1 act on this and I mean this is obvious in the hyperbolic model but it's not so obvious in the upper half space Mullen can it can be very well described in this cliff
17:04
algebra modeling there exists a set which I call this the V stands for volunteers who more than 100 years ago introduced this theory this view of the and minus 2 which is a subset of matrices 2 by 2 matrices With interest indeed Clifford algebra C and minus 2 acting on VGA minus 1 by fractional linear transformations if I take a matrix G the 2 by 2 matrix with certain not all entries but said so not all Of these matrices are allowed to I have to take a certain subset but there to by 2 matrices with entries in the and minus 2 and then there's a phase place the time season plus the universe now this algebras highly noncommutative so the order does make a difference I cannot write a that plus the overseas placed the I'm doing all of this in the in the Clifford algebra which is highly noncommutative GE is ABC News In this SVC and minus 2 and that this view and minus 1 and this makes sense so a B C and D are elements and CNN minus 2 but of course I can embed C and minus 2 In into and minus 1 and then I know what the product With an element in C and minus 1 as it turns out which is not easy to see if we have to show it and that this is an invertible element in this in this such not everything and it underwrites invertible but it turns out that this element will always be it was the whom I this is yesterday I have problem with the definition is a bit more complicated but this is certainly 1 of the requirements and and the definition of SB is rather complicated but I can give you the definition simple cases so here are some basic examples SVideo C 0 is simply a sale to are so there is no extra assumption it's just while the determiners 1 but other than that it's just everything the same holds for S V C 1 this is just a failed to see but already it's easy to it's quite complicated or a bit more complicated this is the 3rd of all matrices G freed 2 by 2 matrices with entries in the such that the following hold ABC that are minors be stars equals 1 babies and to see the stars are in V 2 so they have occured explained isn't in the 2nd and what is star star is the pollution so far is the inclusion that maps X plus I Y last JayZ class kW 2 X plus Y plus JayZ minus kW so in particular stars the identity on this vector space the 2 because the 2 is defined by last managing coordinates and it changes the sign of the last quarter but let's make a reality check up on what the dimension of this and this and this turns out to be a group of what is the dimension of this group well for each entry you have over the real full possibilities because these are Hamilton so this is mention 16 but this means that the last coordinate vanishes so the subtracts 1 dimension another dimension and here you have the random "quotation mark turning and that has to be 1 the subtracts form more dimensions so until the F16 minus 1 minus 1 minus 4 the mention 10 if 10 degrees of freedom over the rails and the dimensions of this for 1 step that's so this is good it's a reality check the
22:22
dimensions of SUB c and minus 2 authority to its 16 minus full minus 1 month's 1 10 and that's the dimensions of this whole thing about all right I'm so the most interesting case is the most interesting case this is the case any free that's the case that actually mentioned yesterday well perhaps the most interesting cases any puts to bed down yeah I'm supposed to talk not to talk on the case in the courts to Sudan page 3 is they failed to seize more general as you 2 and as we have seen in the Upper House Ways model to begin recording exotics Y & R In our 3 such that are as positive and so you can view this as a Hamilton Court turned with last Ishant coordinate this is sitting inside the Hamilton where last vanishing coordinates no of any questions so if you want to have further reading literature on especially on the hyperbolic free space but also on hyperbolic in space and this violent group and Clifford algebras and so on basically everything by this Pulitzer cool about it 10 minute they have several free all the papers and you find everything in great detail in their works where there is a famous book the book treats hyperbole free space in complete detail you find everything you want to know hopefully in this book but so that's that's certainly the most important reference but this had Bolick in space is treated or in in research OK so they are comedians theory is very similar to to that in the end cases very similar to the case and he wants to simply because it's a rank 1 road and then In particular so there was there was 1 Laplacian eigenvalues 1 spectral parameters and 1 has similar for instance when a similar balance awards from a new there is a couldn't itself formula which has a very similar shape as the original cause it's a formula you can find this in many works you could find this and work of makeover the me Tallil warlock and there is also a little long paper by cocktail the Piechoski Shapiro and cyanide so the Artmedia theories fairly similar to the classical case that this serious quite different because had a theory so there I had cooperators if you take if you taken an arithmetic subgroup you can define the cooperative's in in the usual way but the hacker theory is is a bit different from because over Q P it's all in 1 may have large rock not necessarily but depending on it can have large rank and than the theory is a little different so I mean it's actually said is that they had cooperated with the exists I change the picture completely and had a theory is that is a very important part and they had a theory is more complicated because of this whole and 1 over Q P you may have rancour and plus 1 over to douse bracket it's already in the
27:41
case in equal to 3 in the case and equal studiously 3 over to maybe 1 . 5 but if you take the girls bracket that 1 but then equals 3 the rank already be too and so is if you have a given round than the at least morally and in some sense very precisely the head guy duress generated by as many elements as the ranks said and done so you see this so if if any 3 then the rank can be as large as 2 and that you can see this in the classical picture but if you if you viewed this as automorphic forms over a over an order in in imaginary quadratic field from their army fired Prime Sewell Brumby brands that are not interesting but their Split crimes and in our prize and if you have split Prime's then you get to heckle operators full both copies of the Split branch some examples any equal 3 this peace the prime ideal the rational prime ideal decomposes S P P bar then 1 gets To had cooperated P and TP but of course if peace talks before this happens for half the price and if if peace innards but but then of course you get only 1 OK so why is this interesting and you can define whatever you want but don't does this have any arithmetic significance will certainly the case and equals 3 has a lot of our phonetic significance because it's automorphic forms over menagerie quadratic fields but what about higher like high and so high hyperbolic space what is the art medics significance so he wasn't there many example I'm and that's associated with a feeder series and quadratic falls if you have a positive definite quadratic form you can easily right down generating series for the representation numbers and then you get a feeder series and because the quadratic forms positive definite that the representation numbers are finite and the only nonnegative representation number so we have no problem with convergence and you get and the modular form on the upper half if you have an indefinite quadratic for and then this doesn't really work but because they're infinitely many ruining the reputation numbers are infinite and you have negative proposed potentially you can you can now you can represent negative numbers so it's not really key how to define a feeder series for an indefinite quadratic form and the associated developed the theory the that QB an integral quadratic form with signature NY and somehow we want to define a feeder series attached to this quadratic fall but the naive thing of just writing not generating serious doesn't work so Ziegel introduces the following the introduces the socalled major inside is a major event of humans it is a positive definite fibrotic form well it's a positive definite states symmetric reels NI and matrix but in fact spent just 1 by and plus 1 matrix the matrix are satisfying continue over here are queuing various equal skewed and 1 can show us that if you have 1 of them you can easily write down all of them so if you've won such matrix are then all matrices self the following form these are some of the form George G. transposed RJ for gene it's few so the special orthogonal group attached To the quadratic form cure so more all of the matrix of matrices satisfying this are given by 1 and then you conjugate while some pretty conjugation but with West the matrix is so cute so what that means you transposed hugely costly already and now we are ready to define the corresponding figures series and we define the feeder series attached to the matrix Q as follows it's a it's a make it's a function of 2 arguments it has an argument in the upper half plane and it has an argument and all Q and it's the sum of all victory In general directors of dimension and plus 1 exponential so what you would like to do is something like 8 strums supposed this is not correct what I'm riding number would you would like to do something like age transposed Q H times there this is what you would do so that there is no This is what you would do if Q was positive .period but since cues not positive definite this doesn't make sense it
34:33
doesn't converge if you just take the X according that it's still OK and for the white cord you do something different so you to execute and for the white coordinates plus wife and he g transposed hard for 6 major fix your favorite matron I 8 With this X plus I Y this explains why why in the usual upper half plane energy is in this auction so did this is this is the socalled view feeder series and it turns out it's
35:42
a modular form in both variables it's a modular form In that as the usual modular form on the upper half plane with respect to some Congress subgroup which depends on cue so if you have a certain level and then you have to more out certain Congress subgroup and also modular form In June so in energy in the 2nd argument it's a modular form for something that's isomorphic to all in 1 because Q his signature and 1 In the 1st Variable it's an automorphic falls usual automorphic from the upper house but if you keep the 1st Variable fixed then you get a nice interesting automorphic fall on S and 1 so it is modular In that engine so there is some arithmetic significance
36:39
attached to the forms on on hyperbolic space but any questions this is the kind of the I'm well in what sense I mean certainly it's it's it's from I guess it's not a it's not hospital I so in this sense that so it has to do with Eisner said series and you can be a problem and if you keep G fixed and
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viewed as a function that then you can decompose and took the yes was this was the model for me tho she listens recently said what hold only a day up yet since then with the you have to mark out bias by suitable discrete subgroup of which depends on cue from soared accusing it depends on the arithmetic of Q there is some level if you give you out by some Congress both instead as a in both variables it was a question of the heart of the celebrated as if he gets the misery where of isn't like he was born a year from his yesterday was even years ago relied on my part prime ideals :colon of the year good question and I don't know what else I think at least in this case I worked it out and it turns out the index by "quotation mark tournaments so yet they they indexed by matrices this is called the cause a determinant this expression here and you can if the cause is determined equals a real number and then this corresponds to coach to that to the end cooperated and just as in the half a dozen of the points this has to be opera during the course of German could be any Hamilton quarter but that it doesn't commute so you have to take something from the center and the center's just the reels in this case and I so at least in this case in this description and for the case of seemed to be Hickel operators are but parametrized by cause a determinant being an arm and I'm actually not sure how they're paraphrasing generals and I'd downloads that you find this anywhere in the literature the border is being litigated raised energy about you want and you can do it as a worker would meet again on his mind .period many amended well so show if you if you forget this picture with the bond of Dubai and just go back to it's all in 1 then of course I mean this is this is a wellknown Rubin you can read and sat talking in the In the original paper and everything and I mean you can just write down the the the the "quotation mark sets with their respective representatives but but of course but if you want to have it if you want to decompose the biblical sets into twofinger Cox this is a complete nightmare if you want to do it in general and yeah anywhere think very very few explicit results are in the literature other than in the case and inputs to any equals 3 which is classic but a any other questions OK so this was just a very very brief introduction into hyperbolic space is just to give you an idea of some definitions so that I can live with you at least know how to start and equally briefly I would like to discuss some plastic others and like the group and also give a few basic definitions and then after that we move on to PG something simply to groups OK so let me 1st defined from the Simplot the group and there's great confusion about what some people call this sp too in some people call this experience but I call to end but if you don't like it feel free to call it a skier so these are all matrices then itself to end are such that transposed J. M. sequence Jr we're jaded the mother of all simplistic matrices minus identity identity so this is the identity of dimension and the identity of the mansion and you can write this as explicitly as all matrices ABC deal in block notation with a B C and D Oregon matrices of dimension and such that paid the transpose minus believes it transposed equaled the identity a transpose is being transposed and seeing the transposed decent spots with other words is a symmetric so this is the usual block notation that you find in most of the literature but capital letters always denote dimension and mattresses but in there isn't an upper half space that I also call H but it's not the age that we had in for the hyperbolic space as age it the set of all matrices Z equals explosive Iwai invited matrices but now we're seeing such that is symmetry and my wife is positive definite and you can embed this into the Simplot the group as matrices I X so this is the same thing that you can know from the upper half plane of and vehicle the inverse where these the square root of what so why is positive definite so you can take a square root and so on this is the usual way to embed complex numbers into S L to R where if the unique symmetric matrix such that transco studying the Croats and OK certify call this gene my Group G then this is just the caution GE Margalo a maximal contacts separate all energy acts on this upperhouse space in the usual way group actually the matrix a B C D Back on a point that which is
44:57
in fact the matrix it is the place to be I can see that plus the inverse of again you have to be careful and with the order because matrices are not commutative but aren't as usual we take a call a discrete subgroups for instance we can take its speed to end pull with the integers but we don't have to and this comes with an inner product there isn't enough product the inner product is just what you want this would you would guess the upper house space modulo grammar and then you take if 1 of their if 2 of car times an invariant measure and the invariant measure is the it's the wife over determinant wise to the power and class 1 so the classical case is the case in equals 1 and then you just recover the usual thing so these are the analog of mass forms if you have followed motheaten single modular forms of a certain weight then you have to include determinant wide to some suitable power case as usual all so this looks all very similar to what you probably know from the classical case except that all of numbness and matrices but formally but it's very similar in many respects so why is this interesting again I mean you can't generalize as much as you wish but otherwise this interesting here's the motivation of moderation kind comes again from quadratic falls motivations why do we want to study the new modular forms so if you were given a positive definite symmetric matrix and end by an integral matrix symmetric positive definite and even by even I mean that the diagonal elements are even and and even interim matrix as an interim matrix with even diagonal elements and pick an integer a positive integer or yet positive lesson and or less than or equal to men for a matrix T all of dimension M With a half interval entry and died at all symmetric and positive definite study the representations all of the but so what does this mean well if he happens to be a number so if M equals 1 then this is what we usually do you want to know how many ways are there to write a given number is sum of Foursquare and the you can just as well ask how many ways are there to write about giving quadratic forms a binary quadratic form as the sum of Foursquare so this is not representation of numbers by forms but representation of Forbes platforms but lower dimensional forms and the special case any quotes wonders just numbers but you can't stop priori pick any and between 1 and 2 so we can find the representation of both R A of the it is the number of matrices GDP of dimensions and time skin so in the classical case an equal to 1 business that the usual victory such that 1 half Julie a duly transposed the equals and because of its positive definite and positive while a positive evidence this is a finite number and you get in coach these representation numbers into the series day event is the sum of all TV Our cues story or a a of the trays
50:21
said you need to traced to go back to numbers where is that lives in H M "quotation mark indeed turns out that this fear function has nice property this transform slides see their plans of the 2 the power In over to the CIA of Z 4 gamma some matrix with lower interest CD gained some Holmgren subgroup gumbo of this P 2 Z but he otherwise it makes no sense yes and so it turns out that the day is physiological form also wait In over to and degree for genus but so that there isn't a natural motivation why we want to study such objects because they're connected to the representations of quadratic forms by providing false alarms we have seen that many many of the formulas look exactly the same but the other formulas that don't look the same so many things become more complicated for instance there is typically a formula for the imaginary part of them as that on which you can relate to the Majorie part of that but here formula looks much more complicated it seemed that plus the minus transposed the imaginary part of that season plus the bar in the it's not you recognize of course that everything on that everything's numbers then you get the usual formula imaginary part overseas that the squared up reality but you can't do this really because it's noncommutative and you can imagine that Dad explicit formulas becomes very ugly in this way but there is the usual fundamental domain fundamental domain forecast P 2 bands that modulo H N but which looks very similar in some sense to the wellknown case any equals 1 1 needs Makovsky reduction theory but well yes well this was a case of this was a motivating example I am now we continue with the with the usual theory and we called the genus and it just so happens that the genus here was I so this is perhaps pedagogically not optimal but Koski reduction theory I I tell there is a fundamental domain such that the coordinates of the matrix Texas excite J bounded by onehalf and why I J Armand Koski reduced which means that the off diagonal is bounded by onehalf times the diagonal and the diagnosed nonnegative In fact it strictly positive and we can we have half square root 3 is bounded by y 1 spotted by wire to don by white and the determinant of is roughly the product why 1 of 2 white and in other words the the off
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diagonal but it's at least in terms of the determined rather negligible the move is part conditions as that's right that's right that's not not everything is in the fundamental physicists as viewed set for the for the fundamental them the exact fundamental demand has not been worked out except for the case in equals to so got chilling in his thesis 50 60 years ago but 1 of the last thing students some no doubt down I don't know certain conditions or whatever exact inequalities 19 OK maybe it's 19 whatever it's it's a it's still a manageable number but a pretty large so there is a finite number of conditions for its plea for I bet for higher genus and I don't think it knows of no 1 has ever worked out exact conditions for the Fund for the rest of the solution to the problem of but I think it's just in the case as in the classical case nobody needs that and it is Europe and if you have a nice Siegel domain and you know that everything is inside the steel demand and everything so I I get back this blackboard the people all of them what I'm supposed to stop anywhere but let me the let me just quickly say something about the 40 expansion because that's something that's very important in the itself to our case and it turns out that the free expunging therefore modular forms is much less useful there exists a for expansion of course but it's it's much less useful for a expunging a modular form has a 4 expires the following types want sums over symmetric matrices positive definite or perhaps positive very definite and half integral some coefficient times he'll traced to the and
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it turns out that this coefficient K of satisfy certain cemeteries for instance but OK so this is for a modular form of weight Katie and there is a determined you to power came a you transposed to you you for all you can do Jill and so this is invariants by units if you want but it's less useful for any greater than 2 on greater than or equal to 2 less useful for any greater than 2 in particular they have t the Fourier coefficients has no direct connection To make eigenvalues no direct connection with the guide and others I mean that's something that we're very much used to that for a coefficients of just take eigenvalues this is not the case as soon as the degreesnot not 1 by the degrees 2 or more then there is no direct connection between 40 coefficient and guiding values so before expired this is really that much less useful 1 has the heck abound 18 is bounded by the determinant the of the to the palate to encase the weight I and the conjecture is that this is bounded by determinant key to the Pollack a minus to carry over to minus 10 plus 1 0 full plus Epsilon so if any quotes 1 2 can subtract the house if there is not a lift and I will explain next time what I mean by that is not and this will do this tomorrow but this is not known I mean unless in the case any equals 1 4 Moffat forms what is known is this might the Delta but Delta's tiny shelters 1 over and and we actually expecting something linear but so yeah there's lots of things to do if you want to do if you want to work on this but there are lots of open questions and I guess I have to stop now so that all today the regional commands for what is a loser he will always be there for a year and that when we know what we have seen that the again Island which is was probably beginning I don't know movement certainly in the case and he was free I don't know that there hasn't been much analytic number theory on this basis and I think it's it's it's now the time that 1 introduces the methods of analytic number theory for these types of altered form on Monday the the latest from this it is all well and good if any quotes to them this is the Roman conjecture ride and themselves there was some interest in the Roman legion conjecture of yeah army would anyway with the free expired and you want to have balanced and certainly good to know what the best balance after the coefficients and this is true on average so in some mean square since this is true but individually it's not and I don't think there is some fundamental interest in knowing what the best possible balance of power and 1 of the problems is that this is in fact wrong for certain forms that come from lower dimensional simplex the groups but I'll discuss this tomorrow you so go to was the formation of the yes yes right at the right right they they go into the rights they go into the soap if you right then you have never seen this mass formula and the and the naming this forecast that forms where the feeder series of modular forms right so these go into the error term but these coefficients and if you can buy the argument that certainly would only be who oversaw the use of bulk holding more than the 1 that fits the bill over the decision of whether the the last piece of foam and 1 of the 2 of them on the phone from the day of he said that it this was the 1st time that the political for this fall's said today's vote efficiency President Putin functions with resellers suggests that there really was In the past 2 years and tho wouldn't by this year they are so that they have certainly some intrinsic meaning it that they are not so much related to triggered but perhaps to other arithmetic objects is losses in the center of the story of how I will briefly mention Lfunctions tomorrow but by dad and I and my plan is not to go into most details but but just to give you an overview of the objects that we are dealing with so that you can at least get an idea how to start if you're interested in working on these things more than