Computational geometry of numbers, cohomology of modular groups and
applications
Workshop on computational problems in number theory
Lecture 1 (July 27, 2015) : Motivations and main concepts (v2.0)
Chern Institute of Mathematics, Nankai University
Philippe Elbaz-Vincent
Slides (and complements) on my lectures available at the following URL :
https://www-fourier.ujf-grenoble.fr/~pev/CIM2015
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Roadmap for my lectures
The goal is to start from the motivations, which will use advanced topics that we will define more carefully later, and introduce the tools from the geometry of numbers in order to either solve or get partial informations on the problems presented during the first lecture.
Pitch for Lecture 1 (July 27, 2015) : Motivations from number theory, cohomology of arithmetic groups and K-theory.
Strategy to compute such homotopical/homological objects using the geometry of numbers.
Pitch for Lecture 2 (July 28, 2015) :Explicit computations of Voronoï complexes, the cohomology of modular groups and K-theory.
Pitch for Lecture 3 (July 29, 2015) : Complementary properties, other geometric models and their applications.
Pitch for Lecture 4 (July 30, 2015) : New results toward K8(Z)and the Voronoi complex of GL8(Z). Applications to the Kummer/Vandiver conjecture. Some problems and challenges. 3 / 29
Main references for Lecture 1
[B] Brown, Kenneth ;Cohomology of groups, Springer 1982.
[C] Cohen, Henri ;A Course in Computational Algebraic Number Theory,GTM 138, Springer 1998.
[M] Martinet, Jacques ;Perfect lattices in euclidean spaces, Springer 2003.
[Ca] Camus, Th. ;Computing automorphisms of algebraic lattices, 2015 (Appendix to my lecture notes).
[EGS] E-V, Gangl, H. and C. Soulé ;Perfect forms, K-theory and the cohomology of modular groups, Adv. In Math. 245, 2013, 587-624.
[LS] Lee, R. ; Szczarba, R. H. ;On the torsion in K4(Z) and K5(Z), Duke Math. J.45 (1978), 101–129. with an appendix of C. Soulé.
[So] Soulé, C. ;On the3-torsion in K4(Z).Topology 39, no.2, (2000), 259–265.
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General Motivation for our work
Numerous problems in modern number theory could be solved or at least better understood, if we have a good knowledge on the algebraicK-theory (or motivic cohomology) of integers of number fields or the the cohomology of subgroups ofGLN(Z) (or more generally congruence subgroups of the linear group over the ring of integers of a number field). As a short list, we could mention :
o modular forms and special values ofL functions, o Iwasawa theory and understanding of the “cyclotomy”, o Galois representations (or automorphic representations).
and Geometry of numbers (i.e., theory of lattices) is often a key tool for achieving such kind of computations...
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More specific motivations related to lattices
Classification of lattices,
Detect “high torsion” classes in H∗(Γ,Z) for Γa modular group (in our cases Γ =GLN(Z),SLN(Z)) or prove that there is no such class !
Apply this information to the K-theory of Z. Deduce interesting arithmetical properties ...
+ For any positive integer n we let Sn be the class of finite abelian groups the order of which has only prime factors less than or equal ton. In practice, we will work modulo some Sn.
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What on earth is Algebraic K–Theory ?
To a commutative ringR we can associate (functorially) an infinite family of abelian groupsKn(R) which encodes a huge amount of information on its arithmetic, geometric and algebraic structures.
These groups extend some classical notions and give higher dimensional analogues of some well known results ;
o K0(R) is the classical Grothendieck group (classifying finitely generatedR−modules or vector bundles of finite rank in a geometric setting),
o K1(R) is the group of invertibles ofR,
o K2(R) classifies the universal extensions ofSL(R) and it also classifies the "arithmetic symbols".
We can give a general abstract definition of theKn for n>0, Kn(R) =πn(BGL(R)+),
or also as the homotopy groups of the classifying space of some category associated toR.
Fact : these groups are hard to compute.
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Relationship between the “Kummer/Vandiver” conjecture and the K-theory of Z
Conjecture of Kummer/Vandiver: Let p be an odd prime, ζp=e2πi/p,Cp be thep-Sylow subgroup of the class group of Q(ζp)andC+ the subgroup fixed by the complex conjugation. The Vandiver conjecture is the statement thatC+ =0(for arbitrary p).
In other words,the class number ofZ[ζp+ζp−1]is not divisible by p.
Ernst Kummer (1810-1893) Harry Vandiver (1882-1973)
Fact (Kurihara, 1992) : IfK4n(Z) =0 for all n>0, then the conjecture of Kummer/Vandiver is true. 8 / 29
Special values of ζ functions, K-theory and the Lichtenbaum conjecture
We can also relateζ-functions (or more generally L-functions) and cohomology or arithmetic groups.
Recall that theBernoulli numbers Bn (n∈N) are defined from the following identity of power series :
t
et−1 =X
n∈N
Bntn n!.
Whenn>1, we haveζ(1−n) = (−1)n−1Bnn. The following conjecture of Lichtenbaum relates the K-theory ofZto the negative values of theζ function, for n>0 even :
|ζ(1−n)|=2#K2n−2(Z)
#K2n−1(Z).
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Special values of ζ functions, K-theory and the Lichtenbaum conjecture
A famous result of Soulé (1979) asserts that the denominator (numerator) of|ζ(1−n)|divides#K2n−1(Z)(#K2n−2(Z)). For instance, sinceζ(−11) =691/32760, we deduce that there exists an element of order 691 inK22(Z).
We will see evidence, that this element of “high torsion” should be detected in the cohomology ofGLm(Z) for somem.
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General problem
Find explicit methods for computing (co)homologies of arithmetic groups (mainly subgroups ofGLN(A)with Athe ring of integers of a number field) and the K-theory of number fields (or their ring of integers).
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Topological Excursion
Cellular complexes, or how to simply describe the
"combinatorial" structure of a topological space : It is a generalisation to several dimensions of a graph.
We calln−cella topological space homeomorphic to the open unit ball ofRn and such that the closure is also homeomorphic to the closed unit ball. A cellular complex (or cell complex or also cellular decomposition or cellular space) is a family of setsXn(withn ∈N), such that eachXn is a collection (eventually infinite) of n−cells.
Usually we work with cell complexes with a finite number of cells.
A classical result shows that any (reasonable) topological space can be approximated by such cell complexes.
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Example of the cube
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Computation of the homology of a cell complex
To a cell complex, we can associate a familyCn (n∈N) of free Z−modules and a familydn:Cn→Cn−1 of linear maps.
The moduleCn is the free module with basis then−cells (modulo a choice of orientation).
If the complex is finite, then all the modules are of finite rank and we will denote by {bnλ}λ∈Λn a basis ofCn,Λn being an index set for then−cells. Then the mapdn is defined by
dn(bλn) =X
µ
[bλn:bn−1µ ]bn−1µ ,
and the integer number[bnλ:bn−1µ ]is calledthe incidence number of the celleµn−1 inside the cell eλn.
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Computation of the homology of a cell complex
The relationdn◦dn+1 =0 (i.e., formallyd2 =0) should hold for anyn.
If the complex isregular (i.e., always at most one cell of dimn+1 is between two cells of dimn), then we can build the incidences inductively starting from the 0−cells up to the maximal cells using thed2 =0 condition and moreover the incidence numbers will be 0, or±1.
Thenth homology group of the complex is defined as the quotient ofKer(dn) by Im(dn+1). This construction is functorial (in the category of cell complexes).
As a consequence :We can determine the homology groups effectively by computing theSmith form of the integral matrices of thedn (relatively to the fixed basis).
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How to use such settings for the computation of modular groups ?
IfG is a group acting on a cell spaceX (i.e., G sends n−cells to n−cells), then, under some technical assumptions onX and on the action, we can show that roughlycomputing the homology ofG (as group homology) is the same as computing the homology of the cell spaceX/G.
Hence, ifX/G can be calculated effectively, we can compute explicitly its homology, and from this the homology ofG (similarly for the cohomology). Notice that in general the spaceX/G will not be regular anymore.
+ In practice, we get usually a spectral sequence and its limit allows us to compute the homology (with some coefficients).
Problem :the main difficulty is to find a cell space X such that X/G will be effective.
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What about the modular groups ?
As they are subgroups of someGLN(Z), we have an ’obvious’
action onZN. We can then capture the topology by regardingZN not as a freeZ−module but as a lattice (i.e., an abelian subgroup of full rank ofRN), and see if this leads to some interesting topological construction...
The Voronoï Miracle !Voronoï (1907) has shown that, modulo the action of GLN(Z) and the multiplication by positive real numbers, there is a finite set of particular lattices (called the perfect (euclidean) lattices), and we can endow this set with a structure of cellular complex. Furthermore it is known that this cell complex has the necessary properties for computing the homology of GLN(Z)(or some modular groups).
This approach has been used fruitfully during the past 30 years, in general limited by our ability to perform concretely the
computations. There has been numerous alternate constructions, in particular the one proposed by Avner Ash and his collaborators.
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Several models from the geometry of numbers
AssumingN >3.
Direct cellular decomposition on the Borel/Serre
compactification ofON(R)GLN(R)GLN(Z) (as done by Soulé for SL3(Z)),
Cellular decomposition on the Voronoï model (based on work of Soulé and Lee/Szczarba, mainly used by
Elbaz-Vincent/Gangl/Soulé) ,
Cellular decomposition on the Ash model (simpler than the Voronoï model, mainly used by Ash, McCornell, Gunnels, Yasaki, see also joint work Dutour, Gangl, Gunnells, Yasaki,...), Variations or/and improvements of the above (e.g., Dutour and al.).
Notice that all the above can be seen as higher-dimensional analogues of the Poincaré half-plane (aka Complex upper half-plane...).
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Toward the Voronoï theory of (euclidean) perfect forms and lattices
Our goal is now to define properly the Voronoï cellular complex associated to some families of quadratic forms or lattices.
We will first recall some facts on lattices, then introduce the notion of perfect forms,
and finally describe how we can construct and compute the Voronoï complex and its homology.
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Some quick facts on euclidean lattices
We denote by(|) the euclidean scalar product onRN.
LetN ∈N. An euclidean lattice of RN is a discrete subgroup of maximal rank ofRN.
In other words, ifLis a lattice ofRN, there existe1, . . . ,eN in RN such thatL=⊕iZei. Theei define a basis of the lattice L.
IfL is a lattice ofRN andei a basis of L, the absolute value of the determinant of the matrix
e1 e2 ... eN
(with theei seen as row vectors) is denoted by |L|and does not depend of the choosen basis (exercise). This quantity is called the norm ofL.
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Some quick facts on euclidean lattices
If(v1, . . . ,vN) is an ordered family of vectors of the latticeL(of rankN), the matrix
Gram(v1, . . . ,vN) = (vi|vj)16i,j6N.
is called theGram matrix of(v1, . . . ,vN). We will denote by min(L) the minimum of a latticeL, i.e. min(L) =infx∈L,x6=0(x|x). We will say thatv is a minimal vector ofL if(v|v) =min(L)and we will denote byS(L)or m(L) the set of minimal vectors ofL. The number#S(L) is called thekissing number ofL and we set
s(L) = 12#S(L). A lattice is integralif any one of its Gram matrices is with integer coefficients, and isunimodularif this Gram matrix is integral and of determinant 1.
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The main tool : Voronoï theory of perfect (euclidean) lattices/forms
We have an equivalence between lattices and forms (cf.
[M](§1.7)) :if Lis a lattice, with base ei, ofRN endowed with its euclidean product.
The Gram matrix of the basis of the lattice gives a definite positive form, and if we have a definite positive quadratic form of rankN, there exists a unique base ofRN such that in this base the form is given by the scalar product.
This define an equivalence (a base change of the lattices will give rise to isometric quadratic forms).
As the forms are associated to lattices, they will have a finite number of non-zero minimal vectors. Theperfect forms are the one who are entirely caracterised by their minimum and their minimal vectors.
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Example
The formq(x,y) =x2+y2 has minimum 1 and minimal vectors
±(1,0) and±(0,1). Nevertheless this form is not perfect, because there is an infinite number of definite positive quadratic forms having these minimal vectors.
On the other hand, the formq(x,y) =x2+xy+y2 has also minimum 1 and has exactly 3 minimal vectors, the one above and
±(1,−1). This form is perfect, the associated lattice is the
"honeycomb lattice" (with optimal spheres packing in the plane),
"it is the only one"...
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Voronoï theory of perfect (euclidean) lattices/forms
Ifv is a vector of RN, we denote by bv the symmetric matrix v·vt that we will see inR
N(N+1)
2 . We call it the Gram vector ofv. If h is either a lattice or its associated quadratic form, letm(h)[ ⊂R
N(N+1) 2
be the set of Gram vectors associated to the minimal vectors ofh.
To say that a latticeLis perfect means that the set m(L)[ span R
N(N+1)
2 (cf. [M]§3).
Notice that the characterisation of perfect forms is due to Korkine and Zolotareff (1872-1877).
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The main result of Voronoï on perfect forms
Georgi Voronoï (1868 - 1908)
Perfect forms have been studied by Voronoï at the beginning of the XXth century and quite extensively since 1950, in particularly due to their inti- mate connection with spheres packing problems and coding theory.
Voronoï (1907) has shown that, mo- dulo the action of GLN(Z) and the multiplication by positive real num- bers, there is a finite number of perfect forms. Furthermore, we can associate to the equivalence classes of perfect forms a regular polytope in R
N(N+1)
2 ,
which is compatible with the action of GLN(Z).
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The Voronoï space
LetN >2 be an integer. We let CN be the set of positive definite real quadratic forms inN variables. Given h∈CN, letm(h) be the finite set of minimal vectors ofh, i.e. vectorsv ∈ZN,v 6=0, such thath(v) is minimal. Recall that a form h is calledperfectwhen m(h) determinesh up to scalar : ifh0∈CN is such that
m(h0) =m(h), thenh0 is proportional to h.
Denote byCN∗ the set of non-negative real quadratic forms onRN the kernel of which is spanned by a proper linear subspace ofQN, byXN∗ the quotient of CN∗ by positive real homotheties, and by π:CN∗ →XN∗ the projection. Let XN =π(CN)and
∂XN∗ =XN∗ −XN. Let Γbe eitherGLN(Z) or SLN(Z)(or more generally, a subgroup ofGLN(Z)of finite index). The group Γacts onCN∗ andXN∗ on the right by the formula
h·γ =γthγ , γ ∈Γ, h∈CN∗,
whereh is viewed as a symmetric matrix and γt is the transpose of the matrixγ.
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The Voronoï space
Given a perfect formh, we denote by σ(h) the convex hull ofm(h)[ inXN∗.
Crucial Fact 1 (consequences of Voronoï’s work) :The sets σ(h)and their intersections whenh runs through the set of perfect forms, give a structure of cell complex (i.e., CW-complex),
compatible with the action ofΓ. It describes completelyXN∗ and it is a regular polytope inR
N(N+1)
2 .
Crucial Fact 2 (Soulé, 1978) :The spaceXN∗ is contractile and XN∗/Γis compact (for the cellular topology). It is a "cellular compactification" ofON(R)GLN(R)GLN(Z)homologically equivalent to the one of Borel/Serre.
As a result, we can use it to compute the cohomology ofΓ.
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The Voronoï space
From a practical viewpoint, we can handle explicitly this complex as soon as we have the classification of perfect forms of rankN. This classification is known forN 68, and the work of Jaquet (1991) gives a setP of representatives of the perfect forms of rankN 67 and, for anyh∈ P, the listm(h) of its minimal vectors, a list of
“neighbours” ofh (modulo the action of its stabilizerΓh) and their representatives inP. We deduce from this data the "list of faces"
ofσ(h) moduloΓh.
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Some classical data on perfect lattices
rank 1 2 3 4 5 6 7 8 9
#classes 1 1 1 2 3 7 33 10916 >500000 example Z A2 A3 A4,D4 A5,D5,A25 A6,D6,E6,E∗6 A7,D7,E7 A8,D8,E8 A9,D9
The classification of perfect forms of rank 8 has been done by Matthieu Dutour, Achill Schürmann and Frank Vallentin in october 2005. More details available at
http://fma2.math.uni-magdeburg.de/~latgeo.
They have also shown that in rank 9, there are at least 500000 classes of perfect forms.
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