Martin Kneser's work on quadratic forms and algebraic groups.

<strong>Martin</strong> Kneser’s Work on Quadratic Forms andAlgebraic GroupsRudolf ScharlauInternational Conference on the Algebraic and Arithmetic Theory ofQuadratic FormsLlanquihue, Chile, 19.12.2007<strong>Martin</strong> Kneser (1928 – 2004)- strong approximation and classnumbers- Galois cohomology of algebraicgroups- geometry of numbers, explicitconstructions of lattices12Contents1 A Short History of Quadratic Forms 1884 – 1954 42 <strong>Martin</strong> Kneser: Quadratic forms and arithmetic of algebraicgroups 1955 – 1970 202.1 Strong approximation and class numbers . . . . . . . . 282.2 Local-global principles and Galois cohomology . . . . . 322.3 Siegel’s theorem and Tamagawa numbers . . . . . . . . 341 A Short History of Quadratic Forms 1884 –1954The theory of quadratic forms emerged as a part of (elementary)number theory, dealing with quadratic diophantine equations, initiallyover the rational integers.2.4 Further contributions by M. Kneser . . . . . . . . . . . 373 A particular problem: finite quotients of Bruhat-Tits buildings 4034

Herrmann Minkowski (1864 – 1909)The main questions in modern language were:a. the equivalence problem: when are to quadratic modules(“lattices”) (L,q) und (M,q ′ ) over Z equivalent ?b. The representation problem: for which t ∈ Z does there exist ax ∈ L with q(x) = t ?c. The determination of the representation numbersa(t,L) = |{x ∈ L | q(x) = t}|.He developped the foundations of ageneral theory of quadratic forms overthe rationals and rational integers. Healready proved major results on allthree questions in a modern way.56A brilliant <strong>work</strong> of the very young Minkowski is the prize-winningpaperGrundlagen für eine Theorie der quadratischen Formen mit ganzzahligenKoeffizienten, Mémoires présentés par divers savants al’Académie des Sciences de l’institut national de France, TomeXXIX, No. 2. 1884.In the main part of this paper, he develops the local classification ofintegral quadratic forms. In the context of the prize question on sumsof five squares, this was preparatory, but clearly of independentimportance.In his Königsberg Dissertation from 1885 entitledUntersuchungen über quadratische Formen.Bestimmung derAnzahl verschiedener Formen, die ein gegebenes Genus enthält.Königsberg 1885; Acta Mathematica 7 (1885), 201–258he proves a version of the Maßformel which is already very similar tothe current one. In contrast to the <strong>work</strong>s of previous authors, the“right hand side” is a product of local densities over all primenumbers.In this context, Minkowski also introduces for the first time (more ofless) today’s notion of a genus of quadratic forms (in any numberover variables).78

In those days, the rational theory (classification over Q) still was aby-product of the integral theory. Nevertheless, the following paperpractically contains the main theorem over Q.H. Minkowski (Letter to Hurwitz), Über die Bedingungen, unterwelchen zwei quadratische Formen mit rationalen Koeffizientenineinander rational transformiert werden können, J. reine angew.Math. 106 (1890), 5–26 = Ges. Abh. I, 219–239.To every rational quadratic form, Minkowski associates a system ofinvariants C p = ±1, one for each prime. He shows that theseinvariants, together with the discriminant (a rational square class),determine the rational equivalence class. This result contains thelocal-global principle (for equivalence, not for representations), butthe term is not yet used.end Minkowski910Helmut Hasse (1898 – 1979)Carl Ludwig Siegel (1896 – 1982)One of the leading german algebraicnumber theorists in the 20th century- introduces Hensel’s p-adic numbersinto the theory of quadraticforms- proves the (strong) local-globalprinciple over number fields- analytic number theory- discrete groups, complex analysis- complete solution of the problemof representation numbers of integralquadratic forms1112

Theorem (Minkowski, Siegel) Es sei L ein positiv definites Gitter derDimension l und M = M 1 ,...,M h ein Repräsentantensystem für einGeschlecht positiv definiter Gitter der Dimension m. Dann gilt für dieDarstellungsanzahlen a(L,M k ) von L durch die verschiedenen M k und dielokalen Darstellungsdichten α p (L,M), p prim, die Beziehung1∑|O(M k )| −1 · ∑kka(L,M k )|O(M k )|=γ(m − l)γ(m)Hierbei sind die Werte γ(n) induktiv definiert durch∏αp (L,M).γ(0) = 1, γ(1) = 1 2 , γ(2) = 1 γ(n − 1), γ(n) = für m ≥ 3,2π n · ρ nwo ρ n das Volumen der n-dimensionalen Einheitskugel ist.Über die analytische Theorie der quadratischen Formen I, II, III, Annals ofMathematics 36 (1935), 527–606, 37 (1936), 230–263, 38 (1937), 212–291Ernst Witt (1911 – 1991)- the founder of the moderntheory of quadratic forms overarbitrary fields- the cancellation theorem- the extension theorem forisometriesTheorie der quadratischen Formen inbeliebigen Körpern, J. reine angew.Math. 176 (1937), 31–44 = Coll. Papers,Ges. Abh. 2–151314Witt was a very original mathematician; he made fundamentalcontributions to diverse of topics: Witt index, Witt group, Wittvectors, to name just three. For instance, in the theory of Liealgebras, in modular forms and in algebraic combinatorics he is citedfor some standard results.In particular, I want to mention Witt’s paperEine Identität zwischen Modulformen zweiten Gerades, Abh.Math. Sem. Univ. Hamburg 14 (1941), 323–337 = Coll. Papers,Ges. Abh. 313–328.It is one of the first contributions to ongoing research on “lattices andmodular forms”. He shows that for ˜D 16 und E 8 ⊥ E 8 , not only theordinary theta series, but also the second degree Siegel theta seriescoincide.We shall come back to this later.end Witt1516

<strong>Martin</strong> Eichler (1912 – 1992)- simple algebras over numberfields- spinor norms, spinor genera- first approximation results- modular forms, theta series<strong>Martin</strong> Kneser very clearly acknowledges the influence of Eichler onhis own <strong>work</strong> in the introduction of the 2001 book version of hislectures on quadratic forms:“Für all dies vergleiche man das einflußreiche WerkQuadratische Formen und orthogonale Gruppen, Springer-Verlag 1952.”He makes precise in what sense the book was influential on himself:1718“Schließlich ein persönliches Wort. Es istziemlich genau 50 Jahre her, daß ich alsjunger Assistent nach Münster kam, baldan Eichlers Seminar teilnahm, wo geradedie neuesten Ergebnisse aus seinem BuchQuadratische Formen und orthogonale Gruppenbesprochen wurden. Da ich im Institutmein Arbeitszimmer mit Eichler teilte, hatteich die besten Möglichkeiten, von einer Seminarsitzungzur nächsten die offen gebliebenenFragen zu klären und so die quadratischenFormen an der Quelle zu studieren.”2 <strong>Martin</strong> Kneser: Quadratic forms andarithmetic of algebraic groups 1955 – 1970(M. Kneser 2001, aus der Einleitung von Quadratische Formen)1920

In the mid 1950s, the theory of algebraic groups and the (arithmetic)theory of quadratic forms were still rather unrelated areas of research. Onthe side of groups, the classification of (semi)simple algebraic groups overalgebraically closed fields was known by <strong>work</strong> of Claude Chevalley. JacquesTits had (essentially) introduced the structures later called buildings whichgive a uniform geometrical interpretaion of all these groups, including theexceptional ones.Already by the end of the 1950s, a completely new area of research hademerged, after Armand Borel had proved his fundamental theorem on theexistence and conjugacy of maximal connected solvable subgroups. Thismade the classification of semisimple groups over arbitrary fieldsaccessible, which was then rather quickly carried out mainly by Borel andTits. They used k-split tori and the relative root system to reduce thequestion essentially to the anisotropic kernel, in analogy with the Wittdecomposition of quadratic forms.Over number fields, this approach embedded the earlier studies of algebraswith involution, hermitian forms, Cayley octaves and Jordan algebras intoa uniform theory. In this situation it was perfectly natural (after <strong>work</strong> ofLang and Tate) to introduce non-abelian Galois cohomology (H 0 ,H 1 ,abelian H 2 ) to treat such classification questions. Jean-Pierre Serre’scours at the Collège de France 1962-63, leading to the famous Lecturenotes No. 5 Cohomologie Galoisienne, demonstrates how quickly the newmethod had been established.The theory of semisimple groups over number fields in turn laid thefoundations for a general treatment of arithmetic subgroups of algebraicgroups, whose fundamentals were developed by Borel and Harish-Chandra.Clearly, many substantial results had been obtained (much) earlier mainlyby Siegel, but the frame<strong>work</strong> had dramatically changed.2122<strong>Martin</strong> Kneser’s vita- 1945-50 studies in Tübingen,Göttingen and BerlinWe now want to look at (part of) Kneser’s <strong>work</strong> as embedded in thisgeneral picture.- 1951-1956 assistant at Münsterand Heidelberg, Habilitation- 1957-58 Univ. of Saarbrücken- 1959-62 Prof at München- 1963–1993 Prof at Göttingen2324

Four fundamental <strong>work</strong>s by <strong>Martin</strong> Kneser:1. Klassenzahlen indefiniter quadratischer Formen,Archiv d. Math. 7 (1956), 323–3322. Klassenzahlen definiter quadratischer Formen,Archiv d. Math. 8 (1957), 241–250.3a. Strong approximation. in: Algebraic groups and discontinuoussubgroups. Proceedings, Boulder Co 1965.3b. Starke Approximation in algebraischen Gruppen.I.J. reine angew. Math. 218 (1965), 190 – 203.4. Galois-Kohomologie halbeinfacher algebraischer Gruppenüber p-adischen Körpern I. and II.Math. Z.88 (1965), 40–47, 89 (1965), 250–272contents of the paper in Archiv d. Math. 1956:- proof of the strong approximation theorem for representationsand for the orthogonal group- the adelic orthogonal group is introduced for the first time- the number of spinor genera in a genus is a group index- generation of orthogonal groups by reflections- computation of local spinor norms2526contents of the paper in Archiv d. Math. 1957:- main idea: if (V,q) is isotropic at p, lattice over Z[1/p] behavelike indefinite lattices- technically: apply strong approximation (from the previous paper)to the set of places S = ∞ ∪ {p}- for any two classes in the same spinor genus, there arerepresentatives L,M s.t. Z[1/p]L = Z[1/p]M.- the resulting “neighbour method” is used to calculate the classnumber of I n up to dimension 14.2.1 Strong approximation and class numbersWe have already talked, without details, about the use of strongapproximation for class numbers and representations of quadraticforms. We now generalize the situation to algebraic groups and givecomplete definitions and statements.Notation:2728

k an algebraic number fieldo the ring of integers of kp,l,v,... places (equivalence classes of valutations of k)k p ,o p the completion of k, resp. o at pp ∈ o p a prime element for p, if p is finiteS kSA = A kthe set of all places of ka finite set of places of kthe ring of adeles of k(a l ) l∈Sk a typical element of A, so a l ∈ o l f.a.a. lA(S) ⊂ A the S-integral ideles, so a l ∈ o l for l /∈ SVGLG(R)G(A k )G(o)G(k) ⊂ G(A k )a finite-dimensional vector space over ka linear algebraic group defined over oa lattice in Vfor any over-ring R ⊇ o the group of R-points in Gin particularthe adele-group of G over kthe stabilizer of L in G(k)diagonally embedded2930Definition 1 The G-class number of a lattice L is the number ofG-classes in the G-genus of L.Definition 2 (Strong Approximation) Let G be an algebraicgroup over k and S be a finite set of places of k. We say that strongapproximation holds for the pair (G,S) if G(k)G(A(S)) is dense inG(A k ).Theorem 1 (Kneser 1965, Platonov 1969)Strong approximation holds for all pairs (G,S), where G is simplyconnected almost k-simple and G(k v ) is not compact for at least onev ∈ S.2.2 Local-global principles and Galois cohomologyDefinition 3 The Hasse principle holds for an algebraic group G overk if the canonical mapH 1 (k,G) → ∏v∈S kH 1 (k v ,G)is injective.The following result is fundamental for the use of this method:Theorem 2 (Kneser 1965) If G is a semisimple simply connectedgroup over a local field k of characteristic 0, then H 1 (k,G) = 0.The proof uses the classification and structure theory of such groups,but also a lot of case-by-case investigations.3132

2.4 Further contributions by M. KneserTo finish this partial overview of Kneser’s <strong>work</strong>, I want to mentionthree doctoral dissertations which were supervised by him. This is mypersonal choice.H.-V. Niemeier, Definite quadratische Formen der Diskriminante 1und Dimension 24, J. Number Theory 5 (1973), 142–178.Jürgen Biermann: Gitter mit kleiner Automorphismengruppe inGeschlechtern von Z-Gittern mit positiv-definiter quadratischer Form,Dissertation Göttingen 1981Yuriko Suwa-Bier Positiv definite quadratische Formen mit gleichenDarstellungsanzahlen, Dissertation Göttingen 1984In constrast to a first impression, this is not a question about thetaseries. It requires a carefully chosen (and eventually computer-based)decomposition of the 12-dimensional cone of pairs of reduced positivedefinite 3 × 3-matrices.The solution of this problem was eventually given in the followingdissertation, supervised by F. Grunewald:Alexander Schiemann: Ternäre positiv definite quadratische Formenmit gleichen Darstellungsanzahlen, Dissertation Bonn 19933738Finally, I want to mention three papers by Kneser himself, which dealwith quadratic forms, but outside the main scope of his <strong>work</strong>s.Zur Theorie der Kristallgitter,Math. Annalen 127, 105–106 (1954)Two remarks on extreme forms, Canadian Journal of Math.7, 145–149 (1955)Lineare Relationen zwischen Darstellungsanzahlen quadratischerFormen, Math. Annalen 168, 31–39 (1967)3 A particular problem: finite quotients ofBruhat-Tits buildingsIn the 1980s: Finite group theorists and geometers <strong>work</strong> on the classification ofcertain classes of (locally) finite incidence geometries belonging to a Coxeterdiagram (and more general diagrams), together with a flag transitiveautomorphism group. The maximal flags are called “chambers”, the term“chamber system” is also common. These geometries locally look like finitebuildings.There is an appropriate covering theory for chamber systems (related to groupamalgamations), and the universal 2-cover under rather general assumptions is abuilding. If the diagram belongs to the known list of affineCoxeter-Dynkin-diagrams and the rank is ≥ 4, then this building is known: it as aBruhat-Tits building.3940

A general reference is the following:ko,p,k p ,o pG ⊂ GL(V )a totally real algebraic number fieldas beforesimply connected semisimple, almost simple over kWilliam M. Kantor: Finite geometries via algebraic affine buildings,pp. 37-44 in: Finite Geometries, Buildings and Related Topics (Eds.W. M. Kantor et al.), Oxford University Press, Oxford 1990Other contributors: Timmesfeld, Stroth, Ronan, Meixner.We maintain the general notation introduced preciously; in particular,we consider the following:anisotropic at the infinite placesp a fixed finite place of k s.t. rk p G ≥ 2¯k p := o/po the residue field at p∆ := ∆(G(k p )) the Bruhat-Tits building of G(k p ).La lattice in V s.t. o p L =: L p defines a vertex of ∆∆ 0∼ = ∆(G(¯kp )) the residue (star, link) of L in ∆.Γ := G(o[ 1 p ]) a {p}-arithmetic discrete subgroup of G(k p)Γ 0 := G(o)the finite stabilizer of L in G(k).4142The following unpublished result grew out of discussions betweenWilliam M. Kantor, <strong>Martin</strong> Kneser and myself in the early 90s.Proposition (M. Kneser, unpublished) Under the aboveassumptions, the following properties of the lattice L (resp. thearithmetic groups Γ, Γ 0 ) are equivalent:• Γ acts chamber transitively on ∆.• (i) Γ 0 acts chamber transitively on ∆ 0 ,(ii) h G (L) = 1.Proof: “=⇒”: (i) is obvious from the assumption, since thechambers of ∆ 0 are exactly the chambers of ∆ containing the vertexL. For (ii), we have to show thatG(A k ) = G(k) · G(A(∞)). (1)Since G is isotropic at p, we can use strong approximation for the setof places ∞ ∪ {p}:G(A k ) = G(k) · G(A(∞ ∪ {p})). (2)Since Γ acts chamber transitively on ∆ it acts also vertex transitivelyon the vertices of a given type, which for “type L” translates asG(k p ) = Γ · G(o p ) = G(o[ 1 p ]) · G(o p). (3)4344

Given an arbitrary idele (σ l ) ∈ G(A k ), first use (2) and write it asσ · (τ l ) with σ ∈ G(k) and τ l ∈ G(o l ) for all l ≠ p. Then use (3) andwrite τ p = γ · δ with γ ∈ G(o[ 1 p ]) and δ ∈ G(o p). Now replace theoriginal decomposition of (σ l ) byσ l = (σγ) · (γ −1 τ l ) for all l.Since p is a unit in all o l ,l ≠ p, we have γ ∈ G(o l ) for all l ≠ p andthus γ −1 τ l is still in G(o l ). Furthermore, γ −1 τ p = δ is in G(o p ) byconstruction. Thus the second factor of the new decomposition is inG(o l ) for all l, and therefore the given idele is a member of the righthand side of (1).“⇐=”: Because of assumption (i), we only have to show thetransitivity of Γ on the vertices of “type L”, that is, the vertices inthe orbit G(k p )L ⊂ ∆. But this transitivity is equivalent to (3), ashas already been used. To prove (3), just apply assumption (1) toideles which are 1 outside p: for any given σ p ∈ G(k p ), there existsσ ∈ G(k) and an idele (τ l ) with τ l ∈ G(o l ) for all l s.t. σ p = σ · τ pand σ · τ l = 1 for all l ≠ p. But this means σ ∈ G(o p ) for all l ≠ p,thus σ ∈ G(o[ 1 ]), as desired.□p4546Notice: This proof is completely analogous to the derivation of theneighbour method from strong approximation. See in particularequation (3) and compare Kneser’s 1957 paper.Consequences: Discrete chamber transitive groups on affinebuildings are very rare. Examples had been found in the <strong>work</strong>s ofKantor and Meixner/Wester in the above-mentioned context.A full classification has been announced in[KLT] W.M. Kantor, R. Liebler, J. Tits, On discrete chamber-transitive automorphism groups of affine buildings,Notices of the AMS Vol. 16, No. 1 (1987).The “generic case” of the suggested proof deals with thenon-existence of such a subgroup for almost all algebraic groups G. Itis briefly sketched in that announcement (see also the survey quotedabove). It uses only condition (i) (or rather the chamber transtivityon residues of all types). This condition is already very restrictive, bya theorem of Gary Seitz. A complete proof of the classification to myknowledge did not appear. A proof of the finiteness result based onthe computation of covolumes of S-arithmetic groups has been givenby Prasad and Borel/Prasad (two subsequent papers in Publ. Math.IHES) already in 1989.The class number condition (ii) could be used to shorten an eventualcomplete proof. For orthogonal groups, one could use the results onthe growth of class numbers of the 1972 dissertation of HorstPfeuffer, again a student of Kneser. For other groups, it is apparentlynecessary to <strong>work</strong> out explicitly the above-mentioned formulas of(Borel and) Prasad on covolumes of arithmetic groups.4748