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On length spectra of lattices [Elektronische Ressource] / von Thomas A. Willging

57 pages
On Length Spectra of LatticesZur Erlangung des akademischen Grades einesDoktors der Naturwissenschaftenvon der Fakulta¨t fu¨r Mathematik desKIT(Karlsruher Institut fu¨r Technologie)genehmigteDissertationvonThomas A. Willgingaus StuttgartTag der mundlichen Prufung: 16. November 2010¨ ¨Referent: PD Dr. Stefan Ku¨hnleinKorreferent: Prof. Dr. Frank HerrlichPrefaceThe theory of quadratic forms is a subject in number theory of the purest sort goingback to Fermat, Euler, Lagrange, Gauß and Minkowski to mention but a few. By aquadratic form Q in dimension n we mean a functionnXQ(x) = a xxij i ji,j=1T nof degree 2 with x := (x ,...,x ) ∈R and coefficients a =a ∈R.1 n ij jiAn old natural question is to ask which integers are represented overZ by a given formwith integer coefficients and moreover in how many ways they are represented. In someinstances these questions can be answered but there is no general answer. The centralidea is the so called local-global principle, that means we first check if an integer isrepresented locally over allZ and overR.pAs is well known, the theoryof positivedefinite quadraticformsprovidesan alternativeapproach for studying lattices. There is a one-to-one correspondence between congru-ence classes of lattices and equivalence classes of quadratic forms. Therefore otherclassical questions as for instance the sphere packing problem or the kissing numberproblem can be considered in this context too.
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On Length Spectra of Lattices
Zur Erlangung des akademischen Grades eines
Doktors der Naturwissenschaften
vonderFakulta¨tfu¨rMathematikdes KIT (KarlsruherInstitutf¨urTechnologie) genehmigte
Dissertation
von
Thomas A. Willging aus Stuttgart
Tagderm¨undlichenPru¨fung:16.November2010 Referent:PDDr.StefanKu¨hnlein Korreferent: Prof. Dr. Frank Herrlich
Preface
The theory of quadratic forms is a subject in number theory of the purest sort going back to Fermat, Euler, Lagrange, Gauß and Minkowski to mention but a few. By a quadratic formQin dimensionnwe mean a function n Q(x) =Xaijxixj ij=1 of degree 2 withx:= (x1  xn)TRnand coefficientsaij=ajiR. An old natural question is to ask which integers are represented overZby a given form with integer coefficients and moreover in how many ways they are represented. In some instances these questions can be answered but there is no general answer. The central idea is the so called local-global principle, that means we first check if an integer is represented locally over allZpand overR. As is well known, the theory of positive definite quadratic forms provides an alternative approach for studying lattices. There is a one-to-one correspondence between congru-ence classes of lattices and equivalence classes of quadratic forms. Therefore other classical questions as for instance the sphere packing problem or the kissing number problem can be considered in this context too. The aim of this thesis is to study a related question, Schmutz Schaller’s conjecture, that in dimensions 2 to 8 the quadratic forms associated with the lattices with the best known sphere packings are maximal. We say respectively that these lattices have maximal lengths. This means that theirklength is strictly greater than the-th k-th length of any other lattice in the same dimension with the same covolume. Here it is important that we do not count the multiplicities of these lengths. After we have introduced the basic concepts in Chapter 1 we will see in Chapter 2 that the Schmutz Schaller conjecture does not hold true for dimension 3. Although the statement holds asymptotically, i.e. ifkis big enough, we will explicitly present a counter-example in Section 2.3. It turns out that only its 6-th length is not dominated by the corresponding length of the lattice with the best sphere packing (Proposition 2.3.5). However, it seems that there is nothing but this exception: one lattice, where for one length the conjecture fails. The results of Section 2.3 are published in [Wi].
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PREFACE
To support the guess that there is only this counter-example we will discuss the con-jecture for ternary lattices with bounded multiplicities. Here the main result is, that the conjecture applies for all these lattices with bounded multiplicities (Theorem 2.5.5). It may be surprising that the multiplicities of the lengths do not play a role at all. This is contrary to the usual definition of the length spectrum. But in contrast to the situation mentioned before the lengths of the complete length spectrum have the same asymptotic behaviour for all lattices with fixed covolume. We will use this fact to prove in Chapter 3 that a lattice with maximal complete lengths does not exist in any dimensionn2. In particular we will prove that for any lattice there exists another lattice, arbitrarily close, such that its complete lengths are not dominated by the complete lengths of the first lattice (Theorem 3.2.4). It is then natural to ask, whether there exist two lattices at all, such that the complete lengths of one lattice dominate the complete lengths of the other. It seems that this is a difficult question in general, but we will answer this question in the negative in the special case of even unimodular lattices in the last section.
At this place I would like to thank all persons who have contributed to this thesis, in particular: PDDr.StefanK¨uhnleinfor his guidance and helpful advice throughout the whole time, Prof. Dr. Frank Herrlichfor his support and for kindly agreeing to be Korreferent of this thesis,Ute LuhmandLothar Lempfor their careful proofreading.
Contents
Preface 1 1 Basic concepts 5 1.1 Lattices and quadratic forms . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Local considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4 Sphere packings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.5 Root lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.6 Asymptotic notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2 Conjecture of Schmutz Schaller 13 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 Asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3 A counter-example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.3.1 The face-centered cubic lattice . . . . . . . . . . . . . . . . . . . . 16 2.3.2 The honeycomb lattice . . . . . . . . . . . . . . . . . . . . . . . . 17 2.3.3 Other counter-examples? . . . . . . . . . . . . . . . . . . . . . . . 20 2.4 Irregular ternary forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.4.2 Ramanujan’s form . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.4.3 The first nontrivial genus . . . . . . . . . . . . . . . . . . . . . . . 22 2.5 Ternaries with bounded multiplicities . . . . . . . . . . . . . . . . . . . . 25 2.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.5.2 Orthogonal lattices . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.5.3 General situation . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.5.4 Higher dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3 Complete length spectrum 39 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.2 Maximal complete lengths . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.2.1 Dimensions 2 to 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.2.2 General situation . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.3 Further questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.3.1 Dominating complete lengths . . . . . . . . . . . . . . . . . . . . 44
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3.3.2 Theta functions and modular forms 3.3.3 Even unimodular lattices . . . . . .
Index
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Chapter 1
Basic concepts
1.1 Lattices and quadratic forms AlatticeΓ in the Euclidean standard spaceRnis a subgroup which is generated by a basisB={b1  bn}. n Γ :=i=P1xibi:xiZn The volume of the fundamental parallelotope FB:={Pαibi: 0αi<1}of Γ is called i=1 thecovolumeof Γ. As is well known, the theory of quadratic forms offers an alternative language for study-ing lattices. There is a one-to-one correspondence between congruence classes of lattices and equivalence classes of quadratic forms. Definition 1.1.1. (a) Let Γ be a lattice inRnwith basisB={b1  bn}and define the matrixMB:= (b1||bn the quadratic form). Then QΓB(x) :=xTMBTMBx, | {z } =:AΓB=:(aij) xZn, isassociated Obviously, the matrixwith Γ.AΓBis symmetric and positive definite. Therefore by a “form” we always mean a positive definite quadratic form. (b) Furthermore we call a formQintegralifQ(x)Zfor allxZnand we callQ classically integralifaijZfor alli j∈ {1  n}. (c) LetI quadratic forms Twobe a ring in a field.QandQwith matricesAand Aare called(I-)equivalentor in the sameclassif there exists a matrixSIn×n with det(S)Isuch thatA=STAS. ForI=Ztwo forms are equivalent, if they refer to the same lattice with different bases.
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CHAPTER 1. BASIC CONCEPTS
To put a finer point on that, we exploit that the set of lattices inRnis GLn(Z)\GLn(R) and the set of real quadratic forms is GLn(R)SOn(R). We can thus identify the set of congruence classes of lattices respectively ofZ-equivalence classes of quadratic forms with GLn(Z)\GLn(R)SOn(R)According to this principle we call a lattice Γ(classically) integralif its associated form QΓis (classically) integral,evenifQΓ(x)2Zfor allxZnandarithmeticifQΓis arithmetic, i.e. there exists aλRsuch thatλQΓis integral. As well we define det(QΓ) := det(AΓ) = (cov(Γ))2. In the sequel we will switch freely between the language of lattices and the language of forms.
1.2 Reduction By “reduction” we mean choosing a representative of a class with nice properties, i.e. whose coefficients satisfy certain inequalities, depending on the reduction. In this sense reduction for lattices consists in finding bases so that the scalar products of the elements satisfy these inequalities. Definition 1.2.1.A positive definite formQwith matrixA= (aij) is said to bereduced (in the sense of Minkowski) if fork= 1  n akkQ(x) for all integral vectorsx:= (x1  xn)Twith gcd(xk  xn) = 1and if in additiona1j0 forj= 2  n In every class there exists a reduced form. More precisely we have, cf. [Ca, p.256]: Theorem 1.2.2.Every positive definite form is equivalent to at least one and at most finitely many reduced forms. It is clear that the reduction conditions imply that a11a22annAn additional consequence is |2aij| ≤aii(1i < jn)We remark the fact that ifn3 these consequences are not sufficient for a form to be reduced. However, these conditions will do all we need. For an associated lattice with basis{b1  bn}it follows that |cos((bi bj))|=||bhib|ib|jbij|||b21i||bi||b2i|12(=i6=j)Hence 21cos((bi bj))21respectively 0cos((b1 bj))12
1.3. LOCAL CONSIDERATIONS
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1.3 Local considerations An old problem in the theory of quadratic forms is the question of representing integers by a positive definite integral form. To that end we will use the so calledlocal-global principle. That means we will first check if an integermisrepresentedbyQlocally, i.e. whetherQ(x) =mwithxZpnis solvable at all placesp, wherepis either a prime number (forp-power congruences) or(for the sign, with the usual convention that Z=Rvery satisfactory local-global theory of forms over the ). UnfortunatelyQdoes not hold overZ two forms are. WhileQ-equivalent if and only if they areQp-equivalent for allp(Weak Hasse Principle), there exist forms which areZp-equivalent for allpbut notZ we define: Therefore in Section 2.4. examples-equivalent, cf. Definition 1.3.1.Thegenusgen(Q) of an integral formQis the set of all (integral) forms that arelocally equivalenttoQ, i.e.Zp-equivalent at all placesp. Obviously gen(Q we get the local-global connection: Nowis a disjoint union of classes.) Theorem 1.3.2.LetQbe an integral form which represents an integermlocally. Then mis represented by one form ingen(Q). A proof can be found for instance in [Ca, Chap.9.5] or [Kn, Satz 22.1]. We make note of the trivial consequence: Corollary 1.3.3.If the genus ofQhas only one class, thenmis represented locally by Qif and only ifmis represented byQ. Hence if an integral formQhas only one class in the genus, all numbers that are not represented byQcan be described by congruence conditions. From Definition 1.3.1 we see that for locally equivalent formsQandQthe rational numberd(etd(etQQ))is ap-adic unit for allp, so it must be±are assumed to be positive definite (or all forms 1. Since withp=) it follows that det(Q) = det(Q). Furthermore it is well known that if a prime numberpdoes not divide 2det(Q) there is only oneZpof forms of the same determinant (for simplicity we-equivalence class assume all forms to be classically integral): Theorem 1.3.4.LetQ,Qbe two classically integral forms such thatdet(Q) = det(Q) is ap-adic unit for an odd primep. ThenQandQareZp-equivalent. Sketch of proof.One can show (cf. [Ca, p.116]) thatQisZp-equivalent to a form n Xbijxixjij=1 where|bii|p≤ |b11|pand (sincep6= 2)|bij|p≤ |b11|p. Thereforebb1ij1Zpfor allbijand we can complete the square. SoQisZp-equivalent to a form b11x12+H(x2  xn)
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CHAPTER 1. BASIC CONCEPTS
for some (n1)-dimensional formH. By induction it follows thatQisZp-equivalent to a diagonal formb11x21++bnnxn2where thebiiarep-adic units sincepdet(Q). And such a form is integrally equivalent to x2+x22++xn21+ det(Q)x2n1 Hence there are only finitely many “critical”p this place we will state Atto consider. (and also prove) a further consequence that will be helpful later on: Proposition 1.3.5.LetQbe ann-dimensional classically integral form,n3andp an odd prime not dividingdet(Q). ThenQrepresents all natural numbersmoverZp. This proposition can be verified with Hensel’s Lemma in its simplest variant: Lemma 1.3.6(Hensel).Letf(x)a polynomial in the single variablebe xand suppose that there exists anx0Zpsuch that |f(x0)|p<|f(x0)|p2wheref(x)denotes the (formal) derivative with respect tox there is a. ThenyZp such thatf(y) = 0. For a proof one can refer to [Ca, p.47]. Corollary 1.3.7.Letpbe an odd prime and define the integral formQ(x) :=a1x21+a2x22 such thata1 a26≡0 (modp) for all positive integers. Thenb6≡0 (modp)there exists a yZp2such thata1y21+a2y22=b. Proof.Qis universal inFp represents all it, i.e.binFp, since we have #{a1x21|x1Fp}= #{ba2x22|x2Fp}=p+12 ⇒ {a1x21|x1Fp} ∩ {ba2x22|x2Fp} 6=˜ ˜Fp:a1˜x21=ba2˜x22x1 x2Let nowb6≡0 (modp) and without loss of generalityx˜16≡0 (modp), then withf(x) := a1x2(ba2˜x22) it is |fx1)|pp1<|f(x˜1)|p2=|2a1x˜1|p2= 1Due to Hensel’s Lemma there exists ay1Zpsuch thata1y12+a2x˜2=b. 2 Proof of Proposition 1.3.5.Theorem 1.3.4 one can assume thatAs in the proof of Qis of the form Q(x) =x21++xn21+ det(Q)x2nDue to Corollary 1.3.7 the formx12+x22represents allb≡ −det(Q) (modp) inZp. Hence for allmthere exists anxZnpsuch thatQ(x) =m.To treat the “critical” prime numbersplater on we need one more fact:
1.3. LOCAL CONSIDERATIONS
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Proposition 1.3.8.LetQbe ann-dimensional classically integral form,n2andp a prime. Then Q represents a numbermoverZpif and only ifQrepresentsmmodulo pr+1, wherepris the highest power ofpdividing4m. Proof.Letu=r(ifpis odd) oru=r2 (ifp= 2), wherepuis the highest power of pdividingmand letxZnsuch that xTA x=m+pr+1l wherepl. Clearlyxr+1:=xis a representation inZpnofmmodulopr+1 we will. Now use an induction argument and define the vector xr+2:=xr+112pr+1xr+1(xTr+1A xr+1)1l Sincexr+1Znpandr+ 1> uit follows thatxr+2Znp we have (withtoo. Then T xr+1A xr+1=m+pr+1l) xTr+2A xr+2=m+pr+1l21pr+1l12pr+1l+14p2(r+1)l2(xrT+1A xr+1)1 =m+41p2(r+1)l2(xTr+1A xr+1)1Hence for somep-adic unitlwe get xT+2A xr+2=m41+p2(r+1)ul=mm+pr+2l4 p6= 2 r+pr+2l p= 2Obviously the sequencexr+sconverges to a solution inZnp. A major quantitative result along these lines was given by Siegel [Si] who was the first one to discover the concrete connection between local and global representability. For a detailed historical survey see [CoSl1]. His result allows us to think of our local in-formation as a weighted average of information over the classes in the genus, cf. [Ca, Chap.9.6], [Ha, p.4]. We shall not have occasion to use it later, but it is hard to resist the temptation to praise Siegel’s theorem at this place. In a more general way we look at the representability of forms by forms, i.e. a formM with matrixBinnvariables is represented by ann-dimensional formQwith matrixA, n >1 andnn, if there exists anXZn×nsuch thatXTAX=B we. Furthermore denote bybQ(M) the number of representations ofMbyQand bybq(M) the number of representations ofMbyQmoduloq=pa. Ifpbis the highest power ofpdividing (2det(M))2then the number βp(M) :=q21nq(n2(n+1))12nnnbq(bqM)(M)nn>=nnis independent ofafor alla > b call these numbers the We Hilfsatz 13]. [Si,, cf.local representation densitiesofM. Now we are able to state Siegel’s result [Si, Hauptsatz]: