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44
ЧЕБЫШЕВСКИЙ СБОРНИК
Том 16. Выпуск 4.
УДК 512
КВАДРАТИЧНЫЕ ФОРМЫ,
АЛГЕБРАИЧЕСКИЕ ГРУППЫ И ТЕОРИЯ ЧИСЕЛ
Н. М. Глазунов (г. Киев)
Аннотация
Целью статьи является обзор некоторых важных результатов в теории квадратичных форм и алгебраических групп, которые оказали и оказывают влияние на развитие теории чисел. Статья ориентирована на избранные задачи и не является исчерпывающей. Представлены математические структуры, методы и результаты, в том числе и новые, связанные в той или иной степени с исследованиями В. П. Платонова.
Содержание статьи следующее. Во введении обращено внимание на классические исследования Коркина, Золотарева и Вороного по теории экстремальных форм и напоминаются соответствующие определения. В разделе "Квадратичные формы и решетки" представлены необходимые определения, результаты о решетках и квадратичных формах над полем вещественных чисел и над кольцом целых рациональных чисел. Раздел 3 "Алгебраические группы" содержит представление классов решеток в вещественных пространствах как факторов алгебраических групп, а также вариант критерия Малера компактности таких факторов. Приведен результат о компактности факторов ортогональных групп квадратичных форм, не представляющих рационально нуля, а также определения и понятия, связанные с кватернионными алгебрами над рациональными числами. Приведенные результаты явно или неявно используются в работах В. П. Платонова, а также в разделах 4 и 5. Раздел 4 "Точки Хигнера и их обобщения" содержит краткий обзор новых исследований в направлении нахождения точек Хигнера и их обобщений. В разделе 5 кратко представлены некоторые новые исследования и результаты по принципу Хассе для алгебраических групп. Для чтения статьи может быть полезным знакомство со статьей автора, опубликованной в 3-м выпуске "Чебышевского сборника" за 2015 год.
Я глубоко признателен Н. М. Добровольскому за помощь и поддержку в процессе подготовки статьи к печати.
Ключевые слова: положительно определенная квадратичная форма; тело над полем рациональных чисел; конечномерная алгебра; принцип Хассе; жесткость; точка Хигнера.
Библиография: 31 названий.
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N. M. GLAZUNOV
QUADRATIC FORMS, ALGEBRAIC GROUPS AND NUMBER THEORY
N. M. Glazunov (Kiev)
Abstract
The aim of the article is an overview of some important results in the theory of quadratic forms, and algebraic groups, and which had an impact on the development of the theory of numbers. The article focuses on selected tasks and is not exhaustive. A mathematical structures, methods and results, including the new ones, related in some extent with research of V.P. Platonov. The content of the article is following. In the introduction drawn attention to the classic researches of Korkin, Zolotarev and Voronoi on the theory of extreme forms and recall the relevant definitions. In section 2 "Quadratic forms and lattices"presented the necessary definitions, the results of the lattices and quadratic forms over the field of real numbers and over the ring of rational integers. Section 3 "Algebraic groups"contains a representation of the class of lattices in a real space as factors of algebraic groups, as well as the version of Mahler’s compactness criterion of such factors. Bringing the results of the compactness of factors of orthogonal groups of quadratic forms which do not represent zero rationally, and the definitions and concepts related to the quaternion algebras over rational numbers. These results explicitly or implicitly are used in the works of V. P. Platonov and in sections 4 and 5. Section 4 "Heegner points and their generalizations" provides an overview of new research in the direction of finding Heegner points and their generalizations. Section 5 summarizes some new research and results on the Hasse principle for algebraic groups. For the reading of the article may be a useful another article which has published by the author in the Chebyshevsky sbor-nik, vol. 16, no. 3, in 2015.
I am deeply grateful to N. M. Dobrovolskii for help and support under the preparation of the article for publication.
Keywords: positive definite quadratic form; finite-dimensional associative division algebra over rationals; Hasse principle; rigidity; Heegner point.
Bibliography: 31 titles.
1. Introduction
Korkin and Zolotarev have investigated positively defined quadratic forms [1, 2] and proved [3] that every extreme form is perfect. Voronoi [4] has proved that every extreme form is eutactic and that perfect eutactic form extreme. Recall some definitions. We follow [1, 2, 3, 4, 5, 6, 7, 10] (and references therein).
Let K be a field of characteristic not 2. An n—ary quadratic form over K is the expression of the form
n
f (ж) = f (Ж1, ...,Xn)=Y^ xixj, (1)
i,j=1
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for aij E K, aij = aji. The n x n symmetric matrix A = (aij) is called the matrix of the quadratic form f. The matrix and the row vector x = (x\,... ,xn) give f (x) = = xAxtr (tr denotes the transposition). The determinant d = d(f) = det(f) of the quadratic form f is called the determinant of the symmetric matrix A determined by f. Two quadratic forms f and g are called equivalent if there exists a nonsingular homogeneous linear change of variable i.e. if there exists C E GLn(K) such that g(x) = f (xC). Or in matrix notations B = CACtr and d(g) = d(f)det(C)2.
Let f (x) be a positive defined quadratic form in n variables. For real e, set
m(f; e) = min f (x + e),
X
where minimum being taken over integer x,
m(f) = max m(f; e),
6
M) = m(f )/d1/n.
It is said that f is extreme if n(f) is a local minimum.
Blichfeld have investigated minimum values of quadratic forms and packing of sphere. Recall that by Blichfeld [8] the spheres of the packing cannot be displaced without increasing the total volume occurred by them. Extreme forms correspond to packings that are rigid. A positive quadratic form corresponding to a densest lattice packing is called an absolutely extreme.
It is said that two n— ary forms f and g with matrices A and B are properly or improperly integrally equivalent if their matrices are related by
B = MAMtr
for some M with integer entries and determinant +1 or —1 respectively. We will consider mainly rings Z and Zp.
The norm of the vector x = (x\,... ,xn) is denoted by N (x) = (x, x) = x\+ +... + x2n.
The aim of the article is an overview of some important results in the theory of quadratic forms and algebraic groups, and which had an impact on the development of the theory of numbers. The article focuses on selected tasks and is not exhaustive. A mathematical structures, methods and results, including the new ones, related in some extent with research of V. P. Platonov [12, 13, 14, 11].
Further contents of the article the following.
In section 2 "Quadratic forms and lattices" presented the necessary definitions, the results of the lattices and quadratic forms over the field of real numbers and over the ring of rational integers [15, 16]. Section 3 "Algebraic groups" contains a representation of the class of lattices in a real space as factors of algebraic groups, as well as the version of Mahler’s compactness criterion of such factors. Bringing the results of the compactness of factors of orthogonal groups of quadratic forms which
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N. M. GLAZUNOV
do not represent zero rationally, and the definitions and concepts related to the quaternion algebras over rational numbers. These results explicitly or implicitly are used in the works of V. P. Platonov and in sections 4 and 5. Section 4 "Heegner points and their generalizations" provides an overview of new research in the direction of finding Heegner points and their generalizations. Section 5 summarizes some new research and results on the Hasse principle for algebraic groups. For the reading of the article may be a useful another article which has published by the author[31] in the Chebyshevskii sbornik in 2015.
2. Quadratic forms and lattices
2.1. Quadratic forms over fields and quadratic spaces
If C = (ci,., cn), Ci = (Cli,..., Cni) then bj = f (ci, Cj), where f (u, v) = uAvtr is the bilinear form that corresponds to f.
Let Kn be the linear space of the dimension n2 over K with coordinate (all,..., ann). The moduli space MF(n, K) of quadratic forms (1) is defined in Kn by the system of equations
aij - aji = 0, 1 ^ ^ n.
The dimension of MF(n,K) is equal to n(n2+1). Operation of the group GLn(K) on forms (1) defines an operation on MF(n,K).
To any quadratic form f = f (x) = xAxtr and V = Kn it is possible to associate a map xAxtr : V ^ K and the foregoing bilinear form ф = f (u, v). The space (V, ф) is called the quadratic space [9]. If (V, ф), (V ,ф ) are quadratic spaces then it is said that they are isometric if there exists a linear isomorphism a : V ^ V such that ф (a(u),a(u)) = ф(и, u) for all u G V. There exists a one-to-one correspondence between equivalence classes of n-ary quadratic forms over K and isometry classes of n-dimensional quadratic spaces over K.
2.2. Lattices and quadratic forms over reals
Let
f (x) = f (XU..
i xn) ^ aijxixj i
i,j=1
be a real quadratic form.
Proposition 1. Let f = f (x) be a real positively defined (non degenerate) quadratic form and S = {s} not empty set of integer points (i.e. points with integer coordinates). Then the function f (x) reaches a minimum on the set S. There is point s0 G S with a condition f (s) ^ f (s0) for all s G S.
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PROOF. For the proof of the lemma 1 it is enough to present f as the sum of squares:
f (x)
n / n \
УУ (cij Xj)
i=i \i=1 /
CijXj \ , det(cij) = 0
2
and to notice, that from the boundness of f (x) follows the boundness of \ cijXj \ , i = 1,... ,n. From this as det(cij) = 0 follows boundness of xi,..., xn. The next statements are consequences of discreteness of a lattice.
Corollary. Let {a} be a non empty set of vectors of a lattice Л. Then there is a vector a0 G {a} of the least length.
Let Л be a lattice of points of n-dimensional Euclidean space; ai,... , ak linear independent vectors of the lattice Л (k ^ n). The set of vectors ai,..., ak is called a primitive system of vectors of the lattice if from
aiai + • • • + akak G л,
where ai,... ,ak are real numbers, follows that ai,... ,ak are integers. In particular for k =1 the vector forms primitive system in only case when it is primitive, i.e. it cannot be presented as a = aa , a G Л, a > 1.
Proposition 2. [7] Let Л be a lattice of points of n— dimensional Euclidean space; k ^ n; ai,..., ak linearly independent vectors of the lattice Л. It is necessary and sufficient that ai,..., ak made primitive system of vectors of a lattice Л so, that ai,..., ak was possible to add to a basis of the lattice Л.
For a lattice Л = Л [ab ..., ak] , k ^ n, the set \Л\ = v(^fJi xiai, 0 ^ xi ^ 1) is called the fundamental parallelepiped of the lattice Л of the volume \Л\.
Let Л = Л [ab ..., an] be a full lattice of n—dimensional Euclidean space, Д(Л) the determinant of the lattice. Correspond to the lattice Л with bases [ai,..., an] the positive quadratic form f (x) in n variables with real coefficients:
n
f (x) = f (xi, ...,xn) = (aixi + ... + anxn)2 = У^ (ai, aj )xixj.
i,j=i
Conversaly for a positive definite quadratic form f in n variables with real coefficients it is possible correspond a lattice Л such that Д(Л) = \[d, where d is the determinant of f.
Forms which are equivalent under a integer unimodular transform is united into a single class
2.3. Integral lattices and quadratic forms
A lattice or a quadratic form is called integral if the inner product of any two lattice vectors is an integer, or if the Gram matrix has integer entries.
Any n—dimensional lattice Л = Л [ai,..., an] has a dual lattice Л* given by
Л* = {x G Rn : (x, u) G Z for all u G Л} .
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3. Algebraic groups
For algebraic groups GLn(R) and GLn(Z) the factorgroup
GLn(R)/GLn (Z)
is the set of all n-dimensional lattices in Rn. Let
L C GLn(R)/GLn(Z) be a subset of the set of lattices.
3.1. Compactness criteria
Recall that there is the function
F : GLn(R) ^ R+, F(g) = |det g\, |det g\ : GL.n(Z) 1.
3.1.1. Fundamental parallelepipeds of a lattice and its sublattice
If (Л : Л) = k then |Л'| = k |Л|.
3.1.2. Hadamard inequality
If Л = Л [ai,..., an] , then |Л| ^ |ai| • • • |an|.
3.1.3. Minkowski lemma
Let T be a convex central symmetric body and Л be a full lattice in real space Rn, v(T) > 2n |Л|. Then ЛП T = 0.
3.1.4. Compactness criteria
Recall the definition.
Definition A lattice Л c L is relatively compact if
1) |Л| <b, Л CL;
2) |x| > a, x E Л C L, x = 0.
Below we give a variant of Mahler’s compactness theorem:
Theorem 1. Let lattices from the set L are relatively compact. Then each lattice Л [ab ..., an] that satisfies conditions of the definition 3.1.4 has the bases such that |aj| < C(a,b), i = 1.... ,n.
Sketch of the proof. The proof is by induction on n with applications of Minkowski lemma, Hadamard inequality and fundamental parallelepipeds of lattices and their sublattices as well as their volumes. Let n =1. By denote. Consider n—ball of the radius r > 2(b/Vn, )1/n, where Vn is the volume of the unit n—ball.
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By Minkowski lemma there is an element a = 0 of the lattice, a! G Л, a = m G Z, |ai| < C, a^Q Л = [ai].
By the induction hypothesis there are a1,, an-1 G Л such that |a*| < C and (a1Q + ... + an-1Q^>| Л = [a1,..., an-1]. In the inductive step Minkowski lemma and Hadamard inequality are used. We omit the proof.
Proposition 3. Let f (x) be the quadratic form which do not represents zero rationally, G = O(f) the orthogonal group of the quadratic form. Then
G(R)/G(Z)
is compact.
3.1.5. Finite-dimensional associative division algebras over rationals
Let D = Qa1 + • • • + Qan be finite-dimensional associative division algebra over rationals. For any u G D put Tu(x) = ux, x G D. We have TUlU2 = TU1 Tu2. Put det Tu = N (u).
It is well known that N(uu2) = N(uQN(u2), and N(u) = 0 if and only if u = 0.
Lemma 1. Suppose that GD is the set of all d G D such that N(d) = 1. Then GD = G is the group.
Proof. For x = x1a1 + • • • + xnan we can write the group in the space of dimension n +1 in coordinates (x-]^,..., xn, t) as N(x1a1 + • • • + xnan)t = 1. Denote the group by Gd. When N(d) = 1 we obtain the group GD.
Proposition 4. Put G = GD. Then
G(R)/G(Z)
is compact.
3.1.6. Quaternion algebras over rationals
Let B = {u = x + yi + zj + tk, x,y, z,t G Q} be the quaternion algebra over Q with relations i2 = a, j2 = b,a,b G Q, ij = k, ji = —k.
Denote by u* the conjugate quaternion. Then n(u) = uu* is by definition the reduced norm of u. It is easy to see that N(u) = (x2 — ay2 — bz2 + abt2)2 = n(u)2.
3.2. Algebraic groups and critical lattices
Extreme lattices and Mahler’s compactness theorem are connected with critical lattices [7, 29]. In this subsection we recall the connection between critical lattices and algebraic groups. In the two-dimensional real case after complexification of the space and critical lattices in it we obtain the interesting class of complete onedimensional projective algebraic groups. These are elliptic curves which correspond to the critical lattices [28, 29].
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4. Heegner points and their generalizations
Heegner points on elliptic curves have defined by Birch and Heegner-like points have defined by several authors (see [18] and references therein). There are open problems in the theory of Heegner-Stark (Darmon) and Darmon-like points in elliptic curves E over Q of conductor N with a prime p such that N = pDM, where D is the product of even (possible zero) distinct primes and (D, M) = 1, namely:
Suppose K be a real quadratic field in which all prime dividing M are split, and all primes dividing pD are inert and let Hp = Kp \ Qp be the Kp- points on the p— adic upper half plane.
Then in the case D =1 there is the conjecture by H. Darmon [19] that local points PT E E(Kp) associated to elements т E K f) Hp defined as certain Coleman integrals of a modular form attached to E to be rational over specific ring class field of K, and to behave in many aspects as the classical Heegner points arising from imaginary quadratic fields;
in the case D > 1 there is a construction of Darmon-like points on E(Kp), by means of certain p—adic integrals related to modular forms on quaternion division algebras of discriminant D. Greenberg conjectured that these points behave in many aspects as Heegner points and, in particular, that they are rational over ring class of K.
The authors of the paper [20] developed the (co)homological techniques for effective construction of the quaternionic Darmon points on E(Kp).
They use the reinterpretation of Darmon's theory of modular symbols and mixed period integrals by M. Greenberg [21]and relate p—adic integration to certain overconvergent cohomology classes in order to derive an efficient algorithm for the computation of quaternionic Darmon points.
In the Introduction authors describe the contents of the paper under review and fixes notation.
The second Section is devoted to preliminaries on Hecke operators, the Bruhat-Tits tree, and measures.
In Section 3 Greenberg’s construction of quaternionic p—adic Darmon points is dealt with. The main results of the paper [20] are presented in Sections 4-6.
In Section 4 the explicit algorithms that allow for the effective calculation of the quaternionic p—adic Darmon points are presented (Theorem 4.1, Theorem 4.2). Theorems and their proofs describe the algorithms and contain a correctness proof of the algorithms.
Section 5 treats the integration pairing via overconvergent cohomology. Authors of the paper [20] reduce the problem of computing integrals under consideration to that of computing moments and give an algorithm for computing the moments by means of the overconvergent cohomology lifting techniques by D. Pollack and
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R. Pollack [22]. Finally last section treats results of extensive calculations and numerical evidence obtained by the authors of the paper [20] in support of the conjectured rationality of Greenberg’s Darmon points.
5. Hasse principle
Let M be a class of algebraic varieties over algebraic number field F. The Hasse principle by[16, 17] for the class M is the next assertion: Suppose that V £ M and V is nonempty for all places v of the field F. Then V(F) is nonempty. For algebraic number field there exists an exact sequence 0 ^ Br(F) ^ 0v Br(Fv) ^ Q/Z ^ 0, where Br(F) is the Brauer group of F. The injectivity of Br(F) ^ 0v Br(Fv) is called the Hasse principle for central simple algebra over F.
Let denotes the set of (normalized) discrete valuations of rank 1 over a field K.
Yong in his paper [17] proves the conjecture by Colliot-Thelene, Parimala, Suresh [24] on the triviality of the natural map Hl(K,G) ^ Пv&nK Hl(Kv ,G) (Hasse principle with respect to for G-torsors over K) for groups of some classical types, where K is a field and G is a smooth connected linear algebraic group over K.
The main results of the paper [23] are two theorems. Theorem 1.6: Let K be a function field of p-adic arithmetic surface and G a semisimple simply connected group over K. Assume p = 2 if G contains an almost simple factor of type 2ЛП of even index. If every almost simple factor of G is of type 1ЛП,2 ЛП, Bn, C*, D*, Fred or G2, then the natural map Hl(K, G) ^ ПvenK H 1(Kv, G) has a trivial kernel.
Some results in the direction has also been proven by Preeti [25].
In Theorem 1.7 the author of [23] obtains (with applications of the Rost invariant) sufficient as well as necessary and sufficient conditions for the Hasse principle with respect to QK when the field K is a function field of a local henselian surface with finite residue field of characteristic p.
The important tools in the proofs are theory of involutions and hermitian forms over central simple algebras, Witt groups, exact sequence of Parimala, Sridharan and Suresh by Bayer-Fluckiger and Parimala [26], invariants of hermitian forms, norm principle for spinor norms by Merkurjev [27].
6. Conclusion
Classical and novel results on quadratic forms and algebraic groups which have influenced the development of the theory of numbers are presented.
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REFERENCES
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2. Korkin A., Zolotarev G. 1873, “Sur les formes quadratiques”, Math. Ann., vol. 6, 366-389.
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National Aviation University.
Received 9.11.2015.
