Strong approximation and Hasse principle for integral quadratic forms over affine curves

A fundamental result in the arithmetic of integral quadratic forms is the strong approximation theorem for the spin group Spin of a non-degenerate inde®nite quadratic form in three or more variables. (In practice, one invokes the corresponding version for the kernel O0 of the spinor norm function.) This result is due to the works of Eichler and Kneser [Ei], [Kn] in the 1950s, and has ever since played a central role in the proofs of a number of important theorems for both inde®nite as well as de®nite quadratic forms. On the other hand, it is well-known that such a strong approximation cannot possibly hold for i† the group SO of a inde®nite quadratic form as it is not simply connected, and ii† for the group Spin of a de®nite quadratic form as Spin1 is compact. The purpose of this paper is to show that there exists a version of an ``almost'' strong approximation theorem for de®nite quadratic forms, one forOm and one forO ‡ m m 3† and show that they too have serious applications in the arithmetic theory; e.g., in the asymptotic distributions of classes in a genus and in the representation theory of de®nite forms. It is anticipated that such almost strong approximations would also extend to other classical groups and will be treated in a future article. We give a brief outline of the contents of this paper. Terminology and notations are generally those from [ OM]. For simplicity as well as convenience we shall consider here only positive de®nite integral Z-lattices. A key step in the proof of the strong approximation theorem for O0 is the following result on inde®nite representations with approximation property: Invent. math. 129, 471±487 (1997)

For each positive integer n n , let g Z ( n ) g_\mathbb {Z}(n) be the smallest integer such that if an integral quadratic form in n n variables can be written as a sum of squares of integral linear forms, then it can be written as a sum of g Z ( n ) g_\mathbb {Z}(n) squares of integral linear forms. We show that every positive definite integral quadratic form is equivalent to what we call a balanced Hermite–Korkin–Zolotarev-reduced form and use it to show that the growth of g Z ( n ) g_\mathbb {Z}(n) is at most an exponential of n \sqrt {n} . Our result improves the best known upper bound on g Z ( n ) g_\mathbb {Z}(n) which is on the order of an exponential of n n . We also define an analogous number g O ∗ ( n ) g_{\mathcal O}^*(n) for writing Hermitian forms over the ring of integers O \mathcal O of an imaginary quadratic field as sums of norms of integral linear forms, and when the class number of the imaginary quadratic field is 1 1 , we show that the growth of g O ∗ ( n ) g_{\mathcal O}^*(n) is at most an exponential of n \sqrt {n} . We also improve on results of both Conway and Sloane and Kim and Oh on s s -integrable lattices.

On almost strong approximation for algebraic groups

An integer may be represented by a quadratic form over each ring of p -adic integers and over the reals without being represented by this quadratic form over the integers. More generally, such failure of a local-global principle may occur for the representation of one integral quadratic form by another integral quadratic form. We show that many such examples may be accounted for by a Brauer–Manin obstruction for the existence of integral points on schemes defined over the integers. For several types of homogeneous spaces of linear algebraic groups, this obstruction is shown to be the only obstruction to the existence of integral points.

A treatment by integral matrices is given for composition of pairs of integral quadratic forms with the same discriminant. The forms are associated with a pair of similar 2 × 2 matrices A B with irrational eigen values which generate the maximal order. The most general integral similarity between A B is given by a matrix whose entries are linear forms in two indeterminates with integral coefficients. This matrix is a compositum of two factors of the same nature. By equating determinants a composition of two quadratic forms results. The method can be generalized to n × n matrices.

An integral quadratic form is called strictly n-regular if it primitively represents all quadratic forms in n variables that are primitively represented by its genus. For any $$n \ge 2$$ , it will be shown that there are only finitely many similarity classes of positive definite strictly n-regular integral quadratic forms in $$n + 4$$ variables. This extends the recent finiteness results for strictly regular quaternary quadratic forms by Earnest et al. (J Number Theory 144: 256–266, 2014).

There are close relations between Bernoulli numbers and quadratic fields or quadratic forms. In order to explain those, we give a survey of the ideal theory of quadratic fields and quadratic forms. Since Gauss , it is well known that there is a deep relation between the ideal theory of quadratic fields (i.e. quadratic extensions of the rational number field) and integral quadratic forms. This is obvious for specialists, but textbooks which explain this in detail are rare. In most textbooks, they treat the correspondence of ideals only in the case of maximal orders (the ring of all integers) for which the description is simple and easy. This is very disappointing. We cannot see the whole picture of the theory of quadratic forms by such a restricted treatment and sometimes it causes misunderstanding. We need the full description of this kind of relation when we explain later the relation between L-functions of prehomogeneous vector spaces and the Bernoulli numbers. So it would be a good chance to try to explain the full relation here.

Small representations of integers by integral quadratic forms

SynopsisA formula is given for the number of genera of primitive integral binary quadratic forms of discriminant D which lie in a rational equivalence class. In particular, necessary and sufficient conditions for the genus and rational equivalence class to coincide are given in terms of the prime factorisation of D.

SynopsisWe compute the number of ℚ-classes of primitive integral binary quadratic forms of any discriminant and characterise when it coincides with the number of genera.

Root systems and inflations of non-negative quasi-Cartan matrices

Combinatorial Completions

We show that the theorem of Ellenberg and Venkatesh on representation of integral quadratic forms by integral positive definite quadratic forms is valid under weaker conditions on the represented form.

The aim of this work is to determine what free energy functionals are expressible as quadratic forms of the state functional \(I^t\) which is discussed in earlier papers. The single integral form is shown to include the functional \(\psi _F\) proposed a few years ago, and also a further category of functionals which are easily described but more complicated to construct. These latter examples exist only for certain types of materials. The double integral case is examined in detail, against the background of a new systematic approach developed recently for double integral quadratic forms in terms of strain history, which was used to uncover new free energy functionals. However, while, in principle, the same method should apply to free energies which can be given by quadratic forms in terms of \(I^t\), it emerges that this requirement is very restrictive; indeed, only the minimum free energy can be expressed in such a manner.

We investigate the existence of a weak solution u ∈ W 1,p 0 (Ω) to the degenerate quasi-linear Dirichlet boundary value problem (P) -Δ p u = λ 1 |u| p-2 u + f(x) in Ω; u = 0 on ∂Ω. It is assumed that 1 < p < ∞, p ¬= 2, Ω is a bounded domain in R N , f ∈ L∞ (Q) is a given function, and the number λ 1 stands for the first (smallest) eigenvalue of the positive p-Laplacian -Ap, where Δ p u ≡ div(|⊇u| p-2 ⊇u). The eigenvalue λ 1 being simple, let φ 1 denote the eigenfunction associated with λ 1 . We show the existence of a solution for problem (P) when f satisfies the orthogonality condition ⊃ Ω fφ 1 dx = 0, in which case the set of solutions is bounded in C 1 (Ω) provided p ¬= 2 and f ≢ 0 in Ω. A key role in our proofs is played by a second-order Taylor formula in its integral form (near φ 1 ) which contains certain Gâteaux derivatives and a positive semidefinite quadratic form. This quadratic form compensates for lack of coercivity in the energy functional corresponding to problem (P). When combined with well-known regularity results, its positive semidefiniteness renders a priori estimates from which existence and boundedness (for p ¬= 2) follow.