Rational equivalence of primitive integral binary 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.

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.

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.

A characterization is given for the integral binary quadratic forms for which the set of represented values is closed under products. It is also proved that for an integral binary quadratic lattice over a Dedekind domain, the product of three values represented by the form is again a value represented by the form. This generalizes the trigroup property observed by V. Arnold in the case of integral binary quadratic forms.

Combinatorial Completions

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.

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

Small representations of integers by integral quadratic forms

Let f be a normalized Hecke eigenform of weight k for the full modular group [Formula: see text]. In this paper, we obtain the asymptotic of higher moments of general divisor functions associated to the Fourier coefficients of Rankin–Selberg L-functions [Formula: see text], supported at the integers represented by primitive integral positive-definite binary quadratic forms (reduced forms) of a fixed discriminant [Formula: see text]. We improve previous results in the case when the reduced form is given by [Formula: see text].

W-groups and values of binary forms

Suppose a positive integer $n$ is written as a sum of squares of $m$ integers. What can one say about the value $T$ of the sum of these $m$ integers itself? Which $T$ can be obtained if one considers all possible representations of $n$ as a sum of squares of $m$ integers? Denoting this set of all possible $T$ by $\mathscr{S}_m(n)$, Goldmakher and Pollack have given a simple characterization of $\mathscr{S}_4(n)$ using elementary arguments. Their result can be reinterpreted in terms of Mordell's theory of representations of binary integral quadratic forms as sums of squares of integral linear forms. Based on this approach, we characterize $\mathscr{S}_m(n)$ for all $m\leq 11$ and provide a few partial results for arbitrary $m$. We also show how Mordell's results can be used to study variations of the original problem where the sum of the integers is replaced by a linear form in these integers. In this way, we recover and generalize earlier results by Z.W. Sun et. al..

Let \(h(d)\) be the class number of properly equivalent primitive binary quadratic forms \(ax^2 + bxy + cy^2\) of discriminant \(d = b^2 a - 4ac\). The case of indefinite forms \((d < 0)\) is considered. Assuming that the extended Riemann hypothesis for some fields of algebraic numbers holds, the following results are proved. 1. Let \(\alpha (x)\) be an arbitrarily slow monotonically increasing function such that \(\alpha (x) \to \infty\). Then $$\# \left\{ {p \leqslant \left. x \right|{\text{ }}\left( {\frac{{\text{5}}}{p}} \right) = 1,h(5p^2 ) >(\log p)^{\alpha (p)} } \right\} = o(\pi (x)),$$ where \(\pi (x) = \# \{ p \leqslant x\}\). 2. Let F be an arbitrary sufficiently large positive constant. Then for \(x >x_F\), the relation $$\# \left\{ {p \leqslant \left. x \right|{\text{ }}\left( {\frac{{\text{5}}}{p}} \right) = 1,h(5p^2 ) >F} \right\} \asymp \frac{{\pi (x)}}{F}$$ holds. 3. The relation $$\# \left\{ {p \leqslant \left. x \right|{\text{ }}\left( {\frac{{\text{5}}}{p}} \right) = 1,h(5p^2 ) = 2} \right\} \sim \frac{9}{{19}}A\pi (x)$$ holds, where A is Artin's constant. Hence, for the majority of discriminants of the form \(d = 5p^2\), where \({\left( {\frac{{\text{5}}}{p}} \right) = 1}\) , the class numbers are small. This is consistent with the Gauss conjecture concerning the behavior of \(h(d)\) for the majority of discriminants \(d >0\) in the general case. Bibliography: 22 titles.

The method of Lyapunov functions is used to investigate the stability of systems with distributed and lumped parameters, described by linear partial and ordinary differential equations. The original high-order partial differential equations are represented by a first-order system of partial differential evolution equations and constraint equations; to do that, we introduce additional variables. Passing to the first-order system of partial differential equations and representing ordinary differential equations in the normal Cauchy form, we obtain a possibility to construct the Lyapunov function as a sum of integral and classical quadratic forms and develop general methods of the investigation of the stability of a broad class of systems with distributed and lumped parameters. For example, we consider the stability of the work of the wind-driven lift pump and take into account the elasticity of the shaft transmitting the torque from the wind engine to the pump.