Genera of rationally equivalent integral binary quadratic forms

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.

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.

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.

In this paper, we consider the simultaneous sign changes of coefficients of Rankin–Selberg L-functions associated to two distinct Hecke eigenforms supported at positive integers represented by some certain primitive reduced integral binary quadratic form with negative discriminant D. We provide a quantitative result for the number of sign changes of such sequence in the interval (x, 2x] for sufficiently large x.

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

In this note we prove an elementary result concerning the distribution of the positive integers which are represented by a positive-definite integral binary quadratic form Theorem 1. Let f(X, Y) = aX ~ + b X Y + c y2 be a positive-definite integral binary quadratic form of discriminant - A (= b 2 - 4ac = mi , there exist integers x and y such that n < ax 2 q- bxy + cy 2 < n + 2ml/~A1/4n ~m + m i . P r 0 0 f. Replacing the form f by an equivalent form we may suppose that m~ = c. We define integers x and y by =F(4cn~l/2q I!4cn-Ax2)l/2-bx l x L\~) /' Y= 2c -+1.

W-groups and values of binary forms

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.

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].