Positive Definite Integral Quadratic Forms Research Articles

The circle method and shifted convolution sums involving the divisor function

Let Q(x) be a positive definite integral quadratic form with the determinant D being squarefree, and r(n,Q) denote the number of representations of n by the quadratic form Q. In this paper, we apply the Hardy-Littlewood-Kloosterman circle method to derive the asymptotic formula for the shifted convolution sum of the divisor function d(n) and Fourier coefficients r(n,Q). With more efforts, our method should have a number of applications for other multiplicative functions.

Can we recover an integral quadratic form by representing all its subforms?

Let o be the ring of integers of a number field. If f is a quadratic form over o and g is another quadratic form over o which represents all proper subforms of f, does g represent f? We show that if g is indefinite, then g indeed represents f. However, when f is positive definite and indecomposable, then there exists a g which represents all proper subforms of f but not f itself. Along the way we give a new characterization of positive definite decomposable quadratic forms over o and a number-field generalization of the finiteness theorem of representations of quadratic forms by quadratic forms over Z[13, Theorem 3.3] which asserts that given any infinite set S of classes of positive definite integral quadratic forms over o of a fixed rank, there exists a finite subset S0 of S with the property that a positive definite quadratic form over o represents all classes in S if and only if it represents all classes in S0.

Jutila's circle method and [formula omitted] shifted convolution sums

Let λf(n) be the normalized Fourier coefficients of a Hecke-Maass or Hecke holomorphic cusp form f for congruence group Γ0(N) with level N and nebentypus χN. Let Q(x) be a positive definite integral quadratic form, and r(n,Q) denote the number of representations of n by the quadratic form Q. In this paper, we apply Jutila's circle method to derive a sharp bound for the shifted convolution sum of Fourier coefficients λf(n) and r(n,Q). We generalize and improve previous results without the Ramanujan conjecture.

On the Exceptional Sets of Integral Quadratic Forms

Abstract A collection $\mathcal S$ of equivalence classes of positive definite integral quadratic forms in $n$ variables is called an $n$-exceptional set if there exists a positive definite integral quadratic form, which represents all equivalence classes of positive definite integral quadratic forms in $n$ variables except those in $\mathcal S$. We show that, among other results, for any given positive integers $m$ and $n$, there is always an $n$-exceptional set of size $m$ and there are only finitely many of them.

The number of representations of integers by generalized Bell ternary quadratic forms

For a positive definite ternary integral quadratic form [Formula: see text], let [Formula: see text] be the number of representations of an integer [Formula: see text] by [Formula: see text]. A ternary quadratic form [Formula: see text] is said to be a generalized Bell ternary quadratic form if [Formula: see text] is isometric to [Formula: see text] for some nonnegative integers [Formula: see text]. In this paper, we give a closed formula for [Formula: see text] for a generalized Bell ternary quadratic form [Formula: see text] with [Formula: see text] and class number greater than [Formula: see text] by using the Minkowski–Siegel formula and bases for spaces of cusp forms of weight [Formula: see text] and level [Formula: see text] with [Formula: see text] consisting of eta-quotients.

On locally primitively universal quadratic forms

A positive definite integral quadratic form is said to be almost (primitively) universal if it (primitively) represents all but at most finitely many positive integers. In general, almost primitive universality is a stronger property than almost universality. The two main results of this paper are (1) every primitively universal form nontrivially represents zero over every ring $$\mathbb {Z}_p$$ of p-adic integers, and (2) every almost universal form in five or more variables is almost primitively universal.

Integral quadratic forms avoiding arithmetic progressions

For every positive integer [Formula: see text], it is shown that there exists a positive definite diagonal quaternary integral quadratic form that represents all positive integers except for precisely those which lie in [Formula: see text] arithmetic progressions. For [Formula: see text], all forms with this property are determined.

Binary quadratic forms represented by sums of squares in an essentially unique way

In this article, we determine all positive definite integral binary quadratic forms that are represented by the sum of k integer squares in an essentially unique way for each integer $$k\ge 4$$.

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

Cartan matrices, quasi-Cartan matrices and associated integral quadratic forms and root systems play an important role in such areas like Lie theory, representation theory and algebraic graph theory. We study quasi-Cartan matrices by means of the inflation algorithm, an idea used in Ovsienko's classical proof of the classification of positive definite integral quadratic forms and recently applied in several other classification results. We prove that it is enough to perform linear number of inflations to reduce positive or non-negative principal quasi-Cartan matrix to its canonical form, i.e., to the Cartan matrix of a Dynkin or Euclidean diagram, respectively. Moreover, we show that in the positive case the length of every sequence of inflations has quadratic bound, and that the only other “natural” non-negative class having such universal bound is the class of so-called pos-sincerely principal quasi-Cartan matrices (and in this case the bound is linear). We obtain such low bounds by applying, among others, some new observations on the properties of the reduced root systems (in the sense of Bourbaki) associated with positive quasi-Cartan matrices.

COMPLETENESS OF THE LIST OF SPINOR REGULAR TERNARY QUADRATIC FORMS

Extending the notion of regularity introduced by Dickson in 1939, a positive definite ternary integral quadratic form is said to be spinor regular if it represents all the positive integers represented by its spinor genus (that is, all positive integers represented by any form in its spinor genus). Jagy conducted an extensive computer search for primitive ternary quadratic forms that are spinor regular, but not regular, resulting in a list of 29 such forms. In this paper, we will prove that there are no additional forms with this property.

On a Waring’s problem for integral quadratic and Hermitian 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.

Cubic Algorithm to Compute the Dynkin Type of a Positive Definite Quasi-Cartan Matrix

Inflations algorithm is a procedure that appears implicitly in Ovsienko’s classical proof for the classification of positive definite integral quadratic forms. The best known upper asymptotic bound for its time complexity is an exponential one. In this paper we show that this bound can be tightened to O(n6) for the naive implementation. Also, we propose a new approach to show how to decide whether an admissible quasi-Cartan matrix is positive definite and compute the Dynkin type in just O(n3) operations taking an integer matrix as input.

Multiplicative property of representation numbers of ternary quadratic forms

Let f be a positive definite integral ternary quadratic form and $$\theta (z;f)=\sum _{n=0}^{\infty }a(n;f)q^n$$ its theta function. For any fixed square-free positive integer t with $$a(t;f)\ne 0$$ , we define $$\rho (n;t,f):=a(tn^2;f)/a(t;f)$$ . For the case when $$f=x_1^2+x_2^2+x_3^2$$ and $$t=1$$ , Hurwitz proved that $$\rho (n;t,f)$$ is multiplicative and he gave its expression. Cooper and Lam proved four similar formulas and proposed a conjecture for some other cases. Using the results given in this paper, we can check the multiplicative property of $$\rho (n;t,f)$$ for many cases. All cases in Cooper and Lam’s conjecture are included in ours.

Diagonal quadratic forms representing all binary diagonal quadratic forms

A (positive definite integral) quadratic form is called diagonally 2-universal if it represents all positive definite integral binary diagonal quadratic forms. In this article, we show that, up to equivalence, there are exactly 18 (positive definite integral) quinary diagonal quadratic forms that are diagonally 2-universal. Furthermore, we provide a “diagonally 2-universal criterion” for diagonal quadratic forms, which is similar to “15-Theorem” proved by Conway and Schneeberger.

Interaction of Hecke–Shimura Rings Acting on Theta-Series

The paper considers principal features of interactions (transfer homomorphisms) for the Hecke–Shimura rings of integral symplectic groups and integral orthogonal groups of positive-definite integral quadratic forms both operating on the theta-series of positive-definite quadratic forms by Hecke operators.

On the representation of quadratic forms by quadratic forms

Using the circle method, we show that for a fixed positive definite integral quadratic form $A$, the expected asymptotic formula for the number of representations of a positive definite integral quadratic form $B$ by $A$ holds true, providing that the dimension of $A$ is large enough in terms of the dimension of $B$ and the maximum ratio of the successive minima of $B$, and providing that $B$ is sufficiently large in terms of $A$.

Class numbers of ternary quadratic forms

G.L. Watson [13,14] introduced a set of transformations, called Watson transformations by most recent authors, in his study of the arithmetic of integral quadratic forms. These transformations change an integral quadratic form to another integral quadratic form with a smaller discriminant, but preserve many arithmetic properties at the same time. In this paper, we study the change of class numbers of positive definite ternary integral quadratic formula along a sequence of Watson transformations, thus providing a new and effective way to compute the class number of positive definite ternary integral quadratic forms. Explicit class number formulae for many genera of positive definite ternary integral quadratic forms are derived as illustrations of our method.

Positive definite quadratic forms representing integers of the form an 2+b

For any subset S of positive integers, a positive definite integral quadratic form is said to be S-universal if it represents every integer in the set S. In this article, we classify all binary S-universal positive definite integral quadratic forms in the case when S=S a ={an 2∣n≥2} or S=S a,b ={an 2+b∣n∈ℤ}, where a is a positive integer and ab is a square-free positive integer in the latter case. We also prove that there are only finitely many S a -universal ternary quadratic forms not representing a. Finally, we show that there are exactly 15 ternary diagonal S 1-universal quadratic forms not representing 1.

On primitively universal quadratic forms

In 1999, Manjul Bhargava proved the Fifteen Theorem and showed that there are exactly 204 universal positive definite integral quaternary quadratic forms. We consider primitive representations of quadratic forms and investigate a primitive counterpart to the Fifteen Theorem. In particular, we give an efficient method for deciding whether a positive definite integral quadratic form in four or more variables with odd square-free determinant is almost primitively universal.

Universal forms over $\mathbb{Q}(\sqrt{5})$

In this paper, we find all quaternary universal positive definite integral quadratic forms over \(\mathbb{Q}(\sqrt{5})\) and prove an analogue of Conway and Schneeberger’s 15-Theorem.