Elsevier

Journal of Number Theory

Volume 224, July 2021, Pages 67-94
Journal of Number Theory

General Section
Diagonal odd-regular ternary quadratic forms

https://doi.org/10.1016/j.jnt.2021.01.014Get rights and content

Abstract

A (positive definite primitive integral) quadratic form is called odd-regular if it represents every odd positive integer which is locally represented. In this paper, we show that there are at most 147 diagonal odd-regular ternary quadratic forms and prove the odd-regularities of all but six candidates.

Introduction

In this paper, a quadratic formf(x1,x2,,xn)=1ijnfijxixj(fijZ) always refers to a positive definite (non-classic) integral quadratic form that is primitive in the sense that the coefficients of the quadratic form are coprime, that is,(f11,,fij,,fnn)=1. In terminology introduced by Dickson in 1927 [1], a quadratic form f is called regular if it represents every integer which is represented by its genus. Jones [5] showed that there are exactly 102 diagonal regular ternary quadratic forms in his unpublished thesis (cf. [6]). Watson proved that there are only finitely many equivalence classes of regular ternary quadratic forms in his thesis [20]. Jagy, Kaplansky and Schiemann [4] succeeded Watson's study on regular quadratic forms and provide the list of 913 candidates of regular ternary quadratic forms. All but 22 of them are already proved to be regular at that time. Oh [15] proved the regularities of eight forms among remaining 22 candidates. A conditional proof for the remaining 14 candidates under the Generalized Riemann Hypothesis was given by Lemke Oliver [13].

For a positive integer d and a nonnegative integer a, define a setSd,a={dn+a:nZ0}. In [16], the notion of Sd,a-regularity is introduced. A positive definite integral quadratic form f is called Sd,a-regular if the following two conditions hold;

  • (i)

    f represents every integer in the set Sd,a which is represented by its genus;

  • (ii)

    the genus of f represents at least one integer in Sd,a.

In the same paper, it was proved that there is a polynomial R(x,y)Q(x,y) such that the discriminant df of any Sd,a-regular ternary quadratic form f is bounded by R(d,a). This implies the finiteness of the Sd,a-regular ternary quadratic forms (up to equivalence) for any given d and a. We say that a quadratic form f is odd-regular if it is S2,1-regular.

Kaplansky [9] showed that there are at most 23 ternary quadratic forms that represent all odd positive integers. He proved that 19 of those forms represent all odd positive integers. Jagy [3] dealt with one candidate. The remaining three candidates arex2+2y2+5z2+xz,x2+3y2+6z2+xy+2yz,x2+3y2+7z2+xy+xz. Rouse [19] proved that, under the assumption of Generalized Riemann Hypothesis, each of these three forms represents all odd positive integers. There are three diagonal ternary quadratic forms representing all odd positive integers which arex2+y2+2z2,x2+2y2+3z2,x2+2y2+4z2 and by definition, all of these forms are odd-regular. Note that the above three forms are regular and thus contained in the list of 102 diagonal regular ternary quadratic forms given in [5] (cf. [6]).

In this article, we show that there are at most 147 diagonal odd-regular ternary quadratic forms and prove the regularities of all but six candidates. Since all diagonal regular ternary quadratic forms are odd-regular, we actually show that there are at most 45 diagonal odd-regular ternary quadratic forms which are not regular and prove the odd-regularities of 39 forms. To bound the discriminant of a diagonal odd-regular ternary quadratic forms, we compute some quantities on the representation of primes in an arithmetic progression by a binary quadratic form.

In 1840, Dirichlet conjectured that any binary quadratic form representing an integer in a given arithmetic progression represents infinitely many primes in that arithmetic progression. The conjecture was proved by Meyer [14]. Later in 2003, Halter-Koch [2, Proposition 1] showed that if a binary quadratic form represents an integer in a given arithmetic progression, then the set of primes in the arithmetic progression represented by the binary quadratic form has positive Dirichlet density.

For the remainder of the paper, we adapt the geometric language of quadratic spaces and lattices, generally following [12] and [18]. Let R be the ring of rational integers Z or the ring of p-adic integers Zp for a prime p and let F be the field of fractions of R. An R-lattice is a finitely generated R-submodule of a quadratic space (V,Q) over F. We let B:V×VF be the symmetric bilinear form associated to the quadratic map Q so that B(x,x)=Q(x) for every xV. For a Z-lattice L=Zv1+Zv2++Zvk, the matrix presentation ML of L is the matrixML=(B(vi,vj))1i,jk, and the corresponding quadratic form is defined asi,j=1kB(vi,vj)xixj. The determinant of the matrix B((vi,vj)) is called the discriminant of L and will be denoted by dL. If L admits an orthogonal basis {v1,v2,,vk}, then we simply writeLQ(v1),Q(v2),,Q(vk). For an integer a and a Z-lattice L, we write aL if a is represented by L and

otherwise. The set of all nonnegative integers represented by L will be denoted by Q(L). The scale ideal and norm ideal of L is denoted by s(L) and n(L), respectively. Throughout the paper, we always assume that every Z-lattice is positive definite and primitive in the sense that n(L)=Z. So the scale ideal s(L) is 12Z or Z. We denote by gen(L) the genus of L. The number of isometry classes in gen(L) is called the class number of L and denoted by h(L). In this article, we abuse the notation when we explicitly write down the isometry classes in gen(L). For example, we just writegen(L)={L,M} when the class number h(L) of L is two and the genus mate of L is M. We write agen(L) when a is represented by the genus of L. For an odd prime p, we denote by Δp a non-square unit in Zp×. Any unexplained notations and terminologies can be found in [12] or [18].

Section snippets

Stability

For a positive integer m and a Z-lattice L, letΛm(L)={xL:Q(x+z)Q(z)(modm)for allzL}. Then Λm(L) is a sublattice of L and it is called Watson transformation of L modulo m. We denote by λm(L) the primitive Z-lattice obtained from Λm(L) by scaling the quadratic space QZL by a suitable rational number.

Lemma 2.1

Let p be an odd prime. Let L be a diagonal odd-regular ternary Z-lattice such that the unimodular component of Lp is anisotropic. Then λp(L) is also a diagonal odd-regular ternary Z-lattice.

Proof

LetL=a

Representations of odd primes in an arithmetic progression

For relatively prime positive integers u and v, we define a set P(u,v) to be the set of all odd primes in the arithmetic progression {un+v}n0. Notice that v may be greater than u. A binary quadratic form M is called P(u,v)-universal if it represents every element of P(u,v). For the following two lemmas, we let M be a positive definite primitive integral binary quadratic form ax2+bxy+cy2 and put D=D(M)=b24ac so that D(M)=4dM. The product of all odd prime divisors of D will be denoted by D.

Diagonal stable odd-regular ternary quadratic forms

In this section, we prove that there are exactly eight diagonal stable odd-regular ternary quadratic forms which are not regular. We fix some notations that will be used throughout the section. We denote by qk the k-th odd prime so that P:={q1=3<q2=5<q3=7<} is the set of odd primes. Clearly, we have P=P(8,1)P(8,3)P(8,5)P(8,7). Let L=a,b,c (abc) be a diagonal stable odd-regular ternary quadratic form and let T be the set of all odd primes at which L is anisotropic. It is well known that T

Diagonal odd-regular ternary quadratic forms

In this section, we prove that there are at most 37 diagonal odd-regular ternary quadratic forms which are not stable and not regular. And we show the odd-regularities of 31 forms among 37 candidates. We first introduce the notion of missing primes. Let a,b,c be a diagonal odd-regular ternary quadratic form and let a,b,c be the stable odd-regular ternary quadratic form obtained from a,b,c by taking λq transformations, if necessary, repeatedly. It might happen that there is an odd

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This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2019R1A6A3A01096245).

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