Kathrin Bringmann, Ben Kane
In this paper, we are interested in the interplay between integral ternary quadratic forms and class numbers. This is partially motivated by a question of Petersson.
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In this paper, near-miss identities for the number of representations of some integral ternary quadratic forms with congruence conditions are found and proven. The genus and spinor genus of the corresponding lattice cosets are then classified. Finally, a complete genus and spinor genus classification for all conductor 2 lattice cosets of 2-adically unimodular lattices is given.
In 1997, Kaplansky conjectured that if two positive definite ternary quadratic forms with integer coefficients have perfectly identical integral representations, then they are isometric, both regular, or included either of two families of ternary quadratic forms. In this article, we prove the existence of pairs of ternary quadratic forms representing same integers which are not in the Kaplansky's list.
Small solutions of ternary quadratic congruences with averaging over the moduli
本文研究了三次二次同余式的解的存在性，通过模数平均方法，证明了在特定条件下解的高度上限为Q^{3/8+ε}α₂^{ε}。
An integral quadratic polynomial (with positive definite quadratic part) is called almost universal if it represents all but finitely many positive integers. In this paper, we provide a characterization of almost universal ternary quadratic polynomials with conductor 2.
We give formulas for local densities of diagonal integral ternary quadratic forms at odd primes. Exponential sums and quadratic Gauss sums are used to obtain these formulas. These formulas (along with 2-adic densities and Siegel's mass formula) can be used to compute the representation numbers of certain ternary quadratic forms.
An integral quadratic polynomial (with positive definite quadratic part) is called almost universal if it represents all but finitely many positive integers. In this paper, we introduce the conductor of a quadratic polynomial, and give an effective characterization of almost universal ternary quadratic polynomials with odd prime power conductor.
We find and prove a class of congruences modulo 4 for Andrews' partition with certain ternary quadratic form. We also discuss distribution of $\overline{\mathcal{EO}}(n)$ and further prove that $\overline{\mathcal{EO}}(n)\equiv0\pmod4$ for almost all $n$. This study was inspired by similar congruences modulo 4 in the work by the second author and Garvan.
In this paper, I use Siegel-Weil formula and Kudla matching principle to prove some interesting identities between representation number (of ternary quadratic space) and the degree of Heegner divisors.
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 6 candidates.
The goal of this note is to provide an analysis of the positive integers that are represented everywhere locally, but not globally, by each of the 29 spinor regular positive definite integral ternary quadratic forms that are not regular.
We give details of a formerly known relation between ternary quadratic forms and quaternion orders through the even Clifford algebra. Based on this and classifications of ternary quadratic forms we give a completely explicit classification of quaternion orders in the padic case.
We consider the prehomogeneous vector space of pairs of ternary quadratic forms. For the lattice of pairs of integral ternary quadratic forms and its dual lattice, there are six zeta functions associated with the the prehomogeneous vector space. We present a conjecture which states that there are simple relations among the six zeta functions. We prove that the coefficients coincide on fundamental discriminants.
G.L. Watson \cite{watson1, watson2} 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 discriminants, 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 def…
Small solutions of generic ternary quadratic congruences to general moduli
本文研究一般奇数模数下三元二次同余方程的小解，利用Burgess和Heath-Brown的特征和估计突破指数1/2障碍，在平均意义下对几乎所有系数得到解的渐近公式，并提出最小解大小的精细猜想。
The classification and representations of positive definite ternary quadratic forms of level 4N
本文针对无平方因子整数N，给出4N级正定三元二次型的完整分类，并利用三元二次型、四元数代数与Jacobi形式的关系，建立各类表示加权和的显式公式，进而得到类数公式及权3/2模形式Eisenstein空间的应用。
Equidistribution of Solutions of Ternary Quadratic Congruences Modulo Prime Powers
本文研究素数幂模三元二次同余方程在固定点附近且 $z$ 与 $p$ 互素的解，通过光滑化方法在 $N\geq q^{1/2+\varepsilon}$ 时建立渐近公式并证明解存在。
We discuss an unusual phenomenon in (integral) positive ternary quadratic forms. We also describe an interesting pairing of genera of ternary forms.
A note on Some Partitions Related to Ternary Quadratic Forms
本文提出与三元二次型相关的分拆函数及其上界与性质，作为共轭Bailey对简单方法的应用，进一步验证了该方法的实用性。
This paper treats certain integral lattices with respect to ternary quadratic forms, which are obtained from the data of a non-zero element and a maximal lattice in a quaternary quadratic space. Such a lattice can be described by means of an order associated with the lattice in the even Clifford algebra of the ternary form. This provides a correspondence between the genus of the lattice and that of the order.
Mordell in 1958 gave a new proof of the three squares theorem. We generalize those techniques to characterize the integers represented by the remaining six "Ramanujan-Dickson ternaries" as well as three other ternary forms.
We investigate the density of integer solutions to certain binary inhomogeneous quadratic congruences and use this information to detect almost primes on a singular del Pezzo surface of degree 6.
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.
Quantitative strong approximation for ternary quadratic forms III
本文通过Hardy-Littlewood圆方法的δ变体，研究三次二次型的整数点计数问题，并将其与Brauer-Manin障碍联系起来。
The paper proposes a polynomial formula for solution quadratic congruences in $\mathbb{Z}_p$. This formula gives the correct answer for quadratic residue and zeroes for quadratic nonresidue. The general form of the formula for $p=3 \;\rm{mod}\,4$, $p=5 \;\rm{mod}\,8$ and for $p=9 \;\rm{mod}\,16$ are suggested.
Almost all primes are not needed in Ternary Goldbach
该研究针对三元哥德巴赫猜想，通过可容许同余扩张系统构造素数集$\u2119$，证明即便剔除几乎所有素数，仅用$\u2119$中的素数仍可满足该猜想，深化了对素数加法性质的认识。
An integral quadratic polynomial is called regular if it represents every integer that is represented by the polynomial itself over the reals and over the $p$-adic integers for every prime $p$. It is called complete if it is of the form $Q({\mathbf x} + {\mathbf v})$, where $Q$ is an integral quadratic form in the variables ${\mathbf x} = (x_1, \ldots, x_n)$ and ${\mathbf v}$ is a vector in ${\mathbb Q}^n$. Its conductor is defined to be the smallest positive integer $c$ such that $c{\mathbf v} \in {\mathbb Z}^n$. We prove that for a fixed positive integer $c$, there are only finitely many equ…
We study an asymptotic formula for counting integral points over an equation defined by a non-degenerated indefinite integral ternary quadratic form $f$ representing a non-zero integer $a$ such that $-a\cdot det(f)$ is square over a number field. In particular, we prove that the finite part of this asymptotic formula is given by the product of local density times $1-p^{-1}$ over all finite primes $p$ over $\Bbb Z$.
Mordell in 1958 gave a new proof of the three squares theorem. Those techniques were generalized by Blackwell, et al., in 2016 to characterize the integers represented by the remaining six "Ramanujan-Dickson ternaries". We continue the generalization of these techniques to four additional forms.
Let $f$ be a positive definite integral ternary quadratic form and let $r(k,f)$ be the number of representations of an integer $k$ by $f$. In this article we study the number of representations of squares by $f$. We say the genus of $f$, denoted by $\text{gen}(f)$, is indistinguishable by squares if for any integer $n$, $r(n^2,f)=r(n^2,f')$ for any quadratic form $f' \in \text{gen}(f)$. We find some non trivial genera of ternary quadratic forms which are indistinguishable by squares. We also give some relation between indistinguishable genera by squares and the conjecture given by Cooper and L…
In this paper, we investigate the interplay between positive-definite integral ternary quadratic forms and class numbers. We generalize a result of Jones relating the theta function for the genus of a quadratic form to the Hurwitz class numbers, obtaining an asymptotic formula (with a main term and error term away from finitely many bad square classes $t_j\mathbb{Z}^2$) relating the number of lattices points in a quadratic space of a given norm with a sum of class numbers related to that norm and the squarefree part of the discriminant of the quadratic form on this lattice.
The equidistribution of roots of quadratic congruences with prime moduli depends crucially upon effective bounds for a special Weyl linear form. Duke, Friedlander and Iwaniec discovered a strong estimate for this Weyl linear form when the quadratic polynomial has negative discriminant. Tóth established an analogous but weaker bound when the quadratic polynomial has positive discriminant. We obtain a stronger estimate for the Weyl linear form for quadratics of positive discriminants.
In this paper we generalize the idea of "essentially unique" representations by ternary quadratic forms. We employ the Siegel formula, along with the complete classification of imaginary quadratic fields of class number less than or equal to 8, to deduce the set of integers which are represented in essentially one way by a given form which is alone in its genus. We consider a variety of forms which illustrate how this method applies to any of the 794 ternary quadratic forms which are alone in their genus. As a consequence, we resolve some conjectures of Kaplansky regarding unique representatio…
We consider a system of homogeneous quadratic forms with congruence conditions in $n\geq 3$ variables and prove the existence of two linearly independent integral solutions of bounded height. We also show the existence of small height integral zeros of this system avoiding a given set of hyperplanes.
This paper gives explicit formulas for the formal total mass Dirichlet series for integer-valued ternary quadratic lattices of varying determinant and fixed signature over number fields F where p = 2 splits completely. We prove this by using local genus invariants and local mass formulas to compute the local factors of the theory developed in [11]. When the signature is positive definite these formulas be checked against tables of positive definite ternary quadratic forms over Z, and we have written specialized software which checks these results when the Hessian determinant is \leq 2 \times 1…
A positive quadratic form is $(k,\ell)$-universal if it represents all the numbers $kx+\ell$ where $x$ is a non-negative integer, and almost $(k,\ell)$-universal if it represents all but finitely many of them. We prove that for any $k,\ell$ such that $k\nmid\ell$ there exists an almost $(k,\ell)$-universal diagonal ternary form. We also conjecture that there are only finitely many primes $p$ for which a $(p,\ell)$-universal diagonal ternary form exists (for any $\ell<p$) and we show the results of computer experiments that speak in favor of the conjecture.
We discuss multiplicative properties of the binary quadratic form $a x^2 + b x y + c y^2$ by considering a ring of matrices which is closed under a triple product. We prove that the ring forms a ternary algebra in the sense of Hestenes, and then derive both multiplicative formulas for a large class of binary quadratic forms and a type of multiplication for points on a conic section which generalizes the algebra of rational points on the unit circle.
Some congruences of calibers of real quadratic fields
本文证明了实二次域中 caliber 与 m-caliber 的同余方程，并推论出若干连分数的周期长度。
Ternary Perfect Forms Over Imaginary Quadratic Field
本文枚举了绝对判别式不超过91的虚二次域上的三元完美型，证明其组合类型数量有与判别式无关的上界，并分析了相关数据。
For a natural number $m$, generalized $m$-gonal numbers are defined by the formula $p_m(x)=\frac{(m-2)x^2-(m-4)x}{2}$ with $x\in \mathbb Z$. In this paper, we determine a criterion on $a,b,c,m$ for which the weighted ternary sum $P_{a,b,c,m}:=ap_m(x)+bp_m(y)+cp_m(z)$ is almost universal. We also prove for some $a,b,c,m$ that the form $P_{a,b,c,m}$ is not almost universal, while it represents all possible congruence classes.
For a positive definite integral ternary quadratic form $f$, let $r(k,f)$ be the number of representations of an integer $k$ by $f$. The famous Minkowski-Siegel formula implies that if the class number of $f$ is one, then $r(k,f)$ can be written as a constant multiple of a product of local densities which are easily computable. In this article, we consider the case when the spinor genus of $f$ contains only one class. In this case the above also holds if $k$ is not contained in a set of finite number of square classes which are easily computable (see, for example, \cite{sp1} and \cite {sp2}). …
Non-universality of ternary quadratic forms over fields containing $\sqrt2$
本文证明了在包含√2的四次实数域上，不存在正定的三次经典二次型是普遍的，并探讨了相关二次型的性质。
We estimate the deviation of the number of solutions of the congruence $$ m^2-n^2 \equiv c \pmod q, \qquad 1 \le m \le M, \ 1\le n \le N, $$ from its expected value on average over $c=1, ..., q$. This estimate is motivated by the recently established by D. R. Heath-Brown connection between the distibution of solution to this congruence and the pair correlation problem for the fractional parts of the quadratic function $αk^2$, $k=1,2,...$ with a real $α$.
Covariants and simultaneous diagonalization of pairs of ternary quadratic forms, and binary quartic forms
本文建立了二元四次型与三元二次型对协变间的对应，并给出了任意特征零域上二次型对的标准对角化方法及有理数域上可对角化的精确判据。
In this paper we prove the main conjectures of Berkovich and Jagy about weighted averages of representation numbers over an S-genus of ternary lattices (defined below) for any odd squarefree S \in N. We do this by reformulating them in terms of local quantities using the Siegel-Weil and Conway-Sloane formulas, and then proving the necessary local identities. We conclude by conjecturing generalized formulas valid over certain totally real number fields as a direction for future work.
Let $f$ be a positive definite ternary quadratic form. We assume that $f$ is non-classic integral, that is, the norm ideal of $f$ is $\z$. We say $f$ is {\it strongly $s$-regular } if the number of representations of squares of integers by $f$ satisfies the condition in Cooper and Lam's conjecture in \cite {cl}. In this article, we prove that there are only finitely many strongly $s$-regular ternary forms up to equivalence if the minimum of the non zero squares that are represented by the form is fixed. In particular, we show that there are exactly $207$ non-classic integral strongly $s$-regul…
On indefinite integral ternary quadratic forms
本文解决了 Margulis 和 Serre 分别于 1990 年提出的两个关于不定整系数三元二次型的问题，通过发展处理高分歧的工具来研究按丢番图不变量加权的类求和。
On the analytic theory of isotropic ternary quadratic forms II
本文为特殊三元二次型给出哈塞原理定量补充的独立初等证明，并由此细化渐近结果、得到若干推论。
We characterize the generating function of the number of representations described in the title in terms of the theory of modular forms. Appealing to this characterization we obtain explicit formulas for the representation numbers as examples.
Small Solutions of generic ternary quadratic congruences
本文研究一般三元二次同余的小解，证明在大多数系数下解的渐近公式成立，突破指数1/2障碍，并在固定素数幂及Lindelöf假设下改进指数。
In 1957 N.C. Ankeny provided a new proof of the three squares theorem using geometry of numbers. This paper generalizes Ankeny's technique, proving exactly which integers are represented by $x^2 + 2y^2 + 2z^2$ and $x^2 + y^2 + 2z^2$ as well as proving sufficient conditions for an integer to be represented by $x^2+y^2+3z^2$ and $x^2 + y^2 + 7z^2$.
Let $Q(x,y,z)$ be an integral quadratic form with determinant coprime to some modulus $q$. We show that $q\mid Q$ for some non-zero integer vector $(x,y,z)$ of length $O(q^{5/8+\varepsilon})$, for any fixed $\varepsilon>0$. Without the coprimality condition on the determinant one could not achieve an exponent below $2/3$. The proof uses a bound for short character sums involving binary quadratic forms, which extends a result of Chang.
Kaplansky conjectured that if two positive-definite real ternary quadratic forms have perfectly identical representations over $\mathbb{Z}$, they are constant multiples of regular forms, or is included in either of two families parametrized by $\mathbb{R}^2$ (so called, hexagonal and rhombohedral families). Our results aim to clarify the limitations imposed to such a pair by computational and theoretical approaches. Firstly, the result of an exhaustive search for such pairs of integral quadratic forms is presented, in order to provide a concrete version of the Kaplansky conjecture. The obtaine…
Quantitative strong approximation for ternary quadratic forms II
本文利用 Heath-Brown 的 Hardy–Littlewood 圆法 $\delta$ 变体，证明整三元二次型仿射曲面上的有理点在除去有限位置后的 adelic 空间中等分布，推进 Linnik 问题研究。
A classical result in number theory is Dirichlet's theorem on the density of primes in an arithmetic progression. We prove a similar result for numbers with exactly k prime factors for k>1. Building upon a proof by E.M. Wright in 1954, we compute the natural density of such numbers where each prime satisfies a congruence condition. As an application, we obtain the density of squarefree n not greater than x with k prime factors such that a fixed quadratic equation has exactly 2^k solutions modulo n.
We show that for a fixed integer $n \neq \pm2$, the congruence $x^2 + nx \pm 1 \equiv 0 \pmodα$ has the solution $β$ with $0 < β< α$ if and only if $α/β$ has a continued fraction expansion with sequence of quotients having one of a finite number of possible asymmetry types. This generalizes the old theorem that a rational number $α/β> 1$ in lowest terms has a symmetric continued fraction precisely when $β^2 \equiv \pm 1 \pmodα$.
We prove a quantitative version of Oppenheim's conjecture for generic ternary indefinite quadratic forms. Our results are inspired by and analogous to recent results for diagonal quadratic forms due to Bourgain.
Inspired by Dickson's classification of regular ternary quadratic forms, we prove that there are no primitive regular $m$-gonal forms when $m$ is sufficiently large. In order to do so, we construct sequences of primes that are inert in a certain quadratic field and show that they satisfy a certain inequality bounding the next prime by the product of the previous primes, a question of independent interest.
For each positive integer $n$, let $g_Δ(n)$ be the smallest positive integer $g$ such that every complete quadratic polynomial in $n$ variables which can be represented by a sum of odd squares is represented by a sum of at most $g$ odd squares. In this paper, we analyze $g_Δ(n)$ by studying representations of integral quadratic forms by sums of squares with certain congruence condition. We prove that the growth of $g_Δ(n)$ is at most an exponential of $\sqrt{n}$, which is the same as the best known upper bound on the $g$-invariants of the original quadratic Waring's problem. We also determine …
Let $τ_3(n)$ be the triple divisor function which is the number of solutions of the equation $d_1d_2d_3=n$ in natural numbers. It is shown that $$ \sum_{1\leq n_1,n_2,n_3\leq \sqrt{x}}τ_3(n_1^2+n_2^2+n_3^2)=c_1x^{\frac{3}{2}}(\log x)^2+ c_2x^{\frac{3}{2}}\log x +c_3x^{\frac{3}{2}} +O_{\varepsilon}(x^{\frac{11}{8}+\varepsilon}) $$ for some constants $c_1$, $c_2$ and $c_3$.
Quantitative strong approximation for ternary quadratic forms I
本文研究不定三元整系数二次型在局部条件下的定量强逼近，利用 Heath-Brown 的 δ 圆法得到含次级项、且允许 m 随 B 增长的整数解计数渐近公式。
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