Let $L$ be a positive definite (non-classic) ternary $\z$-lattice and let $p$ be a prime such that a $\frac 12\z_p$-modular component of $L_p$ is nonzero isotropic and $4\cdot dL$ is not divisible by $p$. For a nonnegative integer $m$, let $\mathcal G_{L,p}(m)$ be the genus with discriminant $p^m\cdot dL$ on the quadratic space $L^{p^m}\otimes \q$ such that for each lattice $T \in \mathcal G_{L,p}(m)$, a $\frac 12\z_p$-modular component of $T_p$ is nonzero isotropic, and $T_q$ is isometric to $(L^{p^m})_q$ for any prime $q$ different from $p$. Let $r(n,M)$ be the number of representations of an integer $n$ by a $\z$-lattice $M$. In this article, we show that if $m \le 2$ and $n$ is divisible by $p$ only when $m=2$, then for any $T \in \mathcal G_{L,p}(m)$, $r(n,T)$ can be written as a linear summation of $r(pn,S_i)$ and $r(p^3n,S_i)$ for $S_i \in \mathcal G_{L,p}(m+1)$ with an extra term in some special case. We provide a simple criterion on when the extra term is necessary, and we compute the extra term explicitly. We also give a recursive relation to compute $r(n,T)$, for any $T \in \mathcal G_{L,p}(m)$, by using the number of representations of some integers by lattices in $\mathcal G_{L,p}(m+1)$ for an arbitrary integer $m$.
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.
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 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.
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.
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.
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.
CUNY Graduate Center（纽约市立大学研究生中心）；Oklahoma State University（俄克拉荷马州立大学）；Institute for Advanced Study & Princeton University（普林斯顿高等研究院与普林斯顿大学）；Seoul National University（首尔国立大学）
On representations of integers by indefinite ternary quadratic forms
令$f$是不定三元二次型，并且令$q$是使得$-q det(f)$不是正方形的整数。令$N(T,f,q)$表示方程$f(x)=q$的整数解的个数，其中$x$位于以原点为中心的半径为$T$的球中。我们感兴趣的是$N(T,f,q)$的渐近行为的$T$趋于无穷大。我们从我们与Z.鲁德尼克认为$N(T,f,q)$的增长类似于cE（T，f，q）$ as $ T $ tends to infinity, where $ E（T，f，q）$ is the Hardy-Littlewood expectation (the product of local densities) and $ 0 \le c \le 2 $. We give examples of $ f $ and $ q $ such that $ c$取值为0，1，2.
In this note, we give an elementary proof of the following classical fact. Any positive definite ternary quadratic form over the rational numbers fails to represent infinitely many positive integers. For any ternary quadratic form (positive definite or indefinite), our method constructs certain congruence classes whose elements, up to a square factor, are the only elements not represented over the rational numbers by that form. In the case of a positive definite ternary form, we show that these classes are non-empty. This shows that the minimum number of variables in a positive definite quadra…
For each integer $x$, the $x$-th generalized pentagonal number is denoted by $P_5(x)=(3x^2-x)/2$. Given odd positive integers $a,b,c$ and non-negative integers $r,s$, we employ the theory of ternary quadratic forms to determine when the sum $aP_5(x)+2^rbP_5(y)+2^scP_5(z)$ represents all but finitely many positive integers.
