我们得到了一个上界的最少数量的变量，以确保系统的t二次型（t>=2）在域F有一个非平凡的零。特别地，如果F是一个局部域，则2t^2+3个变量确保非平凡零的存在（如果t是偶数，则2t^2+1），而如果F=Q_p且p>=11，则2t^2- 2 t +5个变量就足够了（如果3整除t，则2t^2- 2 t +1）。改进之处在于更有效地利用了二次型的成对和三重态的溶解度信息，并且参数是完全初等的。
英文摘要：
We derive an upper bound for the least number of variables needed to guarantee that a system of t quadratic forms (t>=2) over a field F has a nontrivial zero. In particular, if F is a local field, then 2t^2+3 variables insure the existence of a nontrivial zero (2t^2+1 if t is even), while if F=Q_p with p>=11, then 2t^2-2t+5 variables suffice (2t^2-2t+1 if 3 divides t). The improvement lies in a more efficient use of information on the solubility of pairs and triplets of quadratic forms, and the arguments are completely elementary.
We determine the minimal number of variables $Γ^*(d, K)$ which guarantees a nontrivial solution for every additive form of degree $d=4$ over the four ramified quadratic extensions $\mathbb{Q}_2(\sqrt{2}), \mathbb{Q}_2(\sqrt{10}), \mathbb{Q}_2(\sqrt{-2}), \mathbb{Q}_2(\sqrt{-10}) $ of $\mathbb{Q}_2$. In all four fields, we prove that $Γ^*(4,K) = 11$. This is the first example of such a computation for a proper extension of $\mathbb{Q}_p$ where the degree is a power of $p$ greater than $p$.
We determine the minimal number of variables $Γ^*(d, K)$ which guarantees a nontrivial solution for every additive form of degree $d=2m$, $m$ odd, $m \ge 3$ over the six ramified quadratic extensions of $\mathbb{Q}_2$. We prove that if $K$ is one of $\{\mathbb{Q}_2(\sqrt{2}), \mathbb{Q}_2(\sqrt{10}), \mathbb{Q}_2(\sqrt{-2}), \mathbb{Q}_2(\sqrt{-10})\}$, $Γ^*(d,K) = \frac{3}{2}d$, and if $K$ is one of $\{\mathbb{Q}_2(\sqrt{-1}), \mathbb{Q}_2(\sqrt{-5})\}$, $Γ^*(d,K) = d+1$. The case $d=6$ was previously known.
We prove that an additive form of degree $d=2m$, $m$ odd, $m\ge3$, over the unramified quadratic extension $\mathbb{Q}_2(\sqrt{5})$ has a nontrivial zero if the number of variables $s$ satisifies $s \ge 4d+1$. If $3 \nmid d$, then there exists a nontrivial zero if $s \ge \frac{3}{2}d + 1$, this bound being optimal. We give examples of forms in $3d$ variables without a nontrivial zero in case that $3 \mid d$.
We develop a version of Ekedahl's geometric sieve for integral quadratic forms of rank at least five. As one ranges over the zeros of such quadratic forms, we use the sieve to compute the density of coprime values of polynomials, and furthermore, to address a question about local solubility in families of varieties parameterised by the zeros.
We prove that an additive form of degree $d=2m$, $m$ odd over any totally ramified extension of $\mathbb{Q}_2$ has a nontrivial zero if the number of variables $s$ satisifies $s \ge \frac{d^2}{4} + 3d + 1$.
Michael Knapp, in a previous work, conjectured that every additive sextic form over $\mathbb{Q}_2(\sqrt{-1})$ and $\mathbb{Q}_2(\sqrt{-5})$ in seven variables has a nontrivial zero. In this paper, we show that this conjecture is true, establishing that $Γ^*(6, \mathbb{Q}_2(\sqrt{-1})) = Γ^*(6, \mathbb{Q}_2(\sqrt{-5})) = 7 $.
In this paper, we imitate a classical construction of a counterexample to the local-global principle of cubic forms of 4 variables which was discovered first by Swinnerton-Dyer (Mathematica (1962)). Our construction gives new explicit families of counterexamples in homogeneous forms of $4, 5, 6, ..., 2n+2$ variables of degree $2n+1$ for infinitely many integers $n$. It is contrastive to Swinnerton-Dyer's original construction that we do not need any concrete calculation in the proof of local solubility.
Let $-d$ be a a negative discriminant and let $T$ vary over a set of representatives of the integral equivalence classes of integral binary quadratic forms of discriminant $-d$. We prove an asymptotic formula for $d \to \infty$ for the average over $T$ of the number of representations of $T$ by an integral positive definite quaternary quadratic form and obtain results on averages of Fourier coefficients of linear combinations of Siegel theta series. We also find an asymptotic bound from below on the number of binary forms of fixed discriminant $-d$ which are represented by a given quaternary f…
A (positive definite and integral) quadratic form is said to be $\textit{prime-universal}$ if it represents all primes. Recently, Doyle and Williams in [2] classified all prime-universal diagonal ternary quadratic forms, and all prime-universal diagonal quaternary quadratic forms under two conjectures proposed by themselves. In this article, we classify all prime-universal diagonal quadratic forms regardless of ranks. Furthermore, we prove, so called, $67$-Theorem for a diagonal quadratic form to be prime-universal.
The classical quadratic Gauss sum can be thought of as an exponential sum attached to a quadratic form on a cyclic group. We introduce an equivariant version of Gauss sum for arbitrary finite quadratic forms, which is an exponential sum twisted by the action of the orthogonal group. We prove that simple arithmetic formulae hold for some basic classes of quadratic forms. In application, such invariant appears in the dimension formula for certain vector-valued modular forms.
In this paper we consider certain quaternary quadratic forms and octonary quadratic forms and by using the theory of modular forms, we find formulae for the number of representations of a positive integer by these quadratic forms.
A positive-definite integral quadratic form is called regular if it represents every positive integer which is locally represented. In this article, we classify all regular diagonal quadratic forms of rank greater than 3.
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.
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.
A (positive definite integral) quadratic form is called almost 2-universal if it represents all (positive definite integral) binary quadratic forms except those in only finitely many equivalence classes. Oh [7] determined all almost 2-universal quinary diagonal quadratic forms remaining three as candidates. In this article, we prove that those three candidates are indeed almost 2-universal.
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 $n \geq 2$, 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-Kim-Meyer (2014).
We study the minimal number of variables required by a totally positive definite diagonal universal quadratic form over a real quadratic field $\mathbb Q(\sqrt D)$ and obtain lower and upper bounds for it in terms of certain sums of coefficients of the associated continued fraction. We also estimate such sums in terms of $D$ and establish a link between continued fraction expansions and special values of $L$-functions in the spirit of Kronecker's limit formula.
A formula for the class number $h$ of the imaginary quadratic field $Q(\sqrt{-p}$ is obtained by counting on a specific way the quadratic residues of a prime number of the form $p=4n-1.$ Formulas for the sum of the quadratic residues are also found.
We prove an upper bound for the length of an arithmetic progression represented by an irreducible integral binary quadratic form or a norm form, which depends only on the form and the progression's common difference. For quadratic forms, this improves significantly upon an earlier result of Dey and Thangadurai.
Using half-integral weight modular forms we give a criterion for the existence of real quadratic $p$-rational fields. For $p=5$ we prove the existence of infinitely many real quadratic $p$-rational fields.
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.
A formula for the sum of quadratic residues modulus a prime $p=4n-1$ is studied. We relate some terms on this formula with roots of quadratics and provide an exhaustive analysis of new concepts based on these roots. A number of formulas for the sum of the quadratic residues are obtained. We finalize the paper by obtaining several identities involving $h(-p)$ the class number of the imaginary quadratic field $\mathbb Q(\sqrt{-p}).$
Using modular forms we determine formulas for the number of representations of a positive integer by diagonal octonary quadratic forms with coefficients $1$, $2$, $3$ or $6$.
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.
We determine explicit formulas for the number of representations of a positive integer $n$ by quaternary quadratic forms with coefficients $1$, $2$, $5$ or $10$. We use a modular forms approach.
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.
We compute the genus of a rational quadratic form in terms of the K-theory of a C*-algebra attached to the adelic orthogonal group of the form. As a corollary, one gets a higher composition law for the rational quadratic forms. As an illustration, we consider the Gauss composition of the binary quadratic forms.
Representations of binary quadratic forms by quaternary quadratic forms
Korea institute for Advanced Study（韩国高等研究院）；Institute for Advanced Study（普林斯顿高等研究院）；Morningside Center of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院晨兴数学中心）
We prove an analogue of the Oppenheim conjecture for a system comprising an inhomogeneous quadratic form and a linear form in $3$ variables using dynamics on the space of affine lattices.
