For special $d$-dimensional hyperbolic shells $E$ with $ d\geq 5$ we show that the number of lattice points in $E$ intersected with a $d$-dimensional cube $C_r$ of edge length $r$, can be approximated by the volume of $E\cap C_r$, as $r$ tends to infinity, up to an error of order ${\mathcal O}(r^{d-2})$. We generalize results and techniques, used by F. Götze (2004), to a large class of {\em indefinite} quadratic forms and we provide explicit bounds for the errors in terms of certain Minkowski minima related to these quadratic forms. Furthermore, we obtain, as in the positive definite case, a result for multivariate diophantine approximation and for the maximal gap between values of such indefinite forms.
The modular transformation behavior of theta series for indefinite quadratic forms is well understood in the case of elliptic modular forms due to Vignéras, who deduced that solving a differential equation of second order serves as a criterion for modularity. In this paper, we will give a generalization of this result to Siegel theta series.
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 $\Bbb Z$.
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）
A number field $k$ admits a binary integral quadratic form which represents all integers locally but not globally if and only if the class number of $k$ is bigger than one. In this case, there are only finitely many classes of such binary integral quadratic forms over $k$. A number field $k$ admits a ternary integral quadratic form which represents all integers locally but not globally if and only if the class number of $k$ is even. In this case, there are infinitely many classes of such ternary integral quadratic forms over $k$. An integral quadratic form over a number field $k$ with more tha…
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.
For any $\varepsilon > 0$ we derive effective estimates for the size of a non-zero integral point $m \in \mathbb{Z}^d \setminus \{0\}$ solving the Diophantine inequality $\lvert Q[m] \rvert < \varepsilon$, where $Q[m] = q_1 m_1^2 + \ldots + q_d m_d^2$ denotes a non-singular indefinite diagonal quadratic form in $d \geq 5$ variables. In order to prove our quantitative variant of the Oppenheim conjecture, we extend an approach developed by Birch and Davenport [BD58b] to higher dimensions combined with a theorem of Schlickewei [Sch85]. The result obtained is an optimal extension of Schlickewei's …
Gauss' classical reduction theory for indefinite binary quadratic forms over $\mathbb{Z}$ has originally been proven by means of purely algebraic and arithmetic considerations. It was later discovered that this reduction theory is closely related to a certain symbolic dynamics for the geodesic flow on the modular surface, and hence can also be deduced geometrically. In this article, we use certain symbolic dynamics for the geodesic flow on Hecke triangle surfaces (also the non-arithmetic ones) to develop reduction theories for the indefinite binary quadratic forms associated to Hecke triangle …
Let $h(d)$ be the class number of indefinite binary quadratic forms of discriminant $d$, and let $\varepsilon_d$ be the corresponding fundamental unit. In this paper, we obtain an asymptotic formula for the $k$-th moment of $h(d)$ over positive discriminants $d$ with $\varepsilon_d\leq x$, uniformly for real numbers $k$ in the range $0<k\leq (\log x)^{1-o(1)}$. This improves upon the work of Raulf, who obtained such an asymptotic for a fixed positive integer $k$. We also investigate the distribution of large values of $h(d)$ when the $d$'s are ordered according to the size of their fundamental…
Values of indefinite quadratic forms at integral points and flows on spaces of lattices
这篇论文主要集中在两个领域最近证明的猜想：A）A。Oppenheim关于实不定二次型在整点的值。B）Dani、Raghunathan和Margulis关于像${\BSL}_n(\R)/{\BSL}_n(\Z)$这样的格空间中轨道闭包的猜想。初看起来，A）属于解析数论，B）属于遍历和李理论，它们似乎完全不相关。在此一并讨论它们，是因为M. S. Raghunathan，也就是说，B）的一个特例产生了A）的一个证明。
We define generalised zeta functions associated to indefinite quadratic forms of signature (g-1,1) -- and more generally, to complex symmetric matrices whose imaginary part has signature (g-1,1) -- and we investigate their properties. These indefinite zeta functions are defined as Mellin transforms of indefinite theta functions in the sense of Zwegers, which are in turn generalised to the Siegel modular setting. We prove an analytic continuation and functional equation for indefinite zeta functions. We also show that indefinite zeta functions in dimension 2 specialise to differences of ray cla…
We establish a quantitative version of Oppenheim's conjecture for generic ternary indefinite quadratic forms using an analytic number theory approach. The statements come with power gains and in some cases are essentially optimal
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.
Let $f(x_1,\ldots,x_s)$ be a translation invariant indefinite quadratic form of integer coefficients with $s\ge 10$. Let $\mathcal{A}\subseteq \mathcal{P}\cap \{1,2,\ldots,X\}$. Let $X$ be sufficiently large. Subject to a rank condition, we prove that there exist distinct primes $p_1,\ldots,p_s\in \mathcal{A}$ such that $f(p_1,\ldots,p_s)=0$ as soon as $|\mathcal{A}|\ge \frac{X}{\log X} (\log\log X)^{-\frac{1}{80}}.$
We translate Davenport's and Heilbronn's work on a quantitative version of the Oppenheim conjecture for indefinite diagonal quadratic forms in 5 variables into the setting of function fields.
InfoMod is a new software and application devoted to the modular group, PSL2(Z). It has algorithms that deals with the classical correspondences among continued fractions, geodesics on the modular surface and binary quadratic forms. In addition the software implements the recently discovered representation of Gauss' indefinite binary quadratic forms and their classes in terms of certain infinite planar graphs (dessins) called çarks. InfoMod illustrates various aspects of these forms, i.e. Gauss' reduction algorithm, the representation problem of forms, ambiguous and reciprocal forms. It can be…
Since the study by Jacobi and Hecke, Hecke-type series have received a lot of attention. Unlike such series associated with indefinite quadratic forms, identities on Hecke-type series associated with definite quadratic forms are quite rare in the literature. Motivated by the works of Liu, we first establish many parameterized identities with two parameters by employing different $q$-transformation formulas and then deduce various Hecke-type identities associated with definite quadratic forms by specializing the choice of these two parameters. As applications, we utilize some of these Hecke-typ…
In a recent preprint, arXiv:1606.05495v1, Alexandrov, Banerjee, Manschot and Pioline introduced generalized error functions and used them to construct indefinite theta series associated to quadratic lattices L of signature (n-2,2). These series are generalizations of those constructed by Zwegers for lattices of signature (n-1,1) and are shown to be `modular completions' of certain nice $q$-series. In this paper, we show that the ABMP-indefinite theta series for signature (n-2,2) can also be obtained as integrals of the form valued theta series introduced in joint work with J. Millson in 1986. …
We prove that the set of CM points on the Shimura curve associated to an Eichler order inside an indefinite quaternion $\mathbb{Q}$-algebra, is in bijection with the set of certain classes of $p$-adic binary quadratic forms, where $p$ is a prime dividing the discriminant of the quaternion algebra. The classes of $p$-adic binary quadratic forms are obtain by the action of a discrete and cocompact subgroup of $\mathrm{PGL}_{2}(\mathbb{Q}_{p})$ arising from the $p$-adic uniformization of the Shimura curve. We finally compute families of $p$-adic binary quadratic forms associated to an infinite fa…
Dirichlet's version of Gauss's reduction theory for indefinite binary quadratic forms includes a map from Gauss-reduced forms to strings of natural numbers. It attaches to a form the minimal period of the continued fraction of a quadratic irrationality associated with the form. When Zagier developed his own reduction theory, parallel to Dirichlet's, he omitted an analogue of this map. We define a new map on Zagier-reduced forms that serves as this analogue. We also define a map from the set of Gauss-reduced forms into the set of Zagier-reduced forms that gives a near-embedding of the structure…
We obtain some Bailey pairs associated with indefinite quadratic forms with the $β_n$ connected to a finite sum. A new general identity is given, which provides identities for $q$-hypergeometric series, including mock theta functions.
We prove an effective equidistribution theorem for orbits of horospherical subgroups of $\mathrm{SO}(2, 2)$ and $\mathrm{SO}(3, 1)$ in quotients of $\mathrm{SL}_4(\mathbb{R})$ with a polynomial error term. In a forthcoming paper, we will use this theorem to prove an effective version of the Oppenheim conjecture for indefinite quadratic forms of signature $(2, 2)$ or $(3, 1)$ with a polynomial error rate.
We obtain truncated restriction estimates of an unexpected form for discrete surfaces \begin{align} S = \{\, ( n_1 , \dots , n_d , R( n_1 , \dots, n_d ) ) \,,\, n_i \in [-N,N] \cap \mathbb{Z} \,\}, \end{align} where $R$ is an indefinite quadratic form with integer matrix.
We obtain estimates for the number of integral solutions in large balls, of inequalities of the form $|Q(x, y)| < ε$, where $Q$ is an indefinite binary quadratic form, in terms of the Hurwitz continued fraction expansions of the slopes of the lines on which $Q$ vanishes. The method is based on a coding of geodesics on the modular surface via Hurwitz expansions of the endpoints of their lifts in the Poincare half-plane.
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…
Markov's theorem classifies the worst irrational numbers with respect to rational approximation and the indefinite binary quadratic forms whose values for integer arguments stay farthest away from zero. The main purpose of this paper is to present a new proof of Markov's theorem using hyperbolic geometry. The main ingredients are a dictionary to translate between hyperbolic geometry and algebra/number theory, and some very basic tools borrowed from modern geometric Teichmüller theory. Simple closed geodesics and ideal triangulations of the modular torus play an important role, and so do the pr…
