Jordan Ellenberg, Akshay Venkatesh
We prove the local-global principle holds for the problem of representations of quadratic forms by quadratic forms, in codimension $\geq 7$. The proof uses the ergodic theory of $p$-adic groups, together with a fairly general observation on the structure of orbits of an arithmetic group acting on integral points of a variety.
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Employing Brüdern's and Wooley's new complification method, we establish an asymptotic Hasse principle for the number of solutions to a system of r_3 cubic and r_2 quadratic diagonal forms, when the number of cubic equations is at least double the number of quadratic forms. The number of variables required is lower than in earlier work. In particular, we achieve essentially square root cancellation for systems of one quadratic and arbitrarily many cubic equations.
After a brief introduction to the classical theory of binary quadratic forms we use these results for proving (most of) the claims made by Pépin in a series of articles on unsolvable quartic diophantine equations, and for constructing families of counterexamples to the Hasse Principle for curves of genus 1 defined by equations of the form $ax^4 + by^4 = z^2$.
Local-global principle for 11-isogenies of elliptic curves is true over quadratic fields
该研究通过确定模曲线$X_{D_{10}}(11)$上的二次点集合，证明二次域上椭圆曲线的11-同源局部-整体原理成立，正面解决了Banwait与Cremona的相关猜想。
Let k be a global field of characteristic not 2. The classical Hasse-Minkowski theorem states that if two quadratic forms become isomorphic over all the completions of k, then they are isomorphic over k as well. It is natural to ask whether this is also true for G-quadratic forms, where G is a finite group. In the case of number fields the Hasse principle for G-quadratic forms does not hold in general, as shown by Jorge Morales. The aim of this paper is to study this question when k is a global field of positive characteristic. We give a sufficient criterion for the Hasse principle to hold, an…
Anisotropy of quadratic forms over global fields of characteristic $\neq$ 2 is diophantine
本文证明了特征不等于2的全局域上各向异性二次型的集合是一个丢番图集，扩展了Koenigsmann的方法并结合类域论和局部-全局原理。
This paper presents an adaptation of recently developed algorithms for quadratic forms over number fields in arXiv:1304.0708 to global function fields of odd characteristics. First, we present algorithm for checking if a given non-degenerate quadratic form is isotropic or hyperbolic. Next we devise a method for computing the dimension of the anisotropic part of a quadratic form. Finally we present algorithms computing two field invariants: the level and the Pythagoras number.
Let $K$ be a number field, $f\in K[x]$ a quadratic polynomial, and $n\in\{1,2,3\}$. We show that if $f$ has a point of period $n$ in every non-archimedean completion of $K$, then $f$ has a point of period $n$ in $K$. For $n\in\{4,5\}$ we show that there exist at most finitely many linear conjugacy classes of quadratic polynomials over $K$ for which this local-global principle fails. By considering a stronger form of this principle, we strengthen global results obtained by Morton and Flynn-Poonen-Schaefer in the case $K=\mathbf Q$. More precisely, we show that for every quadratic polynomial $f\…
We show that the theorem of Ellenberg and Venkatesh on representation of integral quadratic forms by integral positive definite quadratic forms is valid under weaker conditions on the represented form.
Let $ K $ be a global function field of characteristic $ 2 $. For each non-trivial place $ v $ of $ K $, let $ K_{v} $ be the completion of $ K $ at $ v $. We show that if two non-degenerate quadratic forms are similar over every $ K_{v} $, then they are similar over $ K $. This provides an analogue of the version for characteristic not $ 2 $ previously obtained by T.Ono.
Strong Approximation and Hasse Principle for Integral Quadratic Forms over Affine Curves
本文利用 genus 理论研究仿射曲线坐标环上整二次型的表示问题，建立 spin 群强逼近与 Hasse 原理的联系，并给出强逼近失效的例子。
Let $q$ be a unimodular quadratic form over a field $K$. Pfister's famous local--global principle asserts that $q$ represents a torsion class in the Witt group of $K$ if and only if it has signature $0$, and that in this case, the order of Witt class of $q$ is a power of $2$. We give two analogues of this result to systems of quadratic forms, the second of which applying only to nonsingular pairs. We also prove a counterpart of Pfister's theorem for finite-dimensional $K$-algebras with involution, generalizing a result of Lewis and Unger.
Let $A$ and $A'$ be abelian varieties defined over a number field $k$ of dimension $g\geq 1$. For $g\leq 3$, we show that the following local-global principle holds: $A$ and $A'$ are quadratic twists of each other if and only if, for almost all primes $\mathfrak p$ of $k$ of good reduction for $A$ and $A'$, the reductions $A_{\mathfrak p}$ and $A'_{\mathfrak p}$ are quadratic twists of each other. This result is known when $g=1$, in which case it has appeared in works by Kings, Rajan, Ramakrishnan, and Serre. We provide an example that violates this local-global principle in dimension $g=4$.
Galois module structures and the Hasse principle in twist families via the distribution of Selmer groups
本文利用 Abel 簇稀疏二次扭族中 2-Selmer 秩的分布结果，统一研究 Mordell–Weil 群 Galois 模结构及 Hasse 原理失效等问题。
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.
Quantum Drinfeld Modules and Ray Class Fields of Real Quadratic Global Function Fields
本文通过量子 Drinfeld 模研究正特征实二次函数域的射线类域，证明 Weber-Fueter 定理的类比，并借助准晶体为对应数域问题提出猜想。
The Local-Global Conjecture is False for Generalized Circle Packings
本文扩展了Haag等人对Apollonian圆排列中局部-整体猜想的反驳，针对四种积分圆排列类型发现二次不变量，揭示了二次互反律对猜想的阻碍，同时展示了曲率的可预测互反结构。
Universal quadratic forms over semi-global fields
本文研究半全局域上的各向异性全型二次型，计算了其 $m$-不变量与维数集合，并定义了强 $m$-不变量，利用局部-全局原理证明其与强 $u$-不变量的类似性质。
We study the failure of a local-global principle for the existence of $l$-isogenies for elliptic curves over number fields $K$. Sutherland has shown that over $\mathbb{Q}$ there is just one failure, which occurs for $l=7$ and a unique $j$-invariant, and has given a classification of such failures when $K$ does not contain the quadratic subfield of the $l$'th cyclotomic field. In this paper we provide a classification of failures for number fields which do contain this quadratic field, and we find a new `exceptional' source of such failures arising from the exceptional subgroups of $\mbox{PGL}_…
The Kloosterman circle method and weighted representation numbers of quadratic forms
本文发展了一种带凸函数的Kloosterman圆法，为非奇异整二次型的加权表示数提供显式常数的渐近公式，并探讨其在Kleinian球堆积强局部-全局原理证明中的潜在应用。
We prove the Hasse principle and weak approximation for varieties defined over number fields by the nonsingular intersection of pairs of quadratic forms in 8 variables. The argument develops work of Colliot-Thelene, Sansuc and Swinnerton-Dyer, and centres on a purely local problem about forms which split off 3 hyperbolic planes.
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…
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.
Geometrically simple counterexamples to a local-global principle for quadratic twists
针对二次扭局部-整体原理的反例非几何单的问题，研究者利用伽罗瓦上同调与类域论，构造出满足$p\equiv13\pmod{24}$的素数$p$对应的维数为$p-1$的$\mathbb{Q}$上几何单阿贝尔簇反例对。
Let $R$ be a 2-dimensional normal excellent henselian local domain in which 2 is invertible and let $L$ and $k$ be respectively its fraction field and residue field. Let $Ω_R$ be the set of rank 1 discrete valuations of $L$ corresponding to codimension 1 points of regular proper models of $\Spec R$. We prove that a quadratic form $q$ over $L$ satisfies the local-global principle with respect to $Ω_R$ in the following two cases: (1) $q$ has rank 3 or 4; (2) $q$ has rank $\ge 5$ and $R=A[y]$, where $A$ is a complete discrete valuation ring with a not too restrictive condition on the residue fiel…
Let $L$ be a separable quadratic extension of either $\mathbb{Q}$ or $\mathbb{F}_q(t)$. We propose efficient algorithms for finding isomorphisms between quaternion algebras over $L$. Our techniques are based on computing maximal one-sided ideals of the corestriction of a central simple $L$-algebra.
Hasse principles for quadratic forms over function fields
该文研究有限生成域扩张上二次型迷向、等距的Hasse原理，给出小离散赋值集下纯超越扩张的大量迷向Hasse原理反例，并基于已有结果构造了除子赋值下更低维度的反例。
For quadratic forms in $4$ variables defined over the rational function field in one variable over $\mathbb C(\!(t)\!)$, the validity of the local-global principle for isotropy with respect to different sets of discrete valuations is examined.
Let $K/k$ be an extension of number fields, and let $P(t)$ be a quadratic polynomial over $k$. Let $X$ be the affine variety defined by $P(t) = N_{K/k}(\mathbf{z})$. We study the Hasse principle and weak approximation for $X$ in three cases. For $[K:k]=4$ and $P(t)$ irreducible over $k$ and split in $K$, we prove the Hasse principle and weak approximation. For $k=\mathbb{Q}$ with arbitrary $K$, we show that the Brauer-Manin obstruction to the Hasse principle and weak approximation is the only one. For $[K:k]=4$ and $P(t)$ irreducible over $k$, we determine the Brauer group of smooth proper mod…
Hardy and Littlewood's approximate functional equation for quadratic Weyl sums (theta sums) provides, by iterative application, a powerful tool for the asymptotic analysis of such sums. The classical Jacobi theta function, on the other hand, satisfies an exact functional equation, and extends to an automorphic function on the Jacobi group. In the present study we construct a related, almost everywhere non-differentiable automorphic function, which approximates quadratic Weyl sums up to an error of order one, uniformly in the summation range. This not only implies the approximate functional equ…
We prove a local-global principle for twisted flag varieties over a semiglobal field.
We investigate local-global principles for Galois cohomology, in the context of function fields of curves over semi-global fields. This extends work of Kato's on the case of function fields of curves over global fields.
The analytic Hasse Principle for certain singular intersections of quadrics in $\mathbb{P}^9$
本文利用自守方法，证明了在特定奇点条件下至少含10个变量的有理系数二次型对公共零点个数的渐近公式，推广了 Browning 与 Munshi 的工作。
We prove a local-global principle for torsors under the prosolvable geometric fundamental group of an affine curve over a number field.
Let $K$ be a global field of positive characteristic. We prove that the Brauer-Manin obstructions to the Hasse principle, to weak approximation and to strong approximation are the only ones for homogeneous spaces of reductive groups with reductive stabilizers. The methods involve abelianization techniques and arithmetic duality theorems for complexes of tori over K.
We prove a local-global principle for torsors under the prosolvable geometric fundamental group of a hyperbolic curve over a number field.
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