Rainer Schulze-Pillot
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.
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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。
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.
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 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.
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 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\…
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。
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。
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。
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.
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…
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 $g=4$.
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 prove by an elementary method the Riemann hypothesis for the local Euler factor of the zeta function of quadratic orders.
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…
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 this paper, we study the representations of integral quadratic polynomials. Particularly, it is shown that there are only finitely many equivalence classes of positive ternary universal integral quadratic polynomials, and that there are only finitely many regular ternary triangular forms. A more general discussion of integral quadratic polynomials over a Dedekind domain inside a global field is also given.
In this paper, we prove several results on finitely generated dynamical Galois groups attached to quadratic polynomials. First we show that, over global fields, quadratic post-critically finite polynomials are precisely those having an arboreal representation whose image is topologically finitely generated. To obtain this result, we also prove the quadratic case of Hindes' conjecture on dynamical non-isotriviality. Next, we give two applications of this result. On the one hand, we prove that quadratic polynomials over global fields with abelian dynamical Galois group are necessarily post-criti…
On classic $n$-universal quadratic forms over dyadic local fields
本文研究了二进制局部域上经典n-通用二次型的等价条件和最小测试集，确定了其数学性质和表示能力。
Local densities of quadratic polynomials over non-dyadic fields and unramified dyadic fields
本文为非二进域和非分歧二进域上的二次多项式建立了局部密度的精确公式。
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.
We prove a non-minimal modularity lifting theorem for ordinary Galois representations over imaginary quadratic fields, conditional on a local-global compatibility conjecture for ordinary torsion classes.
T. Ikeda and H. Katsurada have developed the theory of the Gross-Keating invariant of a quadratic form in their recent papers [IK1] and [IK2]. In particular, they prove that the local factor of the Fourier coefficients of the Siegel-Eisenstein series is completely determined by the Gross-Keating invariant with extra datum, called the extended GK datum, in [IK2]. On the other hand, such local factor is a special case of the local densities for a pair of two quadratic forms. Thus we propose a general question if the local density can be determined by certain series of the Gross-Keating invariant…
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 $\mbox{PGL}_…
Geometrically simple counterexamples to a local-global principle for quadratic twists
针对二次扭局部-整体原理的反例非几何单的问题，研究者利用伽罗瓦上同调与类域论，构造出满足$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…
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.
Local heights on hyperelliptic curves and quadratic Chabauty
本研究提出一种计算奇素数ℓ≠p处超椭圆曲线局部高度的算法，拓宽了二次Chabauty方法的适用范围，并给出含双非平凡局部高度素数曲线的首次计算实例。
We extend field patching to the setting of Berkovich analytic geometry and use it to prove a local-global principle over function fields of analytic curves. We apply this result to quadratic forms, and combine it with sufficient conditions for local isotropy over a Berkovich curve to obtain applications on the u-invariant. The patching method we adapt was introduced by Harbater and Hartmann, and further developed by these two authors and Krashen. This paper generalizes their results on the local-global principle and quadratic forms.
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