Sara Checcoli, Arno Fehm
We construct infinite Galois extensions $K$ of $\mathbb{Q}$ that satisfy the Northcott property on elements of small height, and where this property can be deduced solely from the splitting behavior of prime numbers in $K$. We also give examples of Galois extensions of $\mathbb{Q}$ which have finite local degree at all prime numbers and do not satisfy the Northcott property.
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We give three examples of fields concerning the Northcott property on elements of small height: The first one has the Northcott property but its Galois closure does not satisfy the Bogomolov property. The second one has the Northcott property and is pseudo-algebraically closed, i.e. every variety has a dense set of rational points. The third one has bounded local degree at infinitely many rational primes but does not have the Northcott property.
On certain infinite extensions of the rationals with Northcott property
一组代数数具有Northcott性质，如果它的每个有界Weil高度的子集是有限的。Northcott定理，有许多丢番图应用，指出有界度的集合具有Northcott性质。Benneri，Dvornicich和Zannier提出了问题，找到领域的无限程度与此属性。Bubieri和Zannier证明了由次数至多为$d$的所有代数数生成的域的最大交换子域$\IQ_{ab}^{(d)}$就是这样的域。本文给出了Northcott性质的一个简单判据，作为应用，我们推出了几个新的例子，例如$\IQ(2^{1/d_1},3^{1/d_2},5^{1/d_3},7^{1/d_4},11^{1/d_5},...)$具有Northcott性质当且仅当$2^{1/d_1},3^{1/d_2},5^{1/d_3},7^{1/d_4},11^{1/d_5},...$趋于无穷大.
On Widmer's criteria for the Northcott property
本文简化了Widmer关于数域无限代数扩张中Northcott性质的新判据，在阿贝尔扩张情形下给出更简洁形式，并据此构造出满足Northcott性质但不满足Widmer判据的域，同时简化了伪代数闭域的构造。
On the Northcott property and other properties related to polynomial mappings
证明了：若$K/\mathbb{Q}$是有限指数的Galois扩张，且$K^{(d)}$是$K$的所有次至多为$d$的扩张的合成，则$K^{(d)}$具有Bogomolov性质，而$K^{(d)}/\mathbb{Q}$的极大阿贝尔子扩张具有Northcott性质.证明了给定任意有限可解群序列$\{G_m\}_m$，存在具有$\text{Gal}(K_m/\mathbb{Q})=G_m$的Galois扩张序列$\{K_m\}_m$，使得域$K_m$的合成具有Northcott性质.特别是，我们提供的例子与一致有界的地方度，但不包含在$\mathbb{Q}^{(d)}$的Northcott属性字段。我们还讨论了Liardet和Narkiewicz为研究多项式映射而引入的性质所涉及的一些问题。利用Northcott性质的结果和Dvornicich和Zannier的结果，我们很容易推导出Narkiewicz提出的一些开放问题的答案。
The Northcott property for special values of Dedekind zeta functions and more general motivic $L$-functions was defined by Pazuki and Pengo. We investigate this property for any complex evaluation of Dedekind zeta functions. The results are more delicate and subtle than what was proven for the function field case in previous work of Li and the authors, since they include some surprising behavior in the neighborhood of the trivial zeros. The techniques include a mixture of analytic and computer assisted arguments.
The Northcott Property for Composites of Number Fields of Bounded Degree
本文通过应用Segal关于profinite群的定理，证明了Galois群指数有限的无限Galois数域扩张具有Northcott性质。
Pazuki and Pengo defined a Northcott property for special values of zeta functions of number fields and certain motivic $L$-functions. We determine the values for which the Northcott property holds over function fields with constant field $\mathbb{F}_q$ outside the critical strip. We then use a case by case approach for some values inside the critical strip, notably $Re (s) < \frac{1}{2} - \frac{\log 2}{\log q}$ and for $s$ real such that $1/2 \leq s \leq 1$, and we obtain a partial result for complex $s$ in the case $1/2< Re(s)\leq 1$ using recent advances on the Shifted Moments Conjecture ov…
Most totally real fields do not have universal forms or Northcott property
本文证明了在配备可构造拓扑的全实域空间中，具有泛型二次型或Northcott性质的域集合是贫集，核心工具是关于二次格表示全正单位平方类数量的新定理。
Northcott property and universality of higher degree forms
本文证明了对于全实数域上的高次型，满足普遍性的全实扩张至多有限个，且在具有Northcott性质的全实无限扩张上不存在普遍型。
In this paper, for an elliptic curve $E$ defined over the algebraic numbers and for any subfield $F$ of algebraic numbers, we say that $E$ has the Northcott property over $F$ if there are at most finitely many $F$-rational points on $E$ of uniformly bounded height, and we say that $E$ has the property (P) over $F$ if for any infinite subset $S$ of $F$-rational points on $E$, $f(S) = S$ for an $F$-endomorphism $f$ of $E$ implies that $f$ is an automorphism. We establish some criteria for both properties and provide typical examples. We also show that the Northcott property implies the property …
On the Northcott property for infinite extensions
本文回顾了代数数子域的 Northcott 性质，提出改进的新判据以解决 Bombieri 的问题，并据此轻松推导出 Bombieri 和 Zannier 关于极大阿贝尔扩张具有该性质的定理。
Universal quadratic forms and Northcott property of infinite number fields
本文证明无限次全实扩张上泛二次型的存在性与全正整数集合的 Northcott 性质不相容，并据此推出在固定素数次全实 Galois 域的合成域及次数为 3d 的合成域上不存在经典泛二次型。
The Geometric Dynamical Northcott Property in the quadratic family
本研究针对二次族的简单情形，利用其动力与参数空间均为1维、参数化简单显式的特点，简化证明了几何Northcott性质相关的定理A，使核心思路更明晰。
In this work we prove a new Northcott property for the Faltings height. Namely we show, assuming the Colmez Conjecture and the Artin Conjecture, that there are finitely many CM abelian varieties over the complex numbers of a fixed dimension which have bounded Faltings height. The technique developed uses new tools from integral p-adic Hodge theory to study the variation of Faltings height within an isogeny class of CM abelian varieties. In special cases, we are moreover able to use the technique to develop new Colmez-type formulas for the Faltings height.
We establish the finiteness of periodic points, that we called Geometric Dynamical Northcott Property, for regular polynomials automorphisms of the affine plane over a function field $\mathbf{K}$ of characteristic zero, improving results of Ingram. For that, we show that when $\mathbf{K}$ is the field of rational functions of a smooth complex projective curve, the canonical height of a subvariety is the mass of an appropriate bifurcation current and that a marked point is stable if and only if its canonical height is zero. We then establish the Geometric Dynamical Northcott Property using a si…
On a local property of infinite Galois extensions implying the Northcott property
本文证明 Bombieri–Zannier 提出的蕴含 Northcott 性质的局部性质在有限扩张下不被保持，且不能从无穷伽罗瓦扩张的伽罗瓦群上读出，并构造了相应反例。
On the Northcott property for special values of L-functions
本文提出研究 L-函数特殊值的 Northcott、Bogomolov 和 Lehmer 性质，建立公理化框架，并证明纯动机 L-函数在临界带左侧具有 Northcott 性质，同时对数域 Dedekind zeta 函数在特殊值 0 和 1 处给出成立与不成立的判别及定量估计。
For an algebraic number $α$ and $γ\in \mathbb{R}$, $h(α)$ be the (logarithmic) Weil height, and $h_γ(α)=(\mathrm{deg}α)^γh(α)$ be the $γ$-weighted (logarithmic) Weil height of $α$. Let $f:\overline{\mathbb{Q}}\to [0,\infty)$ be a function on the algebraic numbers $\overline{\mathbb{Q}}$, and let $S\subset \overline{\mathbb{Q}}$. The Northcott number $\mathcal{N}_f(S)$ of $S$, with respect to $f$, is the infimum of all $X\geq 0$ such that $\{α\in S; f(α)< X\}$ is infinite. This paper studies the set of Northcott numbers $\mathcal{N}_f(\mathcal{O})$ for subrings of $\overline{\mathbb{Q}}$ for th…
The Geometric Dynamical Northcott and Bogomolov Properties
本研究在特征零函数域上建立极化自同态的动力Northcott性质，关联复动力系统稳定性，推广已有一维与高维结果，还提出几何Bogomolov猜想的动力版本。
Height lower bounds for elements of highly composite rings
本文针对$\text{Q}^{(d)}$的整数环合成环元素的高度下界展开研究，证明$\text{Q}^{(3)}$对应的合成环具有Northcott性质，并引入子域大小概念，给出相关子域的大小与Northcott性质的结果。
A note on some Diophantine inequalities over adelic curves
本文在不假设Northcott性质的条件下，为proper adelic曲线上一类作为Roth不等式加强版的丢番图不等式组建立了“大解”数量的上界。
By Northcott's Theorem there are only finitely many algebraic points in affine $n$-space of fixed degree over a given number field and of height at most $X$. For large $X$ the asymptotics of these cardinalities have been investigated by Schanuel, Schmidt, Gao, Masser and Vaaler, and the author. In this paper we study the case where the coordinates of the points are restricted to algebraic integers, and we derive the analogues of Schanuel's, Schmidt's, Gao's and the author's results.
Fields with few small points
本文证明 Northcott 数充分大的代数扩张上，射影簇中次数有界且定义于该域的循环有限，并给出 Abel 簇、射影空间自映射及 CM 点应用。
On a Galois property of fields generated by the torsion of an abelian variety
本文证明数域上阿贝尔簇挠点生成域的伽罗瓦子扩张（伽罗瓦群指数有限）必含于某有限扩张的阿贝尔扩张中，并由此推出这些域具有 Northcott 性质。
The Bogomolov Property through Galois Representations
本文综述了利用伽罗瓦表示研究代数扩张的Bogomolov性质的方法，包括局部p进信息的作用和Sen定理的应用，并介绍了与Amoroso等人的合作成果。
Canonical height functions on the affine plane associated with polynomial automorphisms
设$f: \mathbb{A}^2 \to \mathbb{A}^2$是数域$K$上动力次数为$δ\geq 2$的多项式自同构. (This等价于说$f$是不可三角化的多项式自同构。）然后，我们构造了与$f$相关的定义在$\mathbb{A}^2(\bar{K})$上的正则高度函数。这些函数满足Northcott有限性，并且$\mathbb{A}^2(\bar{K})$上的$\bar{K}$值点是$f$-周期的当且仅当它的高度为零.作为正则高度函数的一个应用，我们给出了无限$f$-轨道上有界高度点的个数的一个估计。
Let $F(s)=\sum_n a_n/λ_n^s$ be a general Dirichlet series which is absolutely convergent on $\Re(s)>1$. Assume that $F(s)$ has an analytic continuation and satisfies a growth condition, which gives rise to certain invariants namely the degree $d_F$ and conductor $α_F$. In this paper, we show that there are at most $2d_F$ general Dirichlet series with a given degree $d_F$, conductor $α_F$ and residue $ρ_F$ at $s=1$. As a corollary, we get that elements in the extended Selberg class with positive Dirichlet coefficients are determined by their degree, conductor and the residue at $s=1$.
We prove that two finite endomorphisms of the unit disk with degree at least two have orbits with infinite intersections if and only if they have a common iteration.
Hilbert property for low-genus families and degree-one del Pezzo surfaces
本文首次研究了无锥面纤维的最小一度del Pezzo曲面的希尔伯特性质，通过推广椭圆曲面结果并利用特定构造证明有理点密度。
Due to Narkiewicz a field $F$ has property (P) if for no polynomial $f\in F[x]$ of degree at least two there is an infinite $f$-invariant subset of $F$. We present a new example of an algebraic extension of $\mathbb{Q}$ satisfying (P). This is the first example in which we can find points of arbitrarily small positive Weil-height. Moreover, we study the possibility of property (P) for the field generated by all symmetric Galois extensions of $\mathbb{Q}$. In particular we prove that there are no infinite backward orbits of non linear polynomials in this field.
Hopf Galois theory for finite separable field extensions was introduced by Greither and Pareigis. They showed that all Hopf Galois extensions of degree up to 5 are either Galois or almost classically Galois and they determined the Hopf Galois character of a separable extension according to the Galois group (or the degree) of its Galois closure. In this paper we study degree 6 separable extensions as well as intermediate extensions for degrees 4,5 and 6. We present an example of a non almost classically Galois Hopf Galois extension of the field of rational numbers of the smallest possible degre…
Let $p$ be a prime and $K$ a number field of degree $p$. We count the number of elliptic curves, up to $\bar{K}$-isomorphism, having a prescribed property, where this property is either that the curve contains a fixed torsion group as a subgroup, or that it has an isogeny of prescribed degree.
Weak weak approximation and the Hilbert property for degree-two del Pezzo surfaces
本文证明含特定有理点的二次 del Pezzo 曲面在数域上满足弱弱逼近，在 Hilbert 域上满足 Hilbert 性质，基于单有理性和圆锥纤维结果。
We say that a field extension $L/F$ has the descent property for isometry (resp. similarity) of quadratic or symmetric bilinear forms if any two forms defined over $F$ that become isometric (resp. similar) over $L$ are already isometric (resp. similar) over $F$. The famous Artin-Springer theorem states that anisotropic quadratic or symmetric bilinear forms over a field stay anisotropic over an odd degree field extension. As a consequence, odd degree extensions have the descent property for isometry of quadratic as well as symmetric bilinear forms. While this is well known for nonsingular quadr…
Given an elliptic surface over a number field, we study the collection of fibres whose Mordell-Weil rank is greater than the generic rank. Under suitable assumptions, we show that this collection is not thin. Our results apply to quadratic twist families and del Pezzo surfaces of degree $1$.
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