Harry Schmidt
We give a short proof of Manin-Mumford in the multiplicative group based on the pigeon-hole principle and the so-called structure theorem for anomalous subvarieties. The arguments appear to be new and perhaps applicable in other situations.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
The Manin-Mumford conjecture in characteristic zero was first proved by Raynaud. Later, Hrushovski gave a different proof using model theory. His main result from model theory, when applied to abelian varieties, can be rephrased in terms of algebraic geometry. In this paper we prove that intervening result using classical algebraic geometry alone. Altogether, this yields a new proof of the Manin-Mumford conjecture using only classical algebraic geometry.
We show that Ribet sections are the only obstruction to the validity of the relative Manin-Mumford conjecture for one dimensional families of semi-abelian surfaces. Applications include special cases of the Zilber-Pink conjecture for curves in a mixed Shimura variety of dimension four, as well as the study of polynomial Pell equations with non-separable discriminants.
In the article [PR1] {\it On Hrushovski's proof of the Manin-Mumford conjecture} (Proceedings of the ICM 2002), R. Pink and the author gave a short proof of the Manin-Mumford conjecture, which was inspired by an earlier model-theoretic proof by Hrushovski. The proof given in [PR1] uses a difficult unpublished ramification-theoretic result of Serre. It is the purpose of this note to show how the proof given in [PR1] can be modified so as to circumvent the reference to Serre's result. J. Oesterlé and R. Pink contributed several simplifications and shortcuts to this note.
We give a short proof of the "prime-to-$p$ version" of the Manin-Mumford conjecture for an abelian variety over a number field, when it has supersingular reduction at a prime dividing $p$, by combining the methods of Bogomolov, Hrushovski, and Pink-Roessler. Our proof here is quite simple and short, and neither $p$-adic Hodge theory nor model theory is used. The observation is that a power of a lift of the Frobenius element at a supersingular prime acts on the prime-to-$p$ torsion points via nontrivial homothety.
Manin-Mumford in arithmetic pencils
本文将Manin-Mumford定理推广到整数环上的阿贝尔概形，用挠点Galois轨道模素数的特殊0-圈替代挠点，并刻画了其平坦部分与垂直部分的不可约分量结构。
We present a new proof of the Manin-Mumford conjecture about torsion points on algebraic subvarieties of abelian varieties. Our principle, which admits other applications, is to view torsion points as rational points on a complex torus and then compare (i) upper bounds for the number of rational points on a transcendental analytic variety (Bombieri-Pila-Wilkie) and (ii) lower bounds for the degree of a torsion point (Masser), after taking conjugates. In order to be able to deal with (i), we discuss (Thm. 2.1) the semi-algebraic curves contained in an analytic variety supposed invariant for tra…
We prove a version of the Manin-Mumford conjecture for semiabelian varieties over fields of positive characteristic. The proof presented here contains the details of the proof sketched by the author in the article "Diophantine geometry from model theory," BSL 7 (2001) no. 1, 37 - 57; using the model theory of difference fields along the lines of Hrushovski. Algebraic proofs have been presented by Pink & Roessler and Pillay.
We prove Zhang's Dynamical Manin-Mumford Conjecture and Dynamical Bogomolov Conjecture for dominant endomorphisms of (P^1)^n. We use the equidistribution theorem for points of small height with respect to an algebraic dynamical system, combined with an analysis of the symmetries of the Julia set for a rational function.
The Relative Manin-Mumford Conjecture
本文在特征0下证明阿贝尔簇族的相对马宁-芒福德猜想，采用皮拉-赞尼尔方法，结合新的高度不等式与退化轨迹研究，还强化结果给出复数域上阿贝尔概法子簇挠点稠密判据，并给出一致马宁-芒福德猜想的无等分布新证明。
On the dynamical Manin-Mumford conjecture for plane polynomial maps
在特征0域上，证明了正则多项式映射的动力Manin-Mumford猜想对避开无穷远处超吸引轨道的不可约曲线成立。
We prove that in positive characteristic, the Manin-Mumford conjecture implies the Mordell-Lang conjecture, in the situation where the ambient variety is an abelian variety defined over the function field of a smooth curve over a finite field and the relevant group is a finitely generated group. In particular, in the setting of the last sentence, we provide a proof of the Mordell-Lang conjecture, which does not depend on tools coming from model theory.
On the Manin-Mumford Theorem for Algebraic Groups
本文将 Manin-Mumford 猜想推广至任意连通代数群，描述了挠点集的 Zariski-闭包，定义了与 Néron-Tate 高度一致的标准高度，并证明了对数域上任意连通代数群的 Bogomolov 猜想推广。
We prove the Dynamical Bogomolov Conjecture for endomorphisms of P^1\times P^1 defined over a number field. We use the equidistribution theorem for points of small height with respect to an algebraic dynamical system, combined with a theorem of Levin regarding symmetries of the Julia set. Using a specialization theorem of Yuan and Zhang, we prove the Dynamical Manin-Mumford Conjecture for endomorhisms of P^1\times P^1 defined over the complex numbers.
We present a reduction of the function field Mordell-Lang conjecture to the function field Manin-Mumford conjecture, in all characteristics, via model theory, but avoiding recourse to the dichotomy theorems for (generalized) Zariski structures. In this version 2, the quantifier elimination result in positive characteristic is extended from simple abelian varieties to all abelian varieties, completing the main theorem in the positive characteristic case. In version 3, some corrections are made to the proof of quantifier elimination in positive characteristic, and the paper is substantially reor…
In this paper we prove a special case of the André-Oort conjecture for Kuga varieties. If $M$ is a Kuga variety fibred over a pure Shimura variety $S$ as an abelian scheme, and $(M_n)$ is a sequence of special subvarieties in $M$ which are faithfully flat over $S$, then the Zariski closure of the union of the $M_n$'s is a finite union of special subvarieties faithfully flat over $S$. The proof is reduced to a variant of the Manin-Mumford conjecture for abelian schemes.
Our results concern analytic functions on the open unit $p$-adic poly-disc in $\mathbb{C}^n_p$ centered at the multiplicative unit and we prove that such functions only vanish at finitely many $n$-tuples of roots of unity $(ζ_1-1,\ldots,ζ_n-1)$ unless they vanish along a translate of the formal multiplicative group. For polynomial functions, this follows from the multiplicative Manin-Mumford conjecture. However we allow for a much wider class of analytic functions; in particular we establish a rigidity result for formal tori. Moreover, our methods apply to Lubin-Tate formal groups beyond just …
A uniform quantitative Manin-Mumford theorem for curves over function fields
本文提出一个关于函数域上曲线的统一定量Manin-Mumford定理，利用Zhang配对、算术Hodge指数定理和Elkies下界，证明了曲线在雅可比嵌入中的torsion点数量上限及Bogomolov型结果。
We prove $p$-adic versions of a classical result in arithmetic geometry stating that an irreducible subvariety of an abelian variety with dense torsion has to be the translate of a subgroup by a torsion point. We do so in the context of certain rigid analytic spaces and formal groups over a $p$-adic field $K$ or its ring of integers $R$, respectively. In particular, we show that the rigidity results for algebraic functions underlying the so-called Manin-Mumford Conjecture generalize to suitable $p$-adic analytic functions. In the formal setting, this approach leads us to uncover purely $p$-adi…
We study the intersection of an algebraic variety with the maximal compact subgroup of a universal vectorial extension of a product of elliptic curves. For this intersection we show a Manin-Mumford type statement. This answers some questions posed by Corvaja-Masser-Zannier which arose in connection with their investigation of the intersection of a curve with real analytic subgroups of various algebraic groups. They prove finiteness in the situation of a single elliptic curve. Using Khovanskii's zero-estimates combined with a stratification result of Gabrielov-Vorobjov and recent work of the au…
The aim of this work is to present a possible adaptation of the Manin-Mumford conjecture to the $T-$modules, a mathematical object which has been introduced in the 1980's by G. Anderson as the natural analogue of the abelian varieties in the context of modules over rings which are contained in positive characteristic function fields. We propose then a generalisation of such an adapted conjecture to a modified general version of Mordell-Lang conjecture for $T-$modules which might correct the one proposed for the first time by L. Denis but no longer compatible with the present results.
Let $A$ be a simple abelian surface over an algebraically closed field $k$. Let $S\subset A(k)$ be the set of torsion points $x$ of $A$ such that there exists a genus $2$ curve $C$ and a map $f: C\to A$ such that $x$ is in the image of $f$, and $f$ sends a Weierstrass point of $C$ to the origin of $A$. The purpose of this note is to show that if $k$ has characteristic zero, then $S$ is finite -- this is in contrast to the situation where $k$ is the algebraic closure of a finite field, where $S=A(k)$, as shown by Bogomolov and Tschinkel. We deduce that if $k=\bar{\mathbb{Q}}$, the Kummer surfac…
A few remarks on a Manin-Mumford conjecture in function field arithmetic and generalized Pila-Wilkie estimates
本文将Pila-Wilkie型有理点计数估计推广到任意特征局部域上的$K^n$子集，整合了特征0下的相关估计，完善了函数域算术的Manin-Mumford猜想相关研究。
The aim of this paper is to prove a theorem of Ax-Lindemann type for complex semi-abelian varieties as an application of a big Picard theorem proved by the author in 1981, and then apply it to prove a theorem of classical Manin-Mumford Conjecture for semi-abelian varieties, which was proved by M. Raynaud 1983, M. Hindry 1988, ....., and Pila-Zannier 2008 by a different method from others, which is most relevant to our's. The present result might be a first instance of a direct connection between the value distribution theory of holomorphic maps and the arithmetic (Diophantine) theory over alge…
We introduce a general strategy for proving quantitative and uniform bounds on the number of common points of height zero for a pair of inequivalent height functions on $\mathbb{P}^1(\overline{\mathbb{Q}}).$ We apply this strategy to prove a conjecture of Bogomolov, Fu, and Tschinkel asserting uniform bounds on the number of common torsion points of elliptic curves in the case of two Legendre curves over $\mathbb{C}$. As a consequence, we obtain two uniform bounds for a two-dimensional family of genus 2 curves: a uniform Manin-Mumford bound for the family over $\mathbb{C}$, and a uniform Bogom…
Generic Manin-Mumford
本文研究代数数集合的乘积在簇中的Zariski稠密性，证明广义拉盖尔多项式根等经典族满足不可能交点定理，并推广了Ihara、Serre、Tate定理及Manin-Mumford猜想。
The Manin conjecture for toric stacks
本文证明分裂环面叠上关于反典范提升线丛的高度有界有理点计数满足 Deligne–Mumford 叠版本的 Manin 猜想。
The paper gives a counter-example to the relative version of the Manin-Mumford conjecture.
Let $f$ be a polynomial automorphism of the affine plane. In this paper we consider the possibility for it to possess infinitely many periodic points on an algebraic curve $C$. We conjecture that this happens if and only if $f$ admits a time-reversal symmetry; in particular the Jacobian $\mathrm{Jac}(f)$ must be a root of unity. As a step towards this conjecture, we prove that the Jacobian of $f$ and all its Galois conjugates lie on the unit circle in the complex plane. Under mild additional assumptions we are able to conclude that indeed $\mathrm{Jac}(f)$ is a root of unity. We use these resu…
We generalize the results of Skorobogatov and Zarhin considering the commutativity of Brauer groups (and Brauer-Manin sets) with taking product of two varieties, by relaxing the condition that varieties are projective.
The Brauer-Manin obstruction on algebraic stacks
本文研究代数堆上的Brauer-Manin集和配对，扩展了Colliot-Thélène和Sansuc的降理论，证明了局部由变分线性群商的堆的Brauer群的 torsion 性，并展示了Brauer-Manin障碍与其它上同调障碍的一致性。
We study differential forms on the universal vector extension $A^\natural$ of an abelian scheme $A$ in characteristic zero, and derive a new construction of the $D$-group scheme structure on $A^\natural$. This gives, in particular, a rather simple description of the Gauss--Manin connection on the de Rham cohomology of $A$ in terms of global algebraic differential forms on $A^\natural$. The key ingredient is the computation of the coherent cohomology of $A^{\natural}$, due to Coleman and Laumon.
Rational points on varieties and the Brauer-Manin obstruction
本讲义介绍了有理点存在性的Brauer-Manin障碍，重点探讨理论与计算之间的相互作用。
These lecture notes describe the current state of affairs for Manin's conjecture in the context of del Pezzo surfaces.
The Mumford-Tate conjecture is first proved for CM abelian varieties by H. Pohlmann [Ann. Math., 1968]. In this note we give another proof of this result and extend it to CM motives.
Let A be an abelian variety of dimension g defined over a number field K. We study the size of the torsion group A(F)_{tors} where F/K is a finite extension and more precisely we study the possible exponent γin the inequality Card(A(F)_{tors})<< [F:K]^γ when F is any extension of K. In the C.M. case we give an exact formula for the best possible exponent in terms of the characters of the Mumford-Tate group--a torus in this case--and discuss briefly the general case. Finally we give applications of this result in direction of a conjecture of Rémond generalising the Manin-Mumford conjecture.
The geometry of preperiodic points in families of maps on $\mathbb{P}^N$
本文研究了复数域上代数映射族的预周期点几何，提出一个猜想性特征化，结合了代数簇相对 Manin-Mumford 猜想和动力学 Manin-Mumford 猜想，并证明了其中的一个蕴含关系。
This note gives an explicit example of transcendental Brauer-Manin obstruction to weak approximation. It has two features which the only previously known example of such obstruction did not have: the class in the Brauer group which is responsible for the obstruction is divisible, and the underlying algebraic variety is an elliptic surface (a K3 surface).
$p$-adic approaches to unlikely intersections
本文介绍 Buium--Coleman 方法证明 Manin--Mumford 猜想及其推广，并以 $p$-adic 方法证明 Bombieri-Masser-Zannier 定理为例。
The generalized Mordell-Lang conjecture (GML) is the statement that the irreducible components of the Zariski closure of a subset of a group of finite rank inside a semi-abelian variety are translates of closed algebraic subgroups. M. McQuillan gave a proof of this statement. We revisit his proof, indicating some simplifications. This text contains a complete elementary proof of the fact that (GML) for groups of torsion points (= generalized Manin-Mumford conjecture), together with (GML) for finitely generated groups imply the full generalized Mordell-Lang conjecture.
The Zilber--Pink conjecture for products of curves with highly degenerate reduction
研究曲线\(X\)的\(n\)重自乘积在其雅可比自乘积内的齐尔伯 - 平克猜想，在\(X\)有适当坏约化等条件下，依据卡茨等人工作给出证明。
Complex solutions of polynomial equations on the unit circle
本文研究单位圆上多项式方程的复数解，给出扎里斯基稠密判定准则并证明可判定性，作为曼宁-芒福德猜想的连续类比，结果与之不同，且证明初等，属阐述性文章。
B. Poonen recently produced smooth threefolds over a number field which do not have a rational point but have no Brauer-Manin obstruction even after descent to a finite 'etale cover. In this note I show that the varieties he produces have zero-cycles of degree 1. ----- B. Poonen a r'ecemment exhib'e des exemples de vari'et'es projectives et lisses de dimension 3 sur un corps de nombres qui n'ont pas de point rationnel et pour lesquelles il n'y a pas d'obstruction de Brauer-Manin apr`es rev^etement fini 'etale. Dans cette note, je montre que les vari'et'es qu'il construit poss`edent des z'ero-c…
On K-stability, height bounds and the Manin-Peyre conjecture
本文探讨复K-半稳定Fano簇高度界与Peyre猜想及Elsenhans-Jahnel上界的关系，提出实点高度不等式类比，并在实射影直线上建立，关联Kähler-Einstein度量。
Rational points over C1 fields
本文简要评述 Manin 1966 年关于完美域上有理曲面的论文及其猜想和后续发展，并给出相关背景与研究文献。
In this note we search the parameter space of Horrocks-Mumford quintic threefolds and locate a Calabi-Yau threefold which is modular, in the sense that the L-function of its middle-dimensional cohomology is associated to a classical modular form of weight 4 and level 55.
In this paper, we survey some Galois-theoretic techniques for studying torsion points on curves. In particular, we give new proofs of some results of A. Tamagawa and the present authors for studying torsion points on curves with "ordinary good" or "ordinary semistable" reduction at a given prime. We also give new proofs of: (1) The Manin-Mumford conjecture: There are only finitely many torsion points lying on a curve of genus at least 2 embedded in its Jacobian by an Albanese map; and (2) The Coleman-Kaskel-Ribet conjecture: If p is a prime number which is at least 23, then the only torsion po…
In this note, we give a necessary and sufficient condition for determining which integers can be written as a sum of two integral squares for certain quadratic fields by using the integral Brauer-manin obstruction (see \cite{CTX}). The condition is computable and originally from the reciprocity law.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。