Jean-Francois Burnol
I associate to a global field K a Lax-Phillips scattering which has the property of causality if and only if the Riemann Hypothesis holds for all the abelian L-functions of K. As a Hilbert space closure problem this provides an adelic variation on a theme initiated by Nyman and Beurling. The adelic aspects are related to previous work by Tate, Iwasawa and Connes.
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Adelic Eisenstein classes and divisibility properties of Stickelberger elements
本文引入了Nori的Eisenstein上同调类及其整细化的adelic细化，用于建立与全实域Abel扩张相关的Stickelberger元素的新可整除性性质。
We prove that for a large class of subvarieties of abelian varieties over global function fields, the Brauer-Manin condition on adelic points cuts out exactly the rational points. This result is obtained from more general results concerning the intersection of the adelic points of a subvariety with the adelic closure of the group of rational points of the abelian variety.
On a relation between Deninger's foliated dynamical systems and Connes-Consani's adelic spaces
本文探讨了Deninger的叶状动力系统与Connes-Consani的adelic空间之间的关系，揭示了类域论的几何视角。
Weak approximation versus the Hasse principle for subvarieties of abelian varieties
本文研究阿贝尔簇子簇的弱逼近失效问题，证明了有限下降作为有理点存在的唯一障碍可推导出弱逼近失效由有限下降解释，并推广至 Mordell-Weil 筛障碍。
We construct an explicit example of a genus $2$ curve $C$ over a number field $K$ such that the adelic Galois representation arising from the action of $\operatorname{Gal}(\overline{K}/K)$ on the Jacobian of $C$ has image $\operatorname{GSp}_4(\widehat{\mathbb Z})$.
This paper is the second part of arXiv:0707.1766. We develope harmonic analysis in some categories of filtered abelian groups and vector spaces over the fields R or C. These categories contain as objects local fields and adelic spaces arising from arithmetical surfaces. Some structure theorems are proven for quotients of the adelic groups of algebraic and arithmetical surfaces.
In this article, we show that for any non-isotrivial family of abelian varieties over a rational base with big monodromy, those members that have adelic Galois representation with image as large as possible form a density-$1$ subset. Our results can be applied to a number of interesting families of abelian varieties, such as rational families dominating the moduli of Jacobians of hyperelliptic curves, trigonal curves, or plane curves. As a consequence, we prove that for any dimension $g \geq 3$, there are infinitely many abelian varieties over $\mathbb Q$ with adelic Galois representation havi…
On the height of the universal abelian variety
本文推广袁-X和张-S.W.的算术相交理论至非nef且不可积的adelic算术除子，用相对有限能量概念证明泛阿贝尔簇上Siegel-Jacobi形式线丛不可积并算出其算术自交数，技术可用于混合Shimura簇等场景。
Adelic Mordell-Lang and the Brauer-Manin obstruction
本文研究整体函数域上阿贝尔簇闭子簇的有理点分布，证明满足条件的阿代尔点是子簇有理点的极限，结合Tate-Shafarevich群有限假设可得有理点在Brauer集中稠密，并将结果推广至全虚数数域情形。
Galois invariants of finite abelian descent and Brauer sets
该文研究全局域上代数簇的有限阿贝尔下降、Brauer-Manin障碍对应的阿德尔点子集的伽罗瓦不变量还原基域障碍集的条件，并将常曲线的相关结果推广到函数域上的等平凡曲线。
Adelic perturbation of rational functions and applications
本文研究有理函数幂级数经几乎拟多项式系数扰动后的 Pólya–Carlson 二分，并据此解决 Byszewski–Cornelissen、Bell–Miles–Ward 猜想及 Royals–Ward 问题的重要情形。
Let $A$ be a $g$-dimensional abelian variety over $\mathbb{Q}$ whose adelic Galois representation has open image in $\text{GSp}_{2g} \widehat{\mathbb{Z}}$. We investigate the endomorphism algebras $\text{End}(A_p) \otimes \mathbb{Q} = \mathbb{Q}( π_p )$ of the reduction of $A$ modulo primes $p$ at which this reduction is ordinary and simple. We obtain conditional and unconditional asymptotic upper bounds on the number of primes at which this "Frobenius field" is a specified number field and, when $A$ is two-dimensional, at which the Frobenius field contains a specified real quadratic number fi…
For a nice algebraic variety $X$ over a number field $F$, one of the central problems of Diophantine Geometry is to locate precisely the set $X(F)$ inside $X(\A_F)$, where $\A_F$ denotes the ring of adèles of $F$. One approach to this problem is provided by the finite descent obstruction, which is defined to be the set of adelic points which can be lifted to twists of torsors for (certain) finite étale group schemes over $F$ on $X$. More recently, Kim proposed an iterative construction of another subset of $X(\A_F)$ which contains the set of rational points. In this paper, we compare the two c…
Let $E/\mathbb Q$ be an elliptic curve, and denote by $N(p)$ the number of $\mathbb{F}_p$-points of the reduction modulo $p$ of $E$. A conjecture of Koblitz, refined by Zywina, states that the number of primes $p \leq X$ at which $N(p)$ is also prime is asymptotic to $C_E \cdot X / \log(X)^2$, where $C_E$ is an arithmetically-defined non-negative constant. Following Miri-Murty (2001) and others, Y.R. Liu (2006) and David-Wu (2012) study the number of prime factors of $N(p)$. We generalize their arguments to abelian varieties $A / \mathbb Q$ whose adelic Galois representation has open image in …
Let F be a global field. In this work, we show that the Brauer-Manin condition on adelic points for subvarieties of a torus T over F cuts out exactly the rational points, if either F is a function field or, if F is the field of rational numbers and T is split. As an application, we prove a conjecture of Harari-Voloch over global function fields which states, roughly speaking, that on any rational hyperbolic curve, the local integral points with the Brauer-Manin condition are the global integral points. Finally we prove for tori over number fields a theorem of Stoll on adelic points of zero-dim…
We reinterpret Kim's non-abelian reciprocity maps for algebraic varieties as obstruction towers of mapping spaces of etale homotopy types, removing technical hypotheses such as global basepoints and cohomological constraints. We then extend the theory by considering alternative natural series of extensions, one of which gives an obstruction tower whose first stage is the Brauer--Manin obstruction, allowing us to determine when Kim's maps recover the Brauer-Manin locus. A tower based on relative completions yields non-trivial reciprocity maps even for Shimura varieties; for the stacky modular c…
Finiteness of Heights in Isogeny Classes of Motives with Semistable Reduction
本文利用积分 $p$-adic Hodge 理论推广了阿贝尔簇的 Faltings 高度，并在假设 adelic Mumford-Tate 猜想下，证明了 motive 同构类中高度的有限性，扩展了 Kisin 和 Mocz 的相关结果。
Knots, primes and class field theory
本文对类域论进行几何推广，通过阿代尔构造搭建关联概形与整体域整数环的几何框架，构造出将有理数域有限阿贝尔扩张映射到阿代尔类空间扇区有限覆叠的函子，建立了周期轨道单值群与素数处弗罗贝尼乌斯伽罗瓦作用的对应关系。
We consider smooth projective curves C/$\mathbb{F}$ over a finite field and their symmetric squares $C^{(2)}$. For a global function field $K/\mathbb{F}$, we study the $K$-rational points of $C^{(2)}$. We describe the adelic points of $C^{(2)}$ surviving Frobenius descent and how the $K$-rational points fit there. Our methods also lead to an explicit bound on the number of $K$-rational points of $C^{(2)}$ satisfying an additional condition. Some of our results apply to arbitrary constant subvarieties of abelian varieties, however we produce examples which show that not all of our stronger conc…
Ramified descent
研究光滑几何连通簇X上的分枝下降问题，探讨数域K上的adele点如何近似于可提升到给定分枝覆盖的点，引入Brauer-Manin障碍，证明其对Hasse原理和弱近似构成障碍，且为阿贝尔覆盖提供非平凡超越障碍的例子。
Arakelov-Green's functions for dynamical systems on projective varieties
本文引入极化动力系统的 Arakelov-Green 函数并证明其 Elkies 型下界，进而得到乘积公式域上 Abel 簇非挠点典范高度的 Lehmer 型下界。
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