This manuscript was written for the CIRM Workshop "La conjecture de Zilber-Pink" held in Luminy, May 2011 and organized by Gaël Rémond and Emmanuel Ullmo
The main emphasis will be on height upper bounds in the algebraic torus G^{n}_{m}. By height we will mean the absolute logarithmic Weil height. Section 3.2 contains a precise definition of this and other more general height functions. The first appendix gives a short overview of known results in the abelian case. The second appendix contains a few height bounds in Shimura varieties.
Let $T$ be an algebraic torus defined over a global field $K$. For any $K$-torsor $X$ under $T$, we relate the Brauer group of $X$ to the adéle class group of $T$ as well as to the Shafarevich Tate group of $T$.
A theorem of Chow concerns homomorphisms of two abelian varieties under a primary field extension base change. In this paper we generalize Chow's theorem to semi-abelian varieties. This contributes to different proofs of a well-known result that every algebraic torus splits over a finite separable field extension. We also obtain the best bound for the degrees of splitting fields of tori.
Let $T$ be an algebraic torus defined over a global field $K$ and split over a finite cyclic extension. In this paper, we determine the Herbrand quotient of the adéle class group of $T$. Our result can be seen as an extension of the so-called first inequality of global class field theory.
We extend the results by R.P. Langlands on representations of (connected) abelian algebraic groups. This is done by considering characters into any divisible abelian topological group. With this we can then prove what is known as the abelian case of the $p$-adic Langlands program.
We give a formula for the class number of an arbitrary CM algebraic torus over $\mathbb{Q}$. This is proved based on results of Ono and Shyr. As applications, we give formulas for numbers of polarized CM abelian varieties, of connected components of unitary Shimura varieties and of certain polarized abelian varieties over finite fields. We also give a second proof of our main result.
在这些讲座中，我们涵盖了基本理论的高度开始的高度在射影空间，高度的多项式和高度的代数曲线。定义了二元形式的最小高度和代数曲线的模高，证明了超椭圆曲线的模高为$\mathcal H (f) \leq c_0 \tilde H (f)$，其中$c_0$为常数，$\tilde H$为相应二元形式的最小高度.对于亏格$g=2$和3，这样的常数是明确确定的。进而计算了亏格为2和亏格为3的高度为1的超椭圆曲线的完整曲线表。
The study of the birational properties of algebraic $k$-tori began in the sixties and seventies with work of Voskresenkii, Endo, Miyata, Colliot-Thélène and Sansuc. There was particular interest in determining the rationality of a given algebraic $k$-tori. As rationality problems for algebraic varieties are in general difficult, it is natural to consider relaxed notions such as stable rationality, or even retract rationality. Work of the above authors and later Saltman in the eighties determined necessary and sufficient conditions to determine when an algebraic torus is stably rational, respec…
Given a number field extension $K/k$ with an intermediate field $K^+$ fixed by a central element of the corresponding Galois group of prime order $p$, we build an algebraic torus over $k$ whose rational points are elements of $K^\times$ sent to $k^\times$ via the norm map $N_{K/K^+}$. The goal is to compute the Tamagawa number of that torus explicitly via Ono's formula that expresses it as a ratio of cohomological invariants. A fairly complete and detailed description of the cohomology of the character lattice of such a torus is given when $K/k$ is Galois. Partial results including the numerat…
In arXiv:1302.4423, Salez proved that every totally real algebraic integer is the eigenvalue of some tree. We define the "arboreal height" of a totally real algebraic integer $λ$ to be the minimal height of a rooted tree having $λ$ as an eigenvalue. In this paper, we prove several results about totally real algebraic integers of arboreal height $2$: We show that all real quadratic integers have arboreal height $\le 2$. We characterize the totally real cubic integers of arboreal height $2$. Finally, we prove that every totally irrational real number field is generated (as a ring over $\mathbb{Q…
We consider the covering map $π:\mathbb{C}^n\to \mathbb{T}$ of a compact complex torus. Given an algebraic variety $X\subseteq \mathbb{C}^n$ we describe the topological closure of $π(X)$ in $\mathbb T$. We obtain a similar description when $\mathbb{T}$ is a real torus and $X\subseteq \mathbb{R}^n$ is a set definable in an o-minimal structure over the reals.
Outer automorphisms of algebraic groups and determining groups by their maximal tori
给出了域上半单代数群存在外自同构的上同调判据。然后将这一准则应用于整体域上D_2n型群的特殊情形，从而完成了“Weakly Suprantifable arithmetic groups and isoprope localized symmetric spaces”（Pub. Math.IHES，2009），并给出了一个新的证明，从另一篇论文的同一作者。
We give the complete stably rational classification of algebraic tori of dimensions $4$ and $5$ over a field $k$. In particular, the stably rational classification of norm one tori whose Chevalley modules are of rank $4$ and $5$ is given. We show that there exist exactly $487$ (resp. $7$, resp. $216$) stably rational (resp. not stably but retract rational, resp. not retract rational) algebraic tori of dimension $4$, and there exist exactly $3051$ (resp. $25$, resp. $3003$) stably rational (resp. not stably but retract rational, resp. not retract rational) algebraic tori of dimension $5$. We ma…
