Arithmetic and reduction of one-dimensional and higher-dimensional Abelian varieties over function fields

函数域上一维和高维阿贝尔簇的算术和约简

• 批准号: 442615504
• 负责人: Dr. Otto Overkamp
• 金额: --
• 依托单位: Mathematical Institute
• 依托单位国家: 德国
• 项目类别: Research Fellowships
• 财政年份: 2020
• 资助国家: 德国
• 起止时间: 2019-12-31 至 2022-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/442615504?language=en
• 关键词: Arithmetic; reduction; dimensional; higher; Abelian

In this project, we shall be concerned with the arithmetic of elliptic curves and higher-dimensional Abelian varieties. Those are geometric objects which have been studied for a long time, particularly since they provide a bridge between arithmetic and geometry. In the present project, all Abelian varieties will be defined over discretely valued fields or function fields in positive characteristic. The research proposal consists of three projects, each of which will shed new light on some aspect of the arithmetic and reduction behaviour of these objects. In the first project, we shall look at the following situation: Let C be a smooth, projective, and geometrically integral curve over a non-perfect field k. Let E be an elliptic curve defined over the function field of C. Then we can consider the minimal proper regular model X of E over C. Then X is a regular surface over k. The goal of the first project is to prove a conjecture which characterises smoothness of such surfaces. Another goal would be proving new results about the behaviour of Néron models under base change.In the second project, we shall also consider elliptic surfaces X, this time defined over a discretely valued field K with algebraically closed residue field. There already exists a criterion which guarantees that X has logarithmic good reduction up to modification; this criterion only uses information contained in the Galois representations on the étale cohomology spaces of X. Such generalisations of the Néron-Ogg-Shafarevich criterion to logarithmic geometry are relatively new, and it is not currently known whether they are sufficient. The goal of this project is the proof of a precise cohomological characterisation of elliptic surfaces over K which have logarithmic good reduction. For certain related surfaces (Kummer surfaces), there already is a precise conjecture, which will probably be accessible using existing methods.For the third project, consider the following situation: Let C be a smooth, projective, and geometrically integral curve, defined this time over a finite field k of characteristic p. Let A be an Abelian variety defined over the function field K of C. If K^sep denotes a separable closure of K, the behaviour of the group A(K^sep) has not been completely understood (as opposed to the group A(K^alg) for an algebraic closure K^alg of K). Recently, Rössler proved that the p-power-torsion subgroup of A(K^sep) can only be infinite if the Néron model of A over C has certain very special properties. In this project, we shall study these properties, with a view towards generalising his newly developed global methods.

在这个项目中，我们将关注椭圆曲线和高维阿贝尔簇的算术。这些几何对象已经研究了很长时间，特别是因为它们提供了算术和几何之间的桥梁。在本项目中，所有的阿贝尔簇将被定义在离散值域或正特征函数域上。该研究提案包括三个项目，每个项目都将揭示这些对象的算术和还原行为的某些方面。在第一个项目中，我们将研究以下情况：设C是非完美域k上的光滑、射影和几何积分曲线。设E是定义在C的函数域上的椭圆曲线。然后我们可以考虑E在C上的极小真正则模型X。则X是K上的正则曲面。第一个项目的目标是证明一个猜想的特点光滑的表面。另一个目标将是证明新的结果的行为Néron模型下基地的变化。在第二个项目中，我们也将考虑椭圆曲面X，这一次定义在一个离散值域K代数封闭的剩余领域。已经存在一个准则，保证X在修改之前具有对数良好约化;这个准则只使用X的étale上同调空间上的伽罗瓦表示中包含的信息。这种将Néron-Ogg-Shafarevich准则推广到对数几何的方法是相对较新的，目前还不知道它们是否足够。这个项目的目标是证明一个精确的上同调特征的椭圆曲面在K有对数良好的减少。对于某些相关曲面（库默曲面），已经有了一个精确的猜想，用现有的方法可能可以得到。对于第三个方案，考虑以下情况：设C是一条光滑的、射影的、几何积分的曲线，此时定义在特征为p的有限域k上。如果K^sep表示K的一个可分闭包，则群A（K^sep）的行为还没有被完全理解（与K的一个代数闭包K^alg的群A（K^alg）相反）。最近，Rössler证明了A（K^sep）的p-幂挠子群只能是无限的，如果C上A的Néron模型具有某些非常特殊的性质。在这个项目中，我们将研究这些性质，以期推广他新开发的全球方法。