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

在这个项目中，我们将关注椭圆曲线和高维阿贝尔族的算术。这些几何对象已经被研究了很长时间，特别是因为它们在算术和几何之间提供了一座桥梁。在本项目中，所有阿贝尔簇都定义在离散值域或具有正特征的函数域上。研究提案由三个项目组成，每个项目都将对这些物体的算术和还原行为的某些方面提供新的了解。在第一个项目中，我们将考虑如下情形：设C是非完美域k上的光滑的、射影的、几何积分的曲线，E是定义在C上的椭圆曲线，那么我们可以考虑E在C上的极小真正则模型X，则X是k上的正则曲面。第一个项目的目的是证明一个刻画这类曲面光滑性的猜想。在第二个项目中，我们还将考虑椭圆曲面X，这一次定义在具有代数闭剩余域的离散值域K上。已有一个判据保证X在修改前具有对数良约化；该判据仅使用X的étal上同调空间上的Galois表示中包含的信息。这种将Néron-Ogg-Shafarevich判据推广到对数几何是相对较新的，目前尚不清楚它们是否充分。这个项目的目的是证明K上具有对数良好约化的椭圆曲面的精确上同调刻画。对于某些相关曲面(Kummer曲面)，已经有了一个精确的猜想，这可能是用现有方法可以达到的。对于第三个方案，考虑如下情况：设C是一条光滑的、射影的、几何积分的曲线，这一次定义在特征为p的有限域k上。设A是定义在C的函数域K上的阿贝尔簇。如果K^Sep表示K的一个可分闭包，则群A(K^Sep)的行为还没有完全被理解(与K的代数闭包K^alg的群A(K^alg)相反)。最近，Rössler证明了仅当A在C上的Néroon模型具有某些非常特殊的性质时，A(K^Sep)的p次方扭子群才是无限的。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局方法。

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.