CLASS INVARIANTS FOR ABELIAN VARIETIES OF HIGHER DIMENSION

本课题的主要研究对象是几何伽罗瓦结构。伽罗瓦结构最早出现在数论中，并激发了许多人的研究。这门学科后来在算术代数几何的背景下得到扩展。更准确地说，马丁·泰勒在1988年提出了一个猜想，即附加在椭圆曲线上的某些伽罗瓦结构不变量应该消失。这是由泰勒和其他人建立的。现在，我的目的是得到关于高维阿贝尔簇的泰勒猜想的正结果。泰勒猜想是在簇到处都有好的约化的背景下提出的。一个自然的问题是：在糟糕的减排情况下会发生什么？在我的博士学位中，我已经展示了如何将泰勒猜想推广到半稳定的阿贝尔变种。此外，我还证明了在某种程度上，对于半稳定椭圆曲线，消失结果仍然成立。由于半稳定案件比良好还原案件更常见，这扩大了调查的范围。在这种情况下，可以预期找到与泰勒猜想相反的例子。该项目可以与国外(来自德国、法国和美国等国)的研究人员建立合作。AMS MSC代码：11Gxx，11Rxx，14Kxx

The principal subject of this research project is geometric Galois structure . Galois structure appeared first in number theory, and has motivated research of many people. This subject was later extended in the context of arithmetic algebraic geometry. More precisely, Martin Taylor has stated in 1988 a conjecture which says that certain Galois structure invariants attached to elliptic curves, should vanish. This has been established by Taylor and others. Now, my aim is to obtain positive results on (an analogue of) Taylor's conjecture for abelian varieties of higher dimension.Taylor's conjecture is stated in the context of varieties having everywhere good reduction. A natural question to ask is : what does happens in the bad reduction case ? In my PhD, I have shown how Taylor's conjecture can be generalized to semi-stable abelian varieties. Also, I have proven that the vanishing result was still true, in some extent, for semi-stable elliptic curves. As the semi-stable case is much more frequent than the good reduction case, this enlarges the field of investigations. In this context, counter-examples to Taylor's conjecture can be expected to be found.This project can establish collaboration with researchers abroad (from countries like Germany, France and the US).AMS MSC code : 11Gxx, 11Rxx, 14Kxx