CLASS INVARIANTS FOR ABELIAN VARIETIES OF HIGHER DIMENSION

本课题主要研究几何伽罗瓦结构。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范围内得到了扩展。更准确地说，马丁·泰勒在1988年提出了一个猜想，认为椭圆曲线上的某些伽罗瓦结构不变量应该消失。泰勒和其他人已经证实了这一点。现在，我的目标是对高维的阿贝尔变的泰勒猜想（类似的）获得肯定的结果。泰勒猜想是在到处都有良好约简的变异的背景下提出的。一个自然的问题是：在坏的还原情况下会发生什么？在我的博士学位上，我展示了泰勒猜想是如何推广到半稳定阿贝尔变体的。此外，我还证明了在某种程度上，对于半稳定椭圆曲线，消失结果仍然是正确的。由于半稳定情况比良好约简情况更频繁，这扩大了研究领域。在这种情况下，可以期望找到泰勒猜想的反例。该项目可以与国外（如德国、法国和美国）的研究人员建立合作关系。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