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DERIVED CATEGORY METHODS IN ARITHMETIC: AN APPROACH TO SZPIRO'S CONJECTURE VIA HOMOLOGICAL MIRROR SYMMETRY AND BRIDGELAND STABILITY CONDITIONS

算术中的派生范畴方法：通过同调镜像对称性和布里奇兰稳定性条件推导SZPIRO猜想

基本信息

批准号：
EP/V047299/1
负责人：
Christian Boehning
金额：
$ 25.78万
依托单位：
University of Warwick
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2021
资助国家：
英国
起止时间：
2021 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FV047299%2F1
关键词：
DERIVED CATEGORY METHODS ARITHMETIC APPROACH

项目摘要

The arithmetic of elliptic curves occupies a central role in number theory and Diophantine geometry. Diophantine geometry studies Diophantine equations, that is, the solution of polynomial equations in integers or rational numbers (in the most basic case), through a combination of techniques from algebraic geometry, algebraic and analytic number theory, and complex geometry. Szpiro's conjecture for elliptic curves over number fields is known to imply the famous abc-conjecture, whose validity in turn yields a large number of other deep results such as Fermat's Last Theorem, Mordell's Conjecture (Falting's theorem), or Roth's theorem about Diophantine approximation of algebraic numbers. Szpiro's conjecture in the arithmetic set-up has an analogue in complex geometry, relating the number of critical points and the number of singular fibres of a non-trivial semistable family of elliptic curves over some base curve (or more generally, curves of higher genus, due to A. Beauville); Szpiro's inequality also has an analogue in symplectic geometry established by Amoros, Bogomolov, Katzarkov, Pantev, whose proof is essentially a topological/group-theoretic argument involving the mapping class group of a torus with one hole. Homological Mirror Symmetry is a principle/yoga having its origin in mathematical physics, whose consequences mathematicians have only started fully to exploit and understand. In particular, it relates symplectic geometry and complex geometry in completely unexpected ways. For example, graded symplectic automorphisms of a torus can be related to autoequivalences of the derived category of coherent sheaves on the mirror elliptic curve, and Dehn twists are seen to correspond to so-called spherical twists. One can then seek to mimic parts of the proof by Amoros, Bogomolov, Katzarkov, Pantev working with derived autoequivalences and using changes in Bridgeland phase as a substitute for the notion of displacement angle in the symplectic situation. It is reasonable to hope that such an argument will still make sense for arithmetic elliptic fibrations and can lead to a proof of Szpiro's conjecture. The goal of the project is to establish foundations and a framework in which Bridgeland stability conditions can be made sense of in arithmetic/Arakelov geometry and in which the programme inspired by Homological Mirror Symmetry outlined above can be carried through. This will also ultimately involve techniques from p-adic geometry and Berkovich spaces.

椭圆曲线的算术在数论和丢番图几何中占有中心地位。丢番图几何研究丢番图方程，即整数或有理数（在最基本的情况下）的多项式方程的解，通过代数几何，代数和解析数论以及复几何的技术组合。Szpiro猜想的椭圆曲线在数域上是众所周知的暗示著名的abc猜想，其有效性反过来又产生了大量其他深刻的结果，如费马大定理，莫德尔猜想（Falting定理），或罗斯定理关于丢番图近似的代数数。Szpiro的猜想在算术上的设置有一个类似的复几何，涉及临界点的数量和奇异纤维的数量的非平凡半稳定族的椭圆曲线在一些基曲线（或更一般地说，曲线的更高的亏格，由于A。Beauville）; Szpiro不等式在辛几何中也有一个类似的例子，由Amoros，Bogomolov，Katzarkov，Pantev建立，其证明本质上是一个拓扑/群论的论点，涉及一个有一个洞的环面的映射类群。同调镜像对称是一个起源于数学物理的原理/瑜伽，数学家们才刚刚开始充分利用和理解它的后果。特别是，它涉及辛几何和复杂的几何在完全意想不到的方式。例如，环面的分次辛自同构可以与镜像椭圆曲线上的相干层的导出范畴的自等价有关，而德恩扭曲被认为对应于所谓的球面扭曲。然后，人们可以试图模仿Amoros，Bogomolov，Katzarkov，Pantev的部分证明，使用导出的自等价，并使用Bridgeland相位的变化作为辛情况下位移角概念的替代品。这是合理的希望，这样的论点将仍然有意义的算术椭圆纤维化，并可能导致证明斯皮罗猜想。该项目的目标是建立基础和框架，使布里奇兰稳定性条件可以在算术/阿拉克洛夫几何中得到理解，并且可以在其中实施上述受同调镜像对称启发的计划。这最终也将涉及到p-adic几何和Berkovich空间的技术。