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核心地位。丢番图几何通过结合代数几何、代数和解析数论以及复几何的技术来研究丢番图方程，即整数或有理数（在最基本情况下）的多项式方程的解。斯皮罗对数域上的椭圆曲线的猜想暗示了著名的 abc 猜想，该猜想的有效性反过来又产生了大量其他深刻的结果，例如费马大定理、莫德尔猜想（法尔廷定理）或关于丢番图近似的罗斯定理代数数。算术设置中的斯皮罗猜想在复杂几何中具有相似之处，将非平凡半稳定椭圆曲线族的临界点数量和奇异纤维数量与某些基曲线（或更一般地说，更高属的曲线）联系起来，归功于 A. Beauville）；斯皮罗不等式在阿莫罗斯、博戈莫洛夫、卡察科夫、潘特夫建立的辛几何中也有类似物，其证明本质上是一种拓扑/群论论证，涉及单孔环面的映射类群。同调镜像对称是起源于数学物理学的原理/瑜伽，数学家们才刚刚开始充分利用和理解其后果。特别是，它以完全意想不到的方式将辛几何和复杂几何联系起来。例如，环面的分级辛自同构可以与镜像椭圆曲线上相干滑轮的派生类别的自等价性相关，并且德恩扭曲被视为对应于所谓的球面扭曲。然后，我们可以尝试模仿 Amoros、Bogomolov、Katzarkov、Pantev 的部分证明，使用导出的自等价性，并使用布里奇兰相位的变化来代替辛情况下的位移角概念。我们有理由希望这样的论证对于算术椭圆纤维振动仍然有意义，并且可以证明斯皮罗猜想。该项目的目标是建立基础和框架，在这个基础和框架中，可以在算术/阿拉克洛夫几何中理解布里奇兰稳定性条件，并且可以在其中执行受上述同调镜像对称启发的程序。这最终还将涉及 p 进几何和伯科维奇空间的技术。