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

椭圆曲线的运算在数论和丢番图几何中占有重要地位。丢番图几何研究丢番图方程，即通过结合代数几何、代数和解析数论以及复几何的技巧来求解整数或有理数(在最基本的情况下)的多项式方程。Szpiro关于数域上椭圆曲线的猜想蕴含着著名的ABC猜想，它的有效性反过来又产生了大量其他深刻的结果，如Fermat最后定理、Mordell猜想(Falting定理)或Roth关于代数数的丢番图逼近的定理。Szpiro在算术结构中的猜想在复杂几何中有类似的结果，它将某一基本曲线(或更一般的更高亏格的曲线，由于A.Beauville)上的非平凡半稳定椭圆曲线族的临界点的个数和奇异纤维的个数联系在一起；Szpiro的猜想在辛几何中也有类似的结果，它的证明实质上是一个涉及单孔环面的映射类群的拓扑/群论论点。同源镜像对称是一种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结果。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在一起。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派生范畴的自等价相关，而Dehn扭曲被认为对应于所谓的球面扭曲。然后，人们可以模仿Amoros、Bogomolov、Katzarkov、Pantev的证明的一部分，利用导出的自等价，并使用Bridgeland相的变化来替代辛情形中的位移角的概念。我们有理由希望，这样的论点对算术椭圆纤颤仍然有意义，并能导致Szpiro猜想的证明。该项目的目标是建立一个基础和框架，在这个框架中，布里奇兰稳定性条件可以在算术/阿拉克洛夫几何中得到解释，并且上面概述的同调镜面对称性的启发方案可以在其中进行。这最终还将涉及p-进几何和Berkovich空间的技术。

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.