Arithmetic Moduli Spaces and Gauge Theory

这个建议是关于丢番图方程的理论，即对代数方程的有理解或积分解的研究。这是数学中最古老的学科之一，可能可以追溯到古巴比伦时期，并在3世纪左右由亚历山大的丢番图系统化。尽管如此，它仍然是数学中一些最困难的问题和广泛的程序的来源，例如费马大定理或伯奇和斯温纳顿-戴尔的猜想。尽管在过去100年左右的时间里，现代算术几何方法取得了很大的进步，但主要的问题仍然没有得到解决，尤其是在计算机上求解方程的算法方法方面。（这又受到某些数理逻辑的不可能定理的阻碍。）本研究通过类比物理学中欧拉-拉格朗日方程的解空间和PI构造的“算术规范场”的非阿基米德几何，提出了将高能物理启发的新思想应用于双变量丢芬图方程的研究。主要目标是将围绕Birch和Swinnerton-Dyer猜想的方法论推广到更高的程度，该猜想与3次方程有关。最终，这项研究应该会在设计一种计算机算法的问题上取得实质性进展，该算法将找到至少为4次和两个未知数的方程的所有有理解。

This proposal is concerned with the theory of Diophantine equations, that is, the study of rational or integral solutions to algebraic equations. This is one of the oldest subjects in mathematics, going back possibly to the ancient Babylonians and systematised by Diophantus of Alexandria around the 3rd century. Nevertheless, it is still the source of some of the most difficult problems and wide-ranging programmes in mathematics, such as Fermat's Last Theorem or the conjectures of Birch and Swinnerton-Dyer. In spite of much progress over the last 100 years or so using the modern methods of arithmetic geometry, the major problems remain unsolved, especially when it comes to algorithmic methods that can find solutions to equations on a computer. (This is furthermore hampered by certain impossibility theorems of mathematical logic.) This research proposes to apply new ideas inspired by high energy physics to the study of Diophantine equations in two variables based on an analogy between the solution space to Euler-Lagrange equations in physics and the non-Archimedean geometry of 'arithmetic gauge fields' constructed by the PI. The main goal is to generalise to higher degree the methodology surrounding the conjecture of Birch and Swinnerton-Dyer, which is concerned with equations of degree 3. Eventually, this research should lead to substantial progress on the problem of devising a computer algorithm that will find all rational solutions to equations of degree at least 4 and two unknowns.