Drinfeld模曲线理论及在编码学中的应用

Drinfeld 模是算术几何中研究 Langlands 猜想引入的主要研究对象，是从经典的椭圆曲线扩展到函数域的平行理论。Drinfeld 模曲线参数化了一族 Drinfeld 模，是这个领域的核心理论。在编码领域中，Drinfeld 模曲线的重要意义在于构造渐近性质良好的代数几何码。本项目研究几个有关 Drinfeld 模曲线的重要课题，具体地：.1) 推导 P^n-torsion 层级结构对应的 Drinfeld 模曲线的显式方程；.2) 定义一种 N-torsion 层级结构的拓扑树，并构造相应的 Drinfeld 模曲线；.3) 计算 (r, k)-型 Drinfeld 模形式的 Fourier 展开式系数。.这些研究具有较好的理论价值和应用前景，为 Drinfeld 模理论和代数几何码下一步研究开辟全新的视角和方向。

Drinfeld modules play the key role in modern studies of arithmetic geometry, particularly that of Langlands conjecture. As an analogue to the classical theory of elliptic curves in the realm of function fields, Drinfeld modular curves parametrize families of Drinfeld modules, and become the main subject of research in this field. Especially in coding theory, Drinfeld modular curves are used to construct algebraic geometric codes with good asymptotic properties. In this project, we plan to carry out the following specific research topic and goals on Drinfeld modular curves:..1) to deduce the explicit equation of Drinfeld modular curves stemming from P^n-torsion level structures;..2) to depict N-Torsions by introducing a type of topological trees, whose nodes correspond to torsion sequences of submodules, and to construct the corresponding Drinfeld module curves;..3) to calculate coefficients of the Fourier expansion of (r, k)-type Drinfeld modular forms...These studies certainly have theoretical value and application prospects, and potentially will open up a new perspective or direction for the next step development of Drinfeld modular theory and algebraic geometric codes.