First Grant for Ambrus Pal

建议的方法来解决的主要目标，取得进展的阿廷-泰特猜想，是联合收割机的方法使用朗兰兹对应的功能领域的工具，p-adic分析性质，特别是结晶上同调。我们的策略可能不仅会在特殊情况下以原始形式显示猜想，而且还会导致改进的版本，从而加深我们对一般情况的理解。特别是，我们打算显示一个广义形式的Gross-Zagier公式的功能领域，并作为一个应用程序，推导出每一个亏格一条曲线定义在一个功能领域有一个可解点。此外，我们将努力证明，一个弱形式的泰特猜想意味着一个精致的版本的阿廷泰特猜想涉及p-adic监管机构。我们还打算通过Langlands程序研究p-adic L-函数的整除性，作为一个密切相关的问题，我们将开始开发后者的p-adic版本，作为我们特定问题的工具，为未来的研究奠定基础。

The proposed approach to tackle the main objective, making progress on the Artin-Tate conjecture, is to combine methods using the Langlands correspondence for function fields with tools of p-adic analytic nature, in particular crystalline cohomology. Our strategy will likely not just show the conjecture in its original form in special cases, but will lead to refined versions as well, therefore deepening our understanding in the general case. In particular we intend to show a generalized form of the Gross-Zagier formula for function fields, and derive as an application that every genus one curve defined over a function field has a solvable point. Moreover we will work towards proving that a weak form of the Tate conjecture implies a refined version of the Artin-Tate conjecture involving p-adic regulators. We also intend to study the divisibility properties of p-adic L-functions via the Langlands program, and as a closely related problem, we'll start to develop a p-adic version of the latter both as a tool for our particular problem, both for laying the foundations for future research.