First Grant for Ambrus Pal

在Artin-Tate猜想上取得进展的主要目标是，提出的方法是将使用函数场的朗兰兹对应的方法与p-进分析性质的工具相结合，特别是晶体上同调。我们的策略可能不仅会在特殊情况下显示猜想的原始形式，还会导致精炼版本，从而加深我们对一般情况的理解。特别地，我们给出了函数域上Gross-Zagier公式的一种推广形式，并作为应用导出了定义在函数域上的每条亏格曲线都有一个可解点。此外，我们将致力于证明Tate猜想的一种弱形式蕴含着Artin-Tate猜想的精化版本，该猜想涉及p-进的调节器。我们还打算通过朗兰兹程序来研究p-进L函数的可除性，并且作为一个密切相关的问题，我们将开始开发后者的p-进版本，作为我们的特殊问题的工具，两者都为以后的研究奠定基础。

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.