p进模形式斜率及p进局部朗兰兹纲领

申请人的研究方向是代数数论，特别专注于p进模形式以及p进局部朗兰兹纲领的研究。对p进模形式斜率的研究起源于90年代Gouvea和Mazur和2000年左右Coleman和Mazur一系列深刻的猜想。他们预测了p进模形式斜率奇妙且神秘的性质，并提出了两个主要的猜想：spectral halo猜想和特征曲线不可约分支有限性的猜想。此前的工作中，申请人与合作者证明了“一半”情况下这两个猜想的正确性。在本项目中，我们计划完整证明这两个猜想，并推广这些关于斜率的结果到希尔伯特模形式或更一般的自守形式。这些推广在算术几何和数论中都有重要的应用。此外，我们还计划进行一些关于p进L函数在特征曲线上整性的研究，以作为p进模形式斜率工作的应用。

The PI's area of research is in algebraic number theory, particularly focusing on the study of p-adic modular forms and p-adic local Langlands program. The study of slopes of p-adic modular forms originated in the pioneer works of Gouvea and Mazur in the 1990s and a series of profound conjectures by Coleman and Mazur in the 2000s. They predicted beautiful yet mysterious properties of slopes of p-adic modular forms, and summarized them into two major conjectures: spectral halo conjecture and the finiteness of irreducible components of the eigencurve. In several joint projects, the PI proved the two conjecture in "half" of the situations. In this proposed project, we plan to complete the proof of the two conjectures in all cases, and to generalize them to p-adic Hilbert modular forms or more general automorphic forms. These generalizations have important applications in arithmetic geometry and in number theory. Moreover, we plan to study certain integrality of p-adic L-functions on the eigencurve, as an application of the study of slopes of p-adic modular forms.