Theory of congruences of Galois representations and modular forms for function fields

2016 Fiscal Year Final Research Report

• Project/Area Number: 26400016
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Kyushu University
• Principal Investigator: Hattori Shin 九州大学, 数理学研究院, 助教 (10451436)
• Project Period (FY): 2014-04-01 – 2017-03-31
• Keywords: Drinfeld保型形式 / 固有値多様体

Outline of Final Research Achievements

The aim of this research project was to construct a theory of v-adic congruences for Drinfeld modular forms. Using Taguchi's duality, I defined the Hodge-Tate-Taguchi map, which is a torsion comparison isomorphism for the Drinfeld setting. With this map, I proved that Drinfeld modular forms with highly congruent Fourier expansions have highly congruent weights, and also that any Drinfeld modular form of tame level n is a v-adic modular form.
I also studied the geometry of eigenvarieties, for future applications to Drinfeld modular forms. I proved the properness of Hilbert eigenvarieties at integral weights and a conjecture of Coleman-Mazur on irreducible components of Coleman-Mazur eigencurves of finite degree.

Free Research Field

整数論