Number theory from a viewpoint of computation of modular forms

2016 Fiscal Year Final Research Report

• Project/Area Number: 26400008
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: University of Toyama
• Principal Investigator: Kimura Iwao 富山大学, 大学院理工学研究部(理学), 准教授 (10313587)
• Project Period (FY): 2014-04-01 – 2017-03-31
• Keywords: モジュラー形式 / コンピューター

Outline of Final Research Achievements

The point of this research is to study some problems on number theory which are related to elliptic modular forms via explicit numerical/symbolic computation using computers. On the first year of this three years project, I mainly concerned on the computation of 2 dimensional complex Artin representation of the Galois group of the rationals associated with the Hecke eigen cuspform of weight 1. It is of special interest if those image are non-solvable. If the conductor of this representation are square free, the explicit bound of order of computations. In the second year, I mainly studied a numerical method to compute Maass waveforms. I found a numerical instability of the method I took makes things complicated. In the last year, I considered zeros of modular forms on Fricke groups and obtained some extension of results which are known by prior study.

Free Research Field

数論