On a deformation of Riemann surfaces of genus two.

• Project/Area Number: 10640193
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Doshisha University
• Principal Investigator: HIORIUCHI Ryutaro Doshisha University, Department of Chemical Engineering and Materials Science, Professor, 工学部・物質化学工学科, 教授 (60065852)
• Project Period (FY): 1998 – 1999
• Project Status: Completed (Fiscal Year 1999)
• Budget Amount: ¥1,300,000 (Direct Cost: ¥1,300,000); Fiscal Year 1999: ¥700,000 (Direct Cost: ¥700,000); Fiscal Year 1998: ¥600,000 (Direct Cost: ¥600,000)
• Keywords: Riemann surface / Algebraic curve / deformation / 被覆面 / ガロア群 / リーマン面の変形 / モノドロミー / ワイアシュトラス点 / フェルマー曲線

Research Abstract

Meanwhile, by Taylor expansion of holomorphic differentials on the Fermat curve, the Weierstrass weight of Leopoldt Weierstrass points, and scrutinize its Wronskian were computed. From it the original equations for general Weierstrass points on the quotient space of the curve by its automorphisms were extracted.
It was shown that a family of Riemann surfaces of genus one (elliptic curves) represented by three-sheeted coverings of the Rie-mann sphere with one totally branched point can be considered as a twenty-seven sheeted covering of the Riemann sphere with four points removed, and can be endowed with complex structure of compact Riemann surface of genus three with twenty-three points removed. The monodromy group is represented by the Galois group of algebraic equation in the parameters of the family. In addition, all the coverings of the family can be obtained from two-sheeted covering of the Riemann sphere with four branch points by attaching another Riemann sphere along curves connecting one of the branch points and a point outside four branch points.