by Yasutaka Ihara (University of Tokyo, Japan)

It is now well-known that the group SL₂(Z[1/p]) and the system of modular curves over F_p² are “closely related”, and that the latter provided first “examples” of curves over finite fields having many rational points. However, the “three basic relationships”, which really justify the former to be called the arithmetic fundamental group of the latter, still do not seem to be so commonly known.

This book consists of two parts; a reproduction of the author's unpublished Lecture Notes (1968,69), and Author's Notes (2008). The former starts with explicit three main conjectural relationships for more general cases and gives various approaches towards their proofs. Though remained formally unpublished, these Lecture Notes had been widely circulated and have stimulated researches in various directions. The main conjectures themselves have also been proved since then. The Author's Notes (2008) gives detailed explanations of these developments, together with open problems.

Published by Mathematical Society of Japan and distributed by World Scientific Publishing Co. for all markets

• The Group Γ and Its ζ-Function
• Detailed Study of Elements of Γ with Parabolic and Elliptic Real Parts, The General Formula for ζΓ (u)
• The G_p-Fields over C
• Full G_p-Subfields Over Algebraic Number Fields
• The Canonical S-Operator and the Canonical Class of Linear Differential Equations of Second Order on Algebraic Function Field L of One Variable over C, and Their Algebraic Characterizations when L is “Arithmetic”
• Unique Existence of an Invariant S-Operator on “Arithmetic” Algebraic Function Fields (including G_p-Fields) over any Field of Characteristic Zero
• Some Properties Γ
• Examples of Γ
• Elliptic Modular Functions Mod p and Γ = PSL₂(Z^(p))
• Non-Abelian Classified Attached to Subgroups of Γ = PSL₂(Z^(p)) with Finite Indices