This book lays the foundation for a theory of uniformization of $p$-adic hyperbolic curves and their moduli. On one hand, this theory generalizes the Fuchsian and Bers uniformizations of complex hyperbolic curves and their moduli to nonarchimedian places. That is why in this book, the theory is referred to as $p$-adic Teichmüller theory, for short. On the other hand, the theory may be regarded as a fairly precise hyperbolic analog of the Serre-Tate theory of ordinary abelian varieties and their moduli.

The theory of uniformization of $p$-adic hyperbolic curves and their moduli was initiated in a previous work by Mochizuki. And in some sense, this book is a continuation and generalization of that work. This book aims to bridge the gap between the approach presented and the classical uniformization of a hyperbolic Riemann surface that is studied in undergraduate complex analysis.

Features:

• Presents a systematic treatment of the moduli space of curves from the point of view of $p$-adic Galois representations.

• Treats the analog of Serre-Tate theory for hyperbolic curves.

• Develops a $p$-adic analog of Fuchsian and Bers uniformization theories.

• Gives a systematic treatment of a "nonabelian example" of $p$-adic Hodge theory.

Readership

Graduate students and research mathematicians working in arithmetic geometry.