Description
Book SynopsisThis book discusses the p-adic modular forms, the eigencurve that parameterize them, and the p-adic L-functions one can associate to them. These theories and their generalizations to automorphic forms for group of higher ranks are of fundamental importance in number theory.
For graduate students and newcomers to this field, the book provides a solid introduction to this highly active area of research. For experts, it will offer the convenience of collecting into one place foundational definitions and theorems with complete and self-contained proofs.
Written in an engaging and educational style, the book also includes exercises and provides their solution.
Trade Review“This book represented hope. If I read it carefully, maybe I would finally get to know what they were all talking about, and gain some real insight into what are obviously very important and influential ideas. While I cannot claim to be an expert by now, my first skim through, skipping all the exercises, has provided me with a satisfying foundation, and I found that revisited passages responded well to a second reading to consolidate what I had learned.” (Neil P. Dummigan, Mathematical Reviews, May, 2023)
“Complete proofs (or detailed references) of all statements are given and many exercises (with their solutions or hints) are included, hence the book may be addressed to graduate students working in this beautiful area of number theory and arithmetic algebraic geometry. This is a welcome addition to the literature in a field.” (Andrzej Dąbrowski, zbMATH 1493.11002, 2022)
Table of Contents- Introduction.-
Part I The ‘Eigen’ Construction.- Eigenalgebras.- Eigenvarieties.-
Part II Modular Symbols and L-Functions.- Abstract Modular Symbols.- Classical Modular Symbols, Modular Forms,
L-functions.- Rigid Analytic Modular Symbols and
p-Adic
L-functions.-
Part III The Eigencurve and its p-Adic L-Functions.- The Eigencurve of Modular Symbols.-
p-Adic
L-Functions on the Eigencurve.- The Adjoint
p-Adic
L-Function and the Ramification Locus of the Eigencurve.- Solutions and Hints to Exercises.
