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ALGEBRA AND RELATED TOPICS
Series on Number Theory and Its Applications - Vol 16
ELEMENTARY MODULAR IWASAWA THEORY
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by Haruzo Hida (University of California, Los Angeles, USA) This book is the first to provide a comprehensive and elementary account of the new Iwasawa theory innovated via the deformation theory of modular forms and Galois representations. The deformation theory of modular forms is developed by generalizing the cohomological approach discovered in the author’s 2019 AMS Leroy P Steele Prize-winning article without using much algebraic geometry. Starting with a description of Iwasawa’s classical results on his proof of the main conjecture under the Kummer – Vandiver conjecture (which proves cyclicity of his Iwasawa module more than just proving his main conjecture), we describe a generalization of the method proving cyclicity to the adjoint Selmer group of every ordinary deformation of a two-dimensional Artin Galois representation. Contents: Cyclotomic Iwasawa Theory; Cuspidal Iwasawa Theory; Cohomological Modular Forms and p-Adic L-Functions; p-Adic Families of Modular Forms; Abelian Deformation; Universal Ring and Compatible System; Cyclicity of Adjoint Selmer Groups; Local Indecomposability of Modular Galois Representation; Analytic and Topological Methods.
Readership: Advanced undergraduate and graduate students, researchers and practitioners in the fields of algebraic/analytic number theory and arithmetic geometry.
440pp Nov 2021 978-981-124-136-9 US$128 £115
Monographs in Number Theory - Vol 9
MODULAR AND AUTOMORPHIC FORMS & BEYOND
by Hossein Movasati (IMPA, Brazil)
The guiding principle in this monograph is to develop a new theory of modular forms which encompasses most of the available theory of modular forms in the literature, such as those for congruence groups, Siegel and Hilbert modular forms, many types of automorphic forms on Hermitian symmetric domains, Calabi – Yau modular forms, with its examples such as Yukawa couplings and topological string partition functions, and even go beyond all these cases. Its main ingredient is the so-called “Gauss – Manin connection in disguise”.
Contents: Introduction; Preliminaries in Algebraic Geometry; Enhanced Schemes; Topology and Periods; Foliations on Schemes; Modular Foliations; Hodge Cycles and Loci; Generalized Period Domain; Elliptic Curves; Product of Two Elliptic Curves; Abelian Varieties; Hypersurfaces; Calabi – Yau Varieties; Transcendence Questions: Introduction; A Biased Overview of Transcendence Theory; The Theorem of Nesterenko; Periods; ReferenceIndex.
Readership: Graduate students and researchers.
