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This book provides an exposition of function field arithmetic with emphasis on recent developments concerning Drinfeld modules, the arithmetic of special values of transcendental functions (such as zeta and gamma functions and their interpolations), diophantine approximation and related interesting open problems. While it covers many topics treated in 'Basic Structures of Function Field Arithmetic' by David Goss, it complements that book with the inclusion of recent developments as well as the treatment of new topics such as diophantine approximation, hypergeometric functions, modular forms, transcendence, automata and solitons. There is also new work on multizeta values and log-algebraicity. The author has included numerous worked-out examples. Many open problems, which can serve as good thesis problems, are discussed.
Contents:
- Number Fields and Function Fields
- Drinfeld Modules
- Explicit Class Field Theory
- Gauss Sums and Gamma Functions
- Zeta Functions
- Higher Rank Theory
- Higher Dimensions and Geometric Tools
- Applications to Gauss Sums, Gamma and Zeta Values
- Diophantine Approximation
- Transcendence Results
- Automata and Algebraicity: Applications
Readership: Graduate students and researchers in algebra, number theory and geometry.
