Curve guide

Author: Li Jinghui
Publisher:
Publish Date: 2002-04-01
Features: The theory of modular curves is the best embodiment of arithmetic algebraic geometry, which is one of the deepest and most fruitful branches of modern number theory. So far, this theory has been scattered across a vast amount of literature in various international languages, with no dedicated monograph on the subject yet available. Therefore, this book is the first monograph on modular curve theory in the international community. The purpose of this book is to enable readers to quickly understand this field and subsequently read the most advanced literature, laying a foundation for in-depth research. The book first introduces the basic knowledge of arithmetic algebraic geometry created by Grothendieck, including representable functors, moduli spaces, Grothendieck topology, sheaves on categories, flat descent, stacks, as well as two important representable functors (i.e., the Hilbert functor and the Picard functor). Building on this foundation, it introduces the arithmetic algebraic geometry definition of modular curves along with elliptic curves, and then discusses the arithmetic algebraic geometry theory corresponding to Fourier expansions, differential forms, singular forms, and Hecke operators in the classical analytic theory of modular forms. This book can serve as a graduate textbook for mathematics departments in universities and can also be used by mathematicians engaged in research in number theory and algebraic geometry.