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This work is a comprehensive treatment of recent developments in the study of elliptic curves and their moduli spaces. The arithmetic study of the moduli spaces began with Jacobi's "Fundamenta Nova" in 1829, and the modern theory was erected by Eichler-Shimura, Igusa, and Deligne-Rapoport. In the past decade mathematicians have made further substantial progress in the field. This book gives a complete account of that progress, including not only the work of the authors, but also that of Deligne and Drinfeld.

Curves, Elliptic --- Moduli theory --- Theory of moduli --- Functions of several complex variables --- Elliptic curves --- Curves, Algebraic --- Geometry, Algebraic --- 511.3 --- Analytic spaces --- Algebraic geometry --- Geometry --- 511.3 Analytical, additive and other number-theory problems. Diophantine approximations --- Analytical, additive and other number-theory problems. Diophantine approximations --- Ordered algebraic structures --- Curves, Elliptic. --- Moduli theory. --- Geometry, Algebraic. --- Abelian variety. --- Addition. --- Algebraic variety. --- Algebraically closed field. --- Ambient space. --- Arithmetic. --- Axiom. --- Barry Mazur. --- Base change. --- Calculation. --- Canonical map. --- Change of base. --- Closed immersion. --- Coefficient. --- Coherent sheaf. --- Cokernel. --- Commutative property. --- Congruence relation. --- Coprime integers. --- Corollary. --- Cusp form. --- Cyclic group. --- Dense set. --- Diagram (category theory). --- Dimension. --- Discrete valuation ring. --- Disjoint union. --- Divisor. --- Eigenfunction. --- Elliptic curve. --- Empty set. --- Factorization. --- Field of fractions. --- Finite field. --- Finite group. --- Finite morphism. --- Free module. --- Functor. --- Group (mathematics). --- Integer. --- Irreducible component. --- Level structure. --- Local ring. --- Maximal ideal. --- Modular curve. --- Modular equation. --- Modular form. --- Moduli space. --- Morphism of schemes. --- Morphism. --- Neighbourhood (mathematics). --- Noetherian. --- One-parameter group. --- Open problem. --- Prime factor. --- Prime number. --- Prime power. --- Q.E.D. --- Regularity theorem. --- Representation theory. --- Residue field. --- Riemann hypothesis. --- Smoothness. --- Special case. --- Subgroup. --- Subring. --- Subset. --- Theorem. --- Topology. --- Two-dimensional space. --- Zariski topology.