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Forms, Quadratic --- Curves, Elliptic --- Number Theory

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Numerical approximation theory --- Geometry --- Curves on surfaces --- Handbooks, manuals, etc.

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Wavelets-congresses --- Approximation theory-congresses --- Curves-computer simulation-congresses --- Surfaces-computer simulation-congresses --- Wavelets (Mathematics) --- Approximation theory --- Curves --- Surfaces

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This book continues the treatment of the arithmetic theory of elliptic curves begun in the first volume. The book begins with the theory of elliptic and modular functions for the full modular group r(1), including a discussion of Hekcke operators and the L-series associated to cusp forms. This is followed by a detailed study of elliptic curves with complex multiplication, their associated Grössencharacters and L-series, and applications to the construction of abelian extensions of quadratic imaginary fields. Next comes a treatment of elliptic curves over function fields and elliptic surfaces, including specialization theorems for heights and sections. This material serves as a prelude to the theory of minimal models and Néron models of elliptic curves, with a discussion of special fibers, conductors, and Ogg's formula. Next comes a brief description of q-models for elliptic curves over C and R, followed by Tate's theory of q-models for elliptic curves with non-integral j-invariant over p-adic fields. The book concludes with the construction of canonical local height functions on elliptic curves, including explicit formulas for both archimedean and non-archimedean fields.

Algebraic geometry --- Number theory --- Arithmetic --- Curves, Algebraic --- Curves, Elliptic --- Arithmétique --- Courbes algébriques --- Courbes elliptiques --- 512.74 --- #KVIV:BB --- Algebraic groups. Abelian varieties --- Arithmetic. --- Curves, Algebraic. --- Curves, Elliptic. --- 512.74 Algebraic groups. Abelian varieties --- Arithmétique --- Courbes algébriques --- Elliptic curves --- Algebraic curves --- Algebraic varieties --- Mathematics --- Set theory --- Calculators --- Numbers, Real

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This book consists of two parts. The first is devoted to the theory of curves, which are treated from both the analytic and algebraic points of view. Starting with the basic notions of the theory of Riemann surfaces the reader is lead into an exposition covering the Riemann-Roch theorem, Riemann's fundamental existence theorem, uniformization and automorphic functions. The algebraic material also treats algebraic curves over an arbitrary field and the connection between algebraic curves and Abelian varieties. The second part is an introduction to higher-dimensional algebraic geometry. The author deals with algebraic varieties, the corresponding morphisms, the theory of coherent sheaves and, finally, the theory of schemes. This book is a very readable introduction to algebraic geometry and will be immensely useful to mathematicians working in algebraic geometry and complex analysis and especially to graduate students in these fields.

Curves, Algebraic. --- Manifolds (Mathematics) --- Schemes (Algebraic geometry) --- Manifolds (Mathematics). --- Schemes (Algebraic geometry).