Presents an elementary and self-contained approach to Abelian varieties, a subject that plays a central role in algebraic and analytic geometry, number theory and complex analysis. Based on notes from a course at Concordia University it should be useful for independent study or graduate courses.
This text presents an elementary and self-contained approach to Abelian varieties, a subject that plays a central role in algebraic and analytic geometry, number theory and complex analysis. The book is based on notes from a course given at Concordia University and would be useful for independent study or as a textbook for graduate courses in complex analysis, Riemann surfaces, number theory, or analytic geometry. Murty works mostly over the complex numbers, discussing the theorem of Abel-Jacobi and Lefschetz's theorem on projective embeddings. After presenting some examples, Murty touches on Abelian varieties over number fields, as well as the conjecture of Tate (Faltings's theorem) and its relation to Mordell's conjecture. References are provided to guide the reader in further study.
Introduction. Riemann surfaces. Riemann-Roch theorem. Abel-Jacobi theorem and period relations. Divisors and theta functions. Dimension of the space of theta functions. Projective embedding and theta functions. Elliptic curves as the intersection of two quadrics. The Fermat curve. Discrete subgroups of ${\rm SL}_2(\Bbb R)$. Riemann surface structure of $\Gamma \backslash {\cal H}*$. The modular curve $X(N)$. Generalities on Abelian varieties. The conjecture of Tate. Finiteness of isogeny classes. Mordell's conjecture. Bibliography. Index
This text presents an elementary and self-contained approach to Abelian varieties, a subject that plays a central role in algebraic and analytic geometry, number theory and complex analysis. The book is based on notes from a course given at Concordia University and would be useful for independent study or as a textbook for graduate courses in complex analysis, Riemann surfaces, number theory, or analytic geometry. Murty works mostly over the complex numbers, discussing the theorem of Abel-Jacobi and Lefschetz's theorem on projective embeddings. After presenting some examples, Murty touches on Abelian varieties over number fields, as well as the conjecture of Tate (Faltings's theorem) and its relation to Mordell's conjecture. References are provided to guide the reader in further study.
Introduction. Riemann surfaces. Riemann-Roch theorem. Abel-Jacobi theorem and period relations. Divisors and theta functions. Dimension of the space of theta functions. Projective embedding and theta functions. Elliptic curves as the intersection of two quadrics. The Fermat curve. Discrete subgroups of ${\rm SL}_2(\Bbb R)$. Riemann surface structure of $\Gamma \backslash {\cal H}*$. The modular curve $X(N)$. Generalities on Abelian varieties. The conjecture of Tate. Finiteness of isogeny classes. Mordell's conjecture. Bibliography. Index