On the passage from local to global in number theory

DOI10.1090/S0273-0979-1993-00414-2zbMath0813.14016arXivmath/9307231OpenAlexW2032634117MaRDI QIDQ3138087

Barry Mazur

Publication date: 30 May 1995

Published in: Bulletin of the American Mathematical Society (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/math/9307231

zbMATH Keywords

rational pointsTate-Shafarevich groupclass field theoryrationality questionssmooth projective varietieslocal-global principlesHasse-Weil \(L\)-function of modular elliptic curvesKolyvagin test classesSelmer groups of elliptic curvesSelmer's curveTate-Shafarevich conjecture

Mathematics Subject Classification ID

Rational points (14G05) Research exposition (monographs, survey articles) pertaining to number theory (11-02) Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) (14G10) Local ground fields in algebraic geometry (14G20) Research exposition (monographs, survey articles) pertaining to algebraic geometry (14-02) Arithmetic ground fields for surfaces or higher-dimensional varieties (14J20) Arithmetic ground fields for abelian varieties (14K15) Global ground fields in algebraic geometry (14G25) Arithmetic algebraic geometry (Diophantine geometry) (11Gxx)

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