Elliptic curves with complex multiplication and the conjecture of Birch and Swinnerton-Dyer

DOI10.1007/BF01389277zbMath0506.14039WikidataQ123160128 ScholiaQ123160128MaRDI QIDQ1836711

Karl Rubin

Publication date: 1981

Published in: Inventiones Mathematicae (Search for Journal in Brave)

Full work available at URL: https://eudml.org/doc/142832

zbMATH Keywords

elliptic curves with complex multiplicationimaginary quadratic fieldBirch--Swinnerton-Dyer conjecture

Mathematics Subject Classification ID

Elliptic curves over global fields (11G05) Special algebraic curves and curves of low genus (14H45) Complex multiplication and moduli of abelian varieties (11G15) (L)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture (11G40)

Related Items (30)

Quadratic Chabauty and 𝑝-adic Gross–Zagier ⋮ Local units, elliptic units, Heegner points and elliptic curves ⋮ Non-vanishing of central \(L\)-values of canonical CM elliptic curves with quadratic twists ⋮ L-functions and division towers ⋮ Fonctions $L$ $p$-adiques à deux variables et ${\bbfZ}_2^p$-extensions ⋮ On the main conjecture of Iwasawa theory for imaginary quadratic fields ⋮ Elliptic curves, \(L\)-functions, and Hilbert's tenth problem ⋮ Nonvanishing of Hecke \(L\)-functions and the Bloch-Kato conjecture ⋮ Weak Leopoldt's conjecture for Hecke characters of imaginary quadratic fields ⋮ A Bernstein-von Mises theorem for discrete probability distributions ⋮ Stark conjectures for CM elliptic curves over number fields. ⋮ Descending Rational Points on Elliptic Curves to Smaller Fields ⋮ Central values of Hecke \(L\)-functions of CM number fields ⋮ On the Birch and Swinnerton-Dyer conjecture ⋮ Heegner cycles and \(p\)-adic \(L\)-functions ⋮ Torsion points on some abelian surfaces of CM-type ⋮ Trivial zeros of -adic -functions at near-central points ⋮ Nonvanishing of Hecke \(L\)-functions for CM fields and ranks of Abelian varieties ⋮ Unnamed Item ⋮ Unnamed Item ⋮ Rubin's conjecture on local units in the anticyclotomic tower at inert primes ⋮ On the critical values of Hecke \(L\)-functions for imaginary quadratic fields ⋮ Unnamed Item ⋮ On the main conjecture of Iwasawa theory for certain non‐cyclotomic Zp‐extensions ⋮ Congruences for special values of L-functions of elliptic curves with complex multiplication ⋮ On the L-functions of canonical Hecke characters of imaginary quadratic fields. II ⋮ On \(L\)-functions of elliptic curves and cyclotomic towers ⋮ On \(L\)-functions of elliptic curves and anticyclotomic towers ⋮ Coates-Wiles towers in dimension two ⋮ Generalized Heegner cycles and p-adic L-functions in a quaternionic setting

Cites Work

• Division values in local fields
• Arithmetic on elliptic curves with complex multiplication. With an appendix by B. Mazur
• On the conjecture of Birch and Swinnerton-Dyer
• Higher explicit reciprocity laws
• Formal complex multiplication in local fields
• Good reduction of abelian varieties
• On the zeta-function of an abelian variety with complex multiplication
• Algorithm for determining the type of a singular fiber in an elliptic pencil
• On the units of algebraic number fields
• Arithmetic on curves of genus 1. VIII. On conjectures of Birch and Swinnerton-Dyer.
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