##
**On the Birch and Swinnerton-Dyer conjecture.**
*(English)*
Zbl 0546.14015

The main result is: Let \(E\) be an elliptic curve with complex multiplication. If \(L(E/\mathbb Q,s)\) has a zero of odd order at \(s=1\), then either the Mordell-Weil group \(E(\mathbb Q)\) has rank at least one, or the \(p\)-primary components of the Tate-Shafarevich group are infinite for all primes \(p\) (\(\neq 2,3)\) where at which \(E\) has ordinary good reduction. This complements a theorem of J. Coates and A. Wiles [Invent. Math. 39, 223–251 (1977; Zbl 0359.14009)] and provides further evidence for the Birch–Swinnerton-Dyer conjectures. The author records a variant of these conjectures suggested by Coates in case the Tate-Shafarevich groups have finite order.

The proof is via a Hecke \(L\)-series associated with a Grössencharacter \(\psi\). The \(L\)-series obeys a functional equation \(L(s)=wL(2-s)\). Both cases \(w=\pm 1\) are needed in the proof. In particular the author shows that \(L(\Psi^{2k+1},k+1)=0\) for only finitely many \(k\) when \(\Psi^{2k+1}\) is a character with \(w=1\). This non-vanishing theorem leads via some intricate work with the anti-cyclotomic \(p\)-extension of the complex multiplication field to a proof that the Selmer groups are infinite in the case \(w=-1\) and thence to the main theorem.

The proof is via a Hecke \(L\)-series associated with a Grössencharacter \(\psi\). The \(L\)-series obeys a functional equation \(L(s)=wL(2-s)\). Both cases \(w=\pm 1\) are needed in the proof. In particular the author shows that \(L(\Psi^{2k+1},k+1)=0\) for only finitely many \(k\) when \(\Psi^{2k+1}\) is a character with \(w=1\). This non-vanishing theorem leads via some intricate work with the anti-cyclotomic \(p\)-extension of the complex multiplication field to a proof that the Selmer groups are infinite in the case \(w=-1\) and thence to the main theorem.

Reviewer: G.Horrocks

### MSC:

11G40 | \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture |

11G15 | Complex multiplication and moduli of abelian varieties |

14G10 | Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) |

14H45 | Special algebraic curves and curves of low genus |

14K22 | Complex multiplication and abelian varieties |

14H52 | Elliptic curves |

14G05 | Rational points |

### Keywords:

elliptic curve with complex multiplication; Mordell-Weil group; Birch Swinnerton-Dyer conjectures; Hecke L-series; Selmer groups### Citations:

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