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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.
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
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