# zbMATH — the first resource for mathematics

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
Full Text: