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General Selmer groups and critical values of Hecke $$L$$-functions. (English) Zbl 0789.14018
Let $$E$$ be an elliptic curve over $$\mathbb{Q}$$ with complex multiplication by the ring of integers of an imaginary quadratic field $$K$$. Let $$\psi$$ be the Grössencharacter attached to the curve $$E$$ over $$K$$ by the theory of complex multiplication. Let $$\Omega$$ be the complex period and let $$- d_ K$$ denote the discriminant of $$K$$. For $$n \geq 1$$ define $$a_ n=n!(2 \pi/ \sqrt{d_ K})^ n \Omega^{-(2n-1)} L(\overline \psi^{2n+1},n+1)$$. The computation of B. H. Gross and D. Zagier in Mem. Soc. Math. Fr., Nouv. Sér. 108, No. 2, 49-54 (1980; Zbl 0462.14015)] shows that, for certain elliptic curves and certain values of $$n$$, the $$p$$-valuation of $$a_ n$$ is even for $$p \neq 2$$.
In the first part of this paper, it is proved that if $$E$$ has good, ordinary reduction at $$p>n+1$$, then the $$p$$-valuation of $$a_ n$$ is even. Next the author proves the following theorem about the Birch and Swinnerton-Dyer conjecture: Let $$E$$ be an elliptic curve over $$\mathbb{Q}$$ with complex multiplication. Either the order of vanishing of $$L(E,s)$$ at $$s=1$$ is congruent modulo 2 to the rank of $$E(\mathbb{Q})$$ or else the $$p$$- primary part of the Tate-Shafarevich group is infinite for all primes $$p$$ where $$E$$ has good, ordinary reduction. This improves a theorem of R. Greenberg[Invent. Math. 72, 241-265 (1983; Zbl 0546.14015)].

##### MSC:
 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) 14H52 Elliptic curves 14G05 Rational points 14H25 Arithmetic ground fields for curves 14G25 Global ground fields in algebraic geometry 14K22 Complex multiplication and abelian varieties
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##### References:
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