General Selmer groups and critical values of Hecke \(L\)-functions
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 \textit{B. H. Gross} and \textit{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 \textit{R. Greenberg}[Invent. Math. 72, 241-265 (1983; Zbl 0546.14015)].
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