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Non-vanishing of certain values of $$L$$-functions. (English) Zbl 0629.12010
Analytic number theory and diophantine problems, Proc. Conf., Stillwater/Okla. 1984, Prog. Math. 70, 223-235 (1987).
[For the entire collection see Zbl 0618.00005.]
In §1, the author describes recent results of himself [Invent. Math. 79, 79–94 (1985; Zbl 0558.12005)] and those of D. Rohrlich [ibid. 75, 409–423, 383–408 (1984; Zbl 0565.14006, Zbl 0565.14008)] concerning non-vanishing of $$L(1/2,\chi)$$ when $$\chi$$ varies over the grossencharacters of an imaginary quadratic field $$K$$; here $$L(s,\chi)$$ denotes the Hecke $$L$$-function assigned to $$\chi$$. A conjecture inspired by these results is stated, and further generalizations thereof are considered.
The second paragraph is devoted to applications: let $$E$$ be an elliptic curve defined over $$\mathbb Q$$ with complex multiplication by the ring of integers of $$K$$ and let $$K_{\infty}$$ be any $$\mathbb Z_ p$$-extension of $$K$$ different from the anticyclotomic one, $$K_{\infty}=\cup_{n\geq 0}K_ n$$, $$K_{n+1}\supset K_ n$$, $$K_ n | K$$ is a cyclic extension of degree $$p^ n$$, then the rank of $$E(K_ n)$$ is bounded as $$n\to \infty$$. Related results and conjectures are discussed.
Finally (§3), a sketch of the proof of the non-vanishing theorems of §1 is given.
Reviewer: B. Z. Moroz

##### MSC:
 11R42 Zeta functions and $$L$$-functions of number fields 11G05 Elliptic curves over global fields 11G15 Complex multiplication and moduli of abelian varieties