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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

11R42 Zeta functions and \(L\)-functions of number fields
11G05 Elliptic curves over global fields
11G15 Complex multiplication and moduli of abelian varieties