Central values of Hecke \(L\)-functions of CM number fields
The authors write: ``In this paper, we prove a formula expressing the central special value of the \(L\)-functions of certain Hecke characters in terms of theta functions. The formula easily implies that the central value is nonnegative and yields a criterion for its positivity. Combining this criterion with the work of Arthaud and Rubin, we show that certain CM elliptic curves have Mordell-Weil rank zero over their field of definition\dots The idea to derive the main formula is as follows. We have the formula of \textit{T. Yang} [J. Reine Angew. Math. 485, 25-53 (1997; Zbl 0867.11037)] expressing the central \(L\)-value as the absolute square of some theta integral on a unitary group of one varible. The task is simply to compute the theta integral\dots To obtain the nonvanishing results, we use a theorem of Shimura concerning special values of \(L\)-functions of conjugate Hecke characters.'' I cite one of the authors' results: Let a rational prime \(p\) and a positive rational integer \(d\) satisfy the following conditions \(p\equiv 7\bmod 8\), \(d\equiv 1\bmod 4\), and \(p> 0.081d^4\). Let us consider the CM elliptic curve \(A(p)^d\) with complex multiplication by \(E= \mathbb{Q} (\sqrt{-p})\), studied by \textit{B. Gross} [Arithmetic on elliptic curves with complex multiplication, Lecture Notes in Mathematics 776 (1980; Zbl 0433.14032)]. Suppose that every rational prime dividing \(d\) splits in \(E\), then the elliptic curve \(A(p)^d\) has Mordell-Weil rank zero over its field definition.
scientific article; zbMATH DE number 1425224
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