Nonvanishing of Hecke \(L\)-functions for CM fields and ranks of Abelian varieties. (English) Zbl 1278.11057
Summary: In this paper we prove a nonvanishing theorem for central values of \(L\)-functions associated to a large class of algebraic Hecke characters of CM number fields. A key ingredient in the proof is an asymptotic formula for the average of these central values. We combine the nonvanishing theorem with work of Y. Tian and S.-W. Zhang [Kolyvagin systems of CM points on Shimura curves, preprint (2008)] to deduce that infinitely many of the CM abelian varieties associated to these Hecke characters have Mordell-Weil rank zero. Included among these abelian varieties are higher-dimensional analogues of the elliptic \(\mathbb Q\)-curves \(A(D)\) of B. H. Gross [Arithmetic on elliptic curves with complex multiplication. Lecture Notes in Mathematics. 776. Berlin etc.: Springer-Verlag (1980; Zbl 0433.14032)].
MSC:
| 11F67 | Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols |
| 11F27 | Theta series; Weil representation; theta correspondences |
| 11F41 | Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces |
| 11G10 | Abelian varieties of dimension \(> 1\) |
| 11G40 | \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture |
| 11R80 | Totally real fields |
Citations:
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