Central values of Hecke \(L\)-functions of CM number fields
type: Item
instance of: scholarly article
title: Central values of Hecke \(L\)-functions of CM number fields
zbMATH Open document ID: 0965.11045
DOI: 10.1215/S0012-7094-99-09817-4
author: Tonghai Yang
author: Fernando Rodriguez-Villegas
published in: Duke Mathematical Journal
publication date: 2000-03-27T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 11R42
Mathematics Subject Classification ID: 11G40
Mathematics Subject Classification ID: 11F67
zbMATH DE Number: 1425224
zbMATH Keywords: Hecke \(L\)-functions
zbMATH Keywords: complex multiplication
zbMATH Keywords: central special value
zbMATH Keywords: theta functions
zbMATH Keywords: CM elliptic curves
zbMATH Keywords: Mordell-Weil rank zero
zbMATH Keywords: nonvanishing
reviewed by: B. Z. Moroz
MaRDI profile type: Publication
cites work: Q3993071
cites work: Arithmetic on elliptic curves with complex multiplication. With an appendix by B. Mazur
cites work: Q3347985
cites work: Theta dichotomy for unitary groups
cites work: Splitting metaplectic covers of dual reductive pairs
cites work: On the L-functions of canonical Hecke characters of imaginary quadratic fields. II
cites work: On some explicit formulas in the theory of Weil representation
cites work: On the square root of special values of certain \(L\)-series
cites work: Square root formulas for central values of Hecke \(L\)-series. II
cites work: Q4203349
cites work: Q4855940
cites work: Q4015225
cites work: Galois conjugacy of unramified twists of Hecke characters
cites work: The non-vanishing of certain Hecke \(L\)-functions at the center of the critical strip
cites work: On the L-functions of canonical Hecke characters of imaginary quadratic fields
cites work: Root Numbers of Hecke L-Functions of CM Fields
cites work: Q4881421
cites work: Elliptic curves with complex multiplication and the conjecture of Birch and Swinnerton-Dyer
cites work: On the factors of the Jacobian variety of a modular function field
cites work: The special values of the zeta functions associated with cusp forms
cites work: On the periods of modular forms
cites work: Q3943047
cites work: Sur la formule de Siegel dans la théorie des groupes classiques
cites work: Theta liftings and Hecke L-functions.
cites work: Common zeros of theta functions and central Hecke L-values of CM number fields of degree 4
cites work: Eigenfunctions of the Weil representation of unitary groups of one variable
full work available at URL: s0012-7094-99-09817-4
OpenAlex ID: W2021233896
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description: scientific article; zbMATH DE number 1425224