On the square root of special values of certain \(L\)-series
type: Item
instance of: scholarly article
title: On the square root of special values of certain \(L\)-series
zbMATH Open document ID: 0773.11034
DOI: 10.1007/BF01243924
author: Fernando Rodriguez-Villegas
published in: Inventiones Mathematicae
publication date: 1992-09-27T00:00:00Z
full work available at URL: https://eudml.org/doc/143953
review text: The main result of the paper is an explicit formula for the value at \(s=1\) of the \(L\)-series \(L(\psi,s)\) attached to certain Hecke characters \(\psi\) of an imaginary quadratic field of discriminant \(p\) where \(p\) is a prime \(\equiv 7\pmod 8\). The two main ingredients of the proof are (i) a formula of Hecke expressing the above \(L\)-series at \(s=1\) as a sum of binary theta series and (ii) a certain factorization lemma that gives the values of these theta series at complex multiplication points in the upper half-plane as the product of values of Jacobi theta functions. The author then applies his formula to the study of the elliptic curve \(A(p)\) over \(\mathbb{Q}(j(1+\sqrt{-p})/2)\) investigated by \textit{B. H. Gross} [Compos. Math. 45, 155-164 (1982; Zbl 0541.14010)] whose \(L\)-series can be expressed in terms of the series \(L(\psi,s)\) as above. He obtains an explicitly computable nonzero rational integer whose square by the Birch- Swinnerton-Dyer conjecture is the predicted order of the Tate-Shafarevich group of \(A(p)\). Finally it is pointed out that results of the type presented in this paper also follow from the work of \textit{J.-L. Waldspurger} [J. Math. Pures Appl., IX. Sér. 60, 375-484 (1981; Zbl 0431.10015)].
Mathematics Subject Classification ID: 11F67
Mathematics Subject Classification ID: 14G10
Mathematics Subject Classification ID: 11G15
Mathematics Subject Classification ID: 14H52
zbMATH DE Number: 59787
zbMATH Keywords: square root of special values
zbMATH Keywords: \(L\)-series
zbMATH Keywords: Hecke characters
zbMATH Keywords: theta series at complex multiplication points
zbMATH Keywords: values of Jacobi theta functions
zbMATH Keywords: elliptic curve
zbMATH Keywords: Tate-Shafarevich group
reviewed by: Winfried Kohnen
MaRDI profile type: Publication
cites work: Arithmetic on elliptic curves with complex multiplication. II
cites work: Q5735284
cites work: Arithmetic on elliptic curves with complex multiplication. With an appendix by B. Mazur
cites work: Q3328654
cites work: Q3904681
cites work: Q3267357
cites work: FINITENESS OF $ E(\mathbf{Q})$ AND $ \textrm{Ø}(E,\mathbf{Q})$ FOR A SUBCLASS OF WEIL CURVES
cites work: Q4075566
cites work: CYCLOTOMIC FIELDS AND MODULAR CURVES
cites work: The non-vanishing of certain Hecke \(L\)-functions at the center of the critical strip
cites work: Tate-Shafarevich groups and L-functions of elliptic curves with complex multiplication
cites work: On Elliptic Curves with Complex Multiplication as Factors of the Jacobians of Modular Function Fields
cites work: On the zeta-function of an abelian variety with complex multiplication
cites work: On the factors of the Jacobian variety of a modular function field
cites work: Q5625265
cites work: \(L\)-functions at \(s=1\). IV: First derivatives at \(s=0\)
cites work: Shimura and Shintani correspondences
cites work: Sur les coefficients de Fourier des formes modulaires de poids demi-entier
cites work: Q3280527
OpenAlex ID: W2075835335
description: scientific article; zbMATH DE number 59787