General Selmer groups and critical values of Hecke \(L\)-functions
type: Item
instance of: scholarly article
title: General Selmer groups and critical values of Hecke \(L\)-functions
zbMATH Open document ID: 0789.14018
DOI: 10.1007/BF01459498
author: Li Guo
published in: Mathematische Annalen
publication date: 1994-06-16T00:00:00Z
full work available at URL: https://eudml.org/doc/165128
review text: Let \(E\) be an elliptic curve over \(\mathbb{Q}\) with complex multiplication by the ring of integers of an imaginary quadratic field \(K\). Let \(\psi\) be the Grössencharacter attached to the curve \(E\) over \(K\) by the theory of complex multiplication. Let \(\Omega\) be the complex period and let \(- d_ K\) denote the discriminant of \(K\). For \(n \geq 1\) define \(a_ n=n!(2 \pi/ \sqrt{d_ K})^ n \Omega^{-(2n-1)} L(\overline \psi^{2n+1},n+1)\). The computation of \textit{B. H. Gross} and \textit{D. Zagier} in Mem. Soc. Math. Fr., Nouv. Sér. 108, No. 2, 49-54 (1980; Zbl 0462.14015)] shows that, for certain elliptic curves and certain values of \(n\), the \(p\)-valuation of \(a_ n\) is even for \(p \neq 2\). In the first part of this paper, it is proved that if \(E\) has good, ordinary reduction at \(p>n+1\), then the \(p\)-valuation of \(a_ n\) is even. Next the author proves the following theorem about the Birch and Swinnerton-Dyer conjecture: Let \(E\) be an elliptic curve over \(\mathbb{Q}\) with complex multiplication. Either the order of vanishing of \(L(E,s)\) at \(s=1\) is congruent modulo 2 to the rank of \(E(\mathbb{Q})\) or else the \(p\)- primary part of the Tate-Shafarevich group is infinite for all primes \(p\) where \(E\) has good, ordinary reduction. This improves a theorem of \textit{R. Greenberg}[Invent. Math. 72, 241-265 (1983; Zbl 0546.14015)].
reviewed by: Li Guo
Mathematics Subject Classification ID: 14G10
Mathematics Subject Classification ID: 14H52
Mathematics Subject Classification ID: 14G05
Mathematics Subject Classification ID: 14H25
Mathematics Subject Classification ID: 14G25
Mathematics Subject Classification ID: 14K22
zbMATH DE Number: 537345
zbMATH Keywords: Selmer groups
zbMATH Keywords: Hecke \(L\)-functions
zbMATH Keywords: elliptic curve
zbMATH Keywords: complex multiplication
zbMATH Keywords: Grössencharacter
zbMATH Keywords: Birch and Swinnerton-Dyer conjecture
zbMATH Keywords: Tate-Shafarevich group
MaRDI profile type: Publication
cites work: Q3828097
cites work: On the Birch and Swinnerton-Dyer conjecture
cites work: Q3972988
cites work: On the critical values of Hecke L-series
cites work: Q4694964
cites work: \(p\)-adic \(L\)-functions for CM fields
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cites work: Arithmétique des courbes elliptiques et théorie d'Iwasawa
cites work: L-functions and division towers
cites work: On the main conjecture of Iwasawa theory for imaginary quadratic fields
cites work: The ``main conjectures'' of Iwasawa theory for imaginary quadratic fields
cites work: On two variable p-adic L-functions
OpenAlex ID: W1998149513
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description: scientific article; zbMATH DE number 537345