On \(L\)-functions of elliptic curves and anticyclotomic towers
type: Item
instance of: scholarly article
title: On \(L\)-functions of elliptic curves and anticyclotomic towers
zbMATH Open document ID: 0565.14008
DOI: 10.1007/BF01388635
author: David E. Rohrlich
published in: Inventiones Mathematicae
publication date: 1984-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143102
review text: Let \(K\) be an imaginary quadratic number field. Let \(P\) be a finite set of prime numbers, and let \(L\) be the maximal anticyclotomic extension of \(K\) unramified outside \(P\). Let \(E\) be an elliptic curve defined over \(\mathbb Q\) which has complex multiplication by the ring of integers in \(K\). Let \(V=\mathbb C\otimes E(L)\) where \(E(L)\) denotes the group of \(L\)-rational points on \(E\). There is a decomposition \(V=\oplus_{\rho}V(\rho)\) where \(\rho\) runs over the distinct characters of \(\text{Gal}(L/K)\). If \(L(s,E/\mathbb Q)\) is the \(L\)-function of \(E\) over \(\mathbb Q\), then \(L(s,E/\mathbb Q)=L(s,\phi)\) where \(\phi\) is the Hecke character of \(K\) determined by \(E\). Let \(X\) be the set of all Hecke characters of the form \(\chi =\phi \rho\) where \(\rho\) is a character of \(\text{Gal}(L/K)\). Let \(W(\chi)\) be the root number in the functional equation for \(L(s,\chi)\). The main theorem in this paper is the following: For all but finitely many \(\chi\) in \(X\), \(\text{ord}_{s=1}L(s,\chi)=0\) if \(W(\chi)=1\) and \(\text{ord}_{s=1}L(s,\chi)=1\) if \(W(\chi)=-1\). This is a generalization of a result of \textit{R. Greenberg} [Invent. Math. 72, 241--265 (1983; Zbl 0546.14015)] who proved the theorem in the case that \(P\) consists of a single odd prime of ordinary reduction for \(E\) and \(W(\chi)=1\). As a corollary, the author obtains that for all but finitely many \(\rho\), \[ W(\phi \rho^{-1})=1\Rightarrow \dim V(\rho)=0, \] and \[ W(\phi \rho^{-1})=-1\Rightarrow \dim V(\rho)\geq 1\quad \text{or}\;\dim V(\rho^{-1})\geq 1. \] An interesting aspect of the proof is that it depends upon Ridout's generalization [\textit{D. Ridout}, Mathematika 5, 40--48 (1958; Zbl 0085.03501)] of the Thue-Siegel-Roth theorem.
reviewed by: Loren D. Olson
Mathematics Subject Classification ID: 11G40
Mathematics Subject Classification ID: 11R23
Mathematics Subject Classification ID: 14G10
zbMATH DE Number: 3900921
zbMATH Keywords: group of rational points
zbMATH Keywords: maximal anticyclotomic extension
zbMATH Keywords: elliptic curve
zbMATH Keywords: complex multiplication
zbMATH Keywords: L-function
zbMATH Keywords: Thue-Siegel-Roth theorem
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3900921