On \(L\)-functions of elliptic curves and anticyclotomic towers
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.
scientific article; zbMATH DE number 3900921
scientific article