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On $$L$$-functions of elliptic curves and anticyclotomic towers. (English) Zbl 0565.14008
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 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 [D. Ridout, Mathematika 5, 40–48 (1958; Zbl 0085.03501)] of the Thue-Siegel-Roth theorem.

MSC:
 11G40 $$L$$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 11R23 Iwasawa theory 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
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References:
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