×

zbMATH — the first resource for mathematics

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)
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] Greenberg, R.: On the Birch and Swinnerton-Dyer conjecture. Invent. math.72, 241-265 (1983) · Zbl 0546.14015
[2] Gross, B.H., Zagier, D.: Points de Heegner et dérivées de fonctionsL. C.R. Acad. Sc. Paris297, Série I 85-87 (1983)
[3] Hecke, E.: Mathematische Werke. Vandenhoeck & Ruprecht, 1959
[4] Kurcanov, P.F.: Elliptic curves of infinite rank over ?-extensions. Mat. Sbornik90, 320-324 (1973) · Zbl 0273.14009
[5] Kurcanov, P.F.: On the rank of elliptic curves over ?-extensions. Mat. Sbornik93, 460-466 (1974) · Zbl 0273.14009
[6] Mazur, B.: Rational points of abelian varieties with values in towers of number fields. Invent. math.18, 183-266 (1972) · Zbl 0245.14015
[7] Ridout, D.: Thep-adic generalization of the Thue-Siegel-Roth theorem. Mathematika5, 40-48 (1958) · Zbl 0085.03501
[8] Roth, K.F.: Rational approximations to algebraic numbers. Mathematika2, 1-20 (1955) · Zbl 0064.28501
[9] Rubin, K.: Elliptic curves with complex multiplication and the conjecture of Birch and Swinnerton-Dyer. Invent. math.64, 455-470 (1981) · Zbl 0506.14039
[10] Rubin, K.: Congruences for special values ofL-functions of elliptic curves with complex multiplication. Invent. math.71, 339-364 (1983) · Zbl 0513.14012
[11] Rubin, K., Wiles, A.: Mordell-Weil groups of elliptic curves over cyclotomic fields. In: Number Theory Related to Fermat’s last theorem. Boston: Birkhäuser 1982 · Zbl 0519.14017
[12] Shimura, G.: On the zeta-function of an abelian variety with complex multiplication. Ann. Math.94, 504-533 (1971) · Zbl 0242.14009
[13] Shimura, G.: The special values of zeta functions associated with cusp forms. Comm. Pure Appl. Math.29, 783-804 (1976) · Zbl 0348.10015
[14] Shimura, G.: On the periods of modular forms. Math. Ann.229, 211-221 (1977) · Zbl 0363.10019
[15] Stevens, G.: Arithmetic on Modular Curves. Progress in Math. vol. 20. Boston: Birkhäuser 1982 · Zbl 0529.10028
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.