zbMATH — the first resource for mathematics

On \(L\)-functions of elliptic curves and cyclotomic towers. (English) Zbl 0565.14006
Let \(f\) be a normalized new form of weight 2, character \(\psi\), and level \(N\). Let \(P\) be a finite set of primes not dividing \(N\), and let \(X\) be the set of all primitive Dirichlet characters which are unramified outside \(P\) and infinity. For \(\chi\in X\), let \(L(s,f,\chi)\) be the \(L\)-function attached to \(f\) and \(\chi\). The main result in this paper is the following theorem:
For all but finitely many \(\chi\in X\), \(L(1,f,\chi)\neq 0\).
Two consequences of this result are given. One is a conjecture of B. Mazur and P. Swinnerton-Dyer [Invent. Math. 25, 1–61 (1974; Zbl 0281.14016)] to the effect that the \(p\)-adic \(L\)-function attached to a Weil curve over an abelian number field is not identically zero.
The other is as follows: Let \(E\) be an elliptic curve defined over \(\mathbb Q\) with complex multiplication by the ring of integers of an imaginary quadratic number field, let \(P\) be a finite set of primes where \(E\) has good reduction. Let \(L\) be the maximal abelian extension of \(\mathbb Q\) unramified outside \(P\) and infinity, and let \(E(L)\) be the group of \(L\)-rational points on \(E\). Then \(E(L)\) is finitely generated.
This generalizes a result of K. Rubin and A. Wiles [Number theory related to Fermat’s last theorem, Proc. Conf., Prog. Math. 26, 237–254 (1982; Zbl 0519.14017)]. If the conjectures of Taniyama-Weil and Birch-Swinnerton-Dyer hold, then the restriction to the case of complex multiplication can be removed.

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)
Full Text: DOI EuDML
[1] Atkin, A.O.L., Lehner, J.: Hecke operators on ?0(m). Math. Ann.185, 134-160 (1970) · Zbl 0185.15502
[2] Estermann, T.: On Kloosterman’s sum. Mathematika8, 83-86 (1961) · Zbl 0114.26302
[3] Greenberg, R.: On the Birch and Swinnerton-Dyer conjecture. Invent. math.72, 241-265 (1983) · Zbl 0546.14015
[4] Igusa, J.: Kroneckerian models of fields of elliptic modular functions. Amer. J. Math.81, 561-577 (1959) · Zbl 0093.04502
[5] Le Veque, W.J.: Topics in number theory, vol. 1 Reading: Addison-Wesley 1956
[6] Li, W.W.: Newforms and functional equations. Math. Ann.212, 285-315 (1975) · Zbl 0286.10016
[7] Manin, Y.I.: Cyclotomic fields and modular curves. Russian Math. Surveys26, 7-78 (1971) · Zbl 0266.14012
[8] Mazur, B.: Rational points of abelian varieties with values in towers of number fields. Invent. math.18, 183-266 (1972) · Zbl 0245.14015
[9] Mazur, B.: On the arithmetic of special values ofL-functions. Invent. math.55, 207-240 (1979) · Zbl 0426.14009
[10] Mazur, B., Swinnerton-Dyer, H.P.F.: Arithmetic of Weil curves. Invent. math.25, 1-61 (1974) · Zbl 0281.14016
[11] Miyake, T.: On automorphic forms onGL 2 and Hecke operators. Annals of Math.94, 174-189 (1971) · Zbl 0215.37301
[12] Ribet, K.: Appendix to: ?Finiteness theorems in geometric classfield theory,? by N. Katz and S. Lang. L’Enseignement Math.27, 315-319 (1981)
[13] Rohrlich, D.E.: OnL-functions of elliptic curves and anticyclotomic towers. Invent. math.75, 383-408 (1984) · Zbl 0565.14008
[14] Rubin, K.: Elliptic curves with complex multiplication and the conjecture of Birch and Swinnerton-Dyer. Invent. math.64, 455-470 (1981) · Zbl 0506.14039
[15] Rubin, K.: Congruences for special values ofL-functions of elliptic curves with complex multiplication. Invent. math.71, 339-364 (1983) · Zbl 0513.14012
[16] 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
[17] Shimura, G.: Introduction to the arithmetic theory of automorphic functions. Publ. Math. Soc. Japan, vol. 11. Tokyo-Princeton 1971 · Zbl 0221.10029
[18] Shimura, G.: On elliptic curves with complex multiplication as factors of the jacobians of modular function fields. Nagoya Math. J.43, 199-208 (1971) · Zbl 0225.14015
[19] Shimura, G.: The special values of the zeta functions associated with cusp forms. Comm. Pure Appl. Math.29, 783-804 (1976) · Zbl 0348.10015
[20] Shimura, G.: On the periods of modular forms. Math. Ann.229, 211-221 (1977) · Zbl 0363.10019
[21] Stevens, G.: Arithmetic on modular curves. Progress in Math., vol. 20. Boston: Birkhäuser 1982 · Zbl 0529.10028
[22] Weil, A.: On some exponential sums. Proc. Nat. Acad. Sci.34, 204-207 (1948) (=Coll. Papers,I, 386-389) · Zbl 0032.26102
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.