×

zbMATH — the first resource for mathematics

General Selmer groups and critical values of Hecke \(L\)-functions. (English) Zbl 0789.14018
Let \(E\) be an elliptic curve over \(\mathbb{Q}\) with complex multiplication by the ring of integers of an imaginary quadratic field \(K\). Let \(\psi\) be the Grössencharacter attached to the curve \(E\) over \(K\) by the theory of complex multiplication. Let \(\Omega\) be the complex period and let \(- d_ K\) denote the discriminant of \(K\). For \(n \geq 1\) define \(a_ n=n!(2 \pi/ \sqrt{d_ K})^ n \Omega^{-(2n-1)} L(\overline \psi^{2n+1},n+1)\). The computation of B. H. Gross and D. Zagier in Mem. Soc. Math. Fr., Nouv. Sér. 108, No. 2, 49-54 (1980; Zbl 0462.14015)] shows that, for certain elliptic curves and certain values of \(n\), the \(p\)-valuation of \(a_ n\) is even for \(p \neq 2\).
In the first part of this paper, it is proved that if \(E\) has good, ordinary reduction at \(p>n+1\), then the \(p\)-valuation of \(a_ n\) is even. Next the author proves the following theorem about the Birch and Swinnerton-Dyer conjecture: Let \(E\) be an elliptic curve over \(\mathbb{Q}\) with complex multiplication. Either the order of vanishing of \(L(E,s)\) at \(s=1\) is congruent modulo 2 to the rank of \(E(\mathbb{Q})\) or else the \(p\)- primary part of the Tate-Shafarevich group is infinite for all primes \(p\) where \(E\) has good, ordinary reduction. This improves a theorem of R. Greenberg[Invent. Math. 72, 241-265 (1983; Zbl 0546.14015)].

MSC:
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
14H52 Elliptic curves
14G05 Rational points
14H25 Arithmetic ground fields for curves
14G25 Global ground fields in algebraic geometry
14K22 Complex multiplication and abelian varieties
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] Coates, J.: Infinite descent on elliptic curves with complex multiplication. In: Artin, M., Tate, J. (eds.), Arithmetic and geometry, papers dedicated to I.R. Shafarevich on the occasion of his 60th birthday. (Prog. Math. vol. 35, pp. 107-136) Boston Basel Stuttgart: Birkh?user 1983
[2] de Shalit, E.: The Iwasawa theory of elliptic curves with complex multiplication. Perspect. Math.3 (1987) · Zbl 0674.12004
[3] Greenberg, R.: On the Birch and Swinnerton-Dyer conjecture. Invent. Math.72, 241-265 (1983) · Zbl 0546.14015
[4] Greenberg, R.: Iwasawa theory forp-adic representations. In: Coates, J. et al. (eds.) Algebraic number theory. (Adv. Stud. Pure Math., vol. 17, pp. 97-137) Boston, MA: Academic Press 1989 · Zbl 0739.11045
[5] Greenberg, R.: Iwasawa theory for motives. In Coates, J., Taylor, M.J. (eds.),L-functions and arithmetic. Proceedings of the Durham Symposium. (Lond. Math. Soc. Lect. Notes Ser., vol. 153, pp. 97-137) Cambridge: Cambridge University Press 1991
[6] Gross, B., Zagier, D.: On the critical values of HeckeL-series. Mem. Soc. Math. Fr.2, 49-54 (1980) · Zbl 0462.14015
[7] Guo, L.: On a Generalization of Tate dualities with application to Iwasawa theory. Compos. Math.85, 125-161 (1993) · Zbl 0789.11063
[8] Katz, N.:p-adicL functions for CM fields. Invent. Math.49, 199-297 (1978) · Zbl 0417.12003
[9] Mazur, B.: Rational points of abelian varieties with values in towers of number fields. Invent. Math.18 183-266 (1972) · Zbl 0245.14015
[10] Miyake, T.: Modular forms. Berlin Heidelberg New York: Springer 1989 · Zbl 0701.11014
[11] Perrin-Riou, B.: Arithm?tique des courbes elliptiques et th?orie d’Iwasawa. Bull. Soc. Math. Fr., Suppl.17 (1984) · Zbl 0599.14020
[12] Rohrlich, D.:L-functions and division towers. Math. Ann.281, 611-632 (1988) · Zbl 0656.14013
[13] Rubin, K.: On the main conjecture of Iwasawa theory for imaginary quadratic fields. Invent. Math.93, 701-713 (1988) · Zbl 0673.12004
[14] Rubin, K.: The ?main conjecture? of Iwasawa theory for imaginary quadratic fields. Invent. Math.103, 25-68 (1991) · Zbl 0737.11030
[15] Yager, R.: On two-variablep-adicL-functions. Ann. Math.115, 411-449 (1982) · Zbl 0496.12010
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.