\(p\)-adic \(L\)-functions and rational points on elliptic curves with complex multiplication. (English) Zbl 0770.11033

The main result is the construction of rational points in \(E(\mathbb{Q})\), \(E\) an elliptic \(CM\)-curve over \(\mathbb{Q}\). More precisely the author uses a method of Perrin-Riou to construct elements in the Selmer group, which come from rational points if the Tate-Shafarevich group is finite (which is true in many cases by results of Kolyvagin and the author, which ultimately depend on the work of Gross-Zagier about Heegner points). The \(p\)-adic height of these points is then related to special values of \(p\)- adic \(L\)-functions.


11G18 Arithmetic aspects of modular and Shimura varieties
14G20 Local ground fields in algebraic geometry
14H52 Elliptic curves
Full Text: DOI EuDML


[1] Bernardi, D., Goldstein, C., Stephens, N.: Notesp-adiques sur les courbes elliptiques. J. Reine Angew. Math.351, 129-170 (1985) · Zbl 0529.14018
[2] Bertrand, D.: Algebraic values ofp-adic elliptic functions. In: Baker, A., Masser, D. (eds.) Transcendence Theory: Advances and Applications, pp. 149-159. London: Academic Press 1977
[3] Bertrand, D.: Valeurs de fonctions th?ta et hauteursp-adiques. In: Bertin, M.-J. (ed.) S?minaire de th?orie des nombres Paris 1980-81. (Prog. Math., vol. 22 pp. 1-11) Boston: Birkh?user 1982
[4] 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
[5] Coates, J., Wiles, A.: Onp-adicL-functions and elliptic units. J. Aust. Math. Soc. Ser. A26, 1-25 (1978) · Zbl 0442.12007
[6] de Shalit, E.: The Iwasawa Theory of Elliptic Curves with Complex Multiplication. (Perspect. Math., vol. 3) Orlando: Academic Press 1987 · Zbl 0674.12004
[7] Greenberg, R.: On the Birch and Swinnerton-Dyer conjecture. Invent. Math.72, 241-265 (1983) · Zbl 0546.14015
[8] Gross, B.: On the conjecture of Birch and Swinnerton-Dyer for elliptic curves with complex multiplication. In: Koblitz, N. (ed.) Number Theory related to Fermat’s Last Theorem. Cambridge, Mass. 1981. (Prog. Math., vol. 26, pp. 219-236) Boston: Birkh?user 1982
[9] Gross, B., Zagier, D.: Heegner points and derivatives of L-series. Invent. Math.84, 225-320 (1986) · Zbl 0608.14019
[10] Katz, N.:p-adic interpolation of real analytic Eisenstein series. Ann. Math.104, 459-571 (1976) · Zbl 0354.14007
[11] Kolyvagin, V.A.: Euler systems. In: The Grothendieck Festschrift, vol. II. (Prog. Math., vol. 87, pp. 435-483) Boston Basel Stuttgart: Birkh?user 1990 · Zbl 0742.14017
[12] Perrin-Riou, B.: Descente infinie et hauteurp-adique sur les courbes elliptiques ? multiplication complexe. Invent. Math.70, 369-398 (1983) · Zbl 0547.14025
[13] Perrin-Riou, B.: Arithm?tique des courbes elliptiques et th?orie d’Iwasawa. Bull. Soc. Math. Fr. Suppl.17 (1984) · Zbl 0599.14020
[14] Perrin-Riou, B.: Points de Heegner et d?riv?es de fonctions Lp-adiques. Invent. Math.89, 455-510 (1987) · Zbl 0645.14010
[15] Robert, G.: Concernant le relation de distribution satisfaite par la fonction ? associ?e ? un r?seau complexe. Invent. Math.100, 231-257 (1990) · Zbl 0729.11029
[16] Rubin, K.: Tate-Shafarevich groups and L-functions of elliptic curves with complex multiplication. Invent. Math.89, 527-560 (1987) · Zbl 0628.14018
[17] Rubin, K.: The ?main conjectures? of Iwasawa theory for imaginary quadratic fields. Invent. Math.103, 25-68 (1991) · Zbl 0737.11030
[18] Shimura, G.: Introduction to the Arithmetic Theory of Automorphic Forms. Princeton: Princeton University Press 1971 · Zbl 0221.10029
[19] Tate, J.: The arithmetic of elliptic curves. Invent. Math.23, 179-206 (1974) · Zbl 0296.14018
[20] Tate, J.: Algorithm for determining the type of a singular fiber in an elliptic pencil. In: Birch, B.J., Kuyk, W. (eds.) Modular Functions of One Variable (IV). (Lect. Notes Math., vol. 476, pp. 33-52) Berlin Heidelberg New York: Springer 1975 · Zbl 1214.14020
[21] Wiles, A.: Higher explicit reciprocity laws. Ann. Math.107, 235-254 (1978) · Zbl 0378.12006
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.