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A new construction of \({\mathfrak p}\)-adic L-functions attached to certain elliptic curves with complex multiplication. (English) Zbl 0608.14015
Although there now exists a vast literature on the p-adic L-functions attached to primes of ordinary reduction of an elliptic curve E, few works have been concerned with the case of supersingular primes. In this paper, the author restricts to the case of complex multiplications, and constructs a one-variable p-adic L-function in both cases, under a standard assumption on the field generated by the torsion points of E. The construction is based on interpolation of Eisenstein series. It would be interesting to compare the p-adic period occuring in this method with those given by Hodge-Tate theory. The article is complemented by a p-adic analogue of the Kronecker limit formula, and by a study of the L- functions having a pole at \(s=0\).
Reviewer: D.Bertrand

MSC:
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
14K22 Complex multiplication and abelian varieties
14H45 Special algebraic curves and curves of low genus
11S40 Zeta functions and \(L\)-functions
14G20 Local ground fields in algebraic geometry
14H20 Singularities of curves, local rings
14H52 Elliptic curves
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References:
[1] J. L. BOXALL, P-adic interpolation of logarithmic derivatives associated to certain Lubin-Tate formal groups, Ann. Inst. Fourier, Grenoble, 36, 3 (1986), to appear. · Zbl 0587.12007
[2] J. L. BOXALL, On p-adic L-functions attached to elliptic curves with complex multiplication (to appear). · Zbl 0608.14015
[3] A. BRUMER, On the units of algebraic number fields, Mathematika, 14 (1967), 121-124. · Zbl 0171.01105
[4] J. COATES and C. GOLDSTEIN, Some remarks on the main conjecture for elliptic curves with complex multiplication, Amer J. Math., 105 (1983), 337-366. · Zbl 0524.14023
[5] J. COATES and A. WILES, On the conjecture of Birch and Swinnerton-Dyer, Inventiones Math., 39 (1977), 223-251. · Zbl 0359.14009
[6] J. COATES and A. WILES, On p-adic L-functions and elliptic units, J. Austral. Math. Soc., ser. A, 26 (1978), 1-25. · Zbl 0442.12007
[7] R. DAMERELL, L-functions of elliptic curves with complex multiplication, Acta Arith., 17 (1970), 287-301. · Zbl 0209.24603
[8] R. GILLARD and G. ROBERT, Groupes d’unités elliptiques, Bull. Soc. Math. France, 107 (1979), 305-317. · Zbl 0434.12003
[9] C. GOLDSTEIN and N. SCHAPPACHER, Séries d’Eisenstein et fonctions L de courbes elliptiques à multiplication complexe, Crelle’s J., 327 (1981), 184-218. · Zbl 0456.12007
[10] K. IWASAWA, Lectures on p-adic L-functions, Annals of Math. Studies, 74 P.U.P. (1972). · Zbl 0236.12001
[11] E. E. KUMMER, Ùber eine allgemeine eigenschaft der rationale entwicklungs coefficienten einer bestimmten gattung analysischer functionen, Crelle’s J., 41 (1851), 368-372, (= Collected Works vol. 1, pp. 358-362 Springer-Verlag (1975)).
[12] T. KUBOTA and H. W. LEOPOLDT, Eine p-adische theorie der zetawerte, Crelle’s J., 214/215 (1964), 328-339. · Zbl 0186.09103
[13] N. KATZ, P-adic interpolation of real-analytic Eisenstein series, Annals of Math., 104 (1976), 459-571. · Zbl 0354.14007
[14] N. KATZ, The Eisenstein measure and p-adic interpolation, Amer. J. Math., 99 (1977), 238-311. · Zbl 0375.12022
[15] N. KATZ, Formal groups and p-adic interpolation, Astérisque, 41-42 (1977), 55-65. · Zbl 0351.14024
[16] N. KATZ, Divisibilities, Congruences and Cartier Duality, J. Fac. Sci. Univ. Tokyo, Ser. 1A, 28 (1982), 667-678. · Zbl 0559.14032
[17] S. LANG, Elliptic functions, Addison Wesley (1973). · Zbl 0316.14001
[18] H. W. LEOPOLDT, Eine p-adische theorie der zetawerte II, Crelle’s J., 274/275 (1975), 224-239. · Zbl 0309.12009
[19] S. LICHTENBAUM, On p-adic L-functions associated to elliptic curves, Inventiones Math., 56 (1980), 19-55. · Zbl 0425.12017
[20] J. LUBIN, One-parameter formal Lie groups over p-adic integer rings, Annals of Math., 80 (1964), 464-484. · Zbl 0135.07003
[21] B. MAZUR and P. SWINNERTON-DYER, Arithmetic of Weil curves, Inventiones Math., 25 (1974), 1-61. · Zbl 0281.14016
[22] B. MAZUR and A. WILES, Class fields of abelian extensions of Q, Invent. Math., 76 (1984), 179-330. · Zbl 0545.12005
[23] G. ROBERT, Unités elliptiques, Bull. Soc. Math. France, Mémoire 36 (1973). · Zbl 0314.12006
[24] K. RUBIN, Congruences for special values of L-functions of elliptic curves with complex multiplication, Invent. Math., 71 (1983), 339-364. · Zbl 0513.14012
[25] J.-P. SERRE, Formes modulaires et fonction Zêta p-adiques, in Springer Lecture Notes in Math., 350 (1973), 191-268. · Zbl 0277.12014
[26] J.-P. SERRE and J. TATE, Good reduction of abelian varieties, Annals of Math., 88 (1968), 492-517. · Zbl 0172.46101
[27] E. DE SHALIT, Ph. D. Thesis, Princeton University (1984).
[28] J. TATE, P-divisible groups, Proc. Conf. on Local Fields, Ed. T. Springer, Springer-Verlag (1967), 158-183. · Zbl 0157.27601
[29] M. M. VISHIK and J. MANIN, P-adic Hecke series of imaginary quadratic fields, Math. USSR Sbornik, 24 (1974), 345-371. · Zbl 0329.12016
[30] L. WASHINGTON, Introduction to cyclotomic fields, Graduate Texts in Math., Springer-Verlag (1982). · Zbl 0484.12001
[31] A. WEIL, Elliptic functions according to Eisenstein and Kronecker, Springer-Verlag (1976). · Zbl 0318.33004
[32] R. YAGER, On the two-variable p-adic L-function, Annals of Math., 115 (1982), 411-449. · Zbl 0496.12010
[33] R. YAGER, P-adic measures on Galois groups, Inventiones Math., 76 (1984), 331-343. · Zbl 0555.12006
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