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p-adic K-theory of elliptic curves. (English) Zbl 0627.14010
The author proves some results on p-adic K-groups of a smooth variety over a number field F. In particular he shows that for a \(curve\quad X\) of genus \(g\) over F the dimension of \(K_ 2(X,{\mathbb{Q}}_ p/{\mathbb{Z}}_ p)\) is at least g[F:\({\mathbb{Q}}]\). The proof relies on comparison with the corresponding étale K-theory and results of W. G. Dwyer and E. M. Friedlander [Trans. Am. Math. Soc. 292, 247-280 (1985; Zbl 0581.14012)]. Examples are given for elliptic curves over an imaginary quadratic field such that \(\dim (K_ 2(X,{\mathbb{Q}}_ p/{\mathbb{Z}}_ p))=[F:{\mathbb{Q}}]\) for primes p satisfying certain regularity conditions.
In the final section the author constructs higher p-adic regulator maps for certain elliptic curves and obtains a result on values of a p-adic L- function which can be seen as p-adic analog to results of Beilinson on the generalized conjecture of Birch and Swinnerton-Dyer.
Reviewer: F.Herrlich

MSC:
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
14G25 Global ground fields in algebraic geometry
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
14H25 Arithmetic ground fields for curves
14H52 Elliptic curves
14H45 Special algebraic curves and curves of low genus
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[1] A. A. Beĭ linson, Higher regulators and values of \(L\)-functions , Current problems in mathematics, Vol. 24, Itogi Nauki i Tekhniki, Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1984, pp. 181-238. · Zbl 0588.14013
[2] S. Bloch, Lectures on algebraic cycles , Duke University Mathematics Series, IV, Duke University Mathematics Department, Durham, N.C., 1980. · Zbl 0436.14003
[3] S. Bloch and A. Ogus, Gersten’s conjecture and the homology of schemes , Ann. Sci. École Norm. Sup. (4) 7 (1974), 181-201 (1975). · Zbl 0307.14008
[4] J. Coates and A. Wiles, On the conjecture of Birch and Swinnerton-Dyer , Invent. Math. 39 (1977), no. 3, 223-251. · Zbl 0359.14009
[5] W. Dwyer and E. Friedlander, Algebraic and Etale \(K\)-theory , to appear in Trans. Amer. Math. Soc.
[6] William G. Dwyer and Eric M. Friedlander, Some remarks on the \(K\)-theory of fields , Applications of algebraic \(K\)-theory to algebraic geometry and number theory, Part I, II (Boulder, Colo., 1983), Contemp. Math., vol. 55, Amer. Math. Soc., Providence, RI, 1986, pp. 149-158. · Zbl 0603.18005
[7] Benedict H. Gross, Arithmetic on elliptic curves with complex multiplication , Lecture Notes in Mathematics, vol. 776, Springer, Berlin, 1980. · Zbl 0433.14032
[8] N. M. Katz, \(p\)-adic interpolation of real analytic Eisenstein series , Ann. of Math. (2) 104 (1976), no. 3, 459-571. JSTOR: · Zbl 0354.14007
[9] S. Lang, Cyclotomic fields , Graduate Texts in Mathematics, vol. 59, Springer-Verlag, New York, 1978. · Zbl 0395.12005
[10] M. M. Vishik and Y. Manin, \(p\)-adic Hecke series of imaginary quadratic fields , Mat. Sb. (N.S.) 95(137) (1974), 357-383, 471. · Zbl 0352.12013
[11] A. Merkurjev and A. Suslin, \(K\)-cohomology of Severi-Brauer varieties and norm residue homomorphism , Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), 1011-1046. · Zbl 0525.18008
[12] J. Neisendorfer, Primary homotopy theory , Mem. Amer. Math. Soc. 25 (1980), no. 232, iv+67. · Zbl 0446.55002
[13] D. Quillen, Higher algebraic \(K\)-theory. I , Algebraic \(K\)-theory, I: Higher \(K\)-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), Springer, Berlin, 1973, 85-147. Lecture Notes in Math., Vol. 341. · Zbl 0292.18004
[14] J.-P. Serre, Facteurs locaux des fonctions zêta des variétés algébriques (définitions et conjectures) , 1969-1970, Sém. Delange-Pisot-Poitou, Exposé 19. · Zbl 0214.48403
[15] C. Soulé, On higher \(p\)-adic regulators , Algebraic \(K\)-theory, Evanston 1980 (Proc. Conf., Northwestern Univ., Evanston, Ill., 1980), Lecture Notes in Math., vol. 854, Springer, Berlin, 1981, pp. 372-401. · Zbl 0488.12008
[16] C. Soulé, Eléments cyclotomiques en \(K\)-théorie , to appear in Astérisque. · Zbl 0865.68016
[17] C. Soulé, Operations on étale \(K\)-theory. Applications , Algebraic \(K\)-theory, Part I (Oberwolfach, 1980), Lecture Notes in Math., vol. 966, Springer, Berlin, 1982, pp. 271-303. · Zbl 0507.14013
[18] C. Soulé, Opérations en \(K\)-théorie algébrique , Canad. J. Math. 37 (1985), no. 3, 488-550. · Zbl 0575.14015
[19] C. Soulé, The rank of étale cohomology of varieties over \(p\)-adic or number fields , Compositio Math. 53 (1984), no. 1, 113-131. · Zbl 0589.14019
[20] C. Soulé, \(K\)-théorie \(p\)-adique des courbes elliptiques , preprint, 1983.
[21] J. Tate, Letter from Tate to Iwasawa on a relation between \(K_2\) and Galois cohomology , Algebraic \(K\)-theory, II: “Classical” algebraic \(K\)-theory and connections with arithmetic (Proc. Conf., Seattle Res. Center, Battelle Memorial Inst., 1972), Springer, Berlin, 1973, 524-527. Lecture Notes in Math., Vol. 342. · Zbl 0284.12004
[22] J. Tate, Duality theorems in Galois cohomology over number fields , Proc. Internat. Congr. Mathematicians (Stockholm, 1962), Inst. Mittag-Leffler, Stockholm-Djursholm, 1962, pp. 288-295. · Zbl 0126.07002
[23] R. W. Thomason, Riemann-Roch for algebraic versus topological \(K\)-theory , J. Pure Appl. Algebra 27 (1983), no. 1, 87-109. · Zbl 0545.14007
[24] R. W. Thomason, Bott stability in algebraic \(K\)-theory , Applications of algebraic \(K\)-theory to algebraic geometry and number theory, Part I, II (Boulder, Colo., 1983), Contemp. Math., vol. 55, Amer. Math. Soc., Providence, RI, 1986, pp. 389-406. · Zbl 0594.18012
[25] R. I. Yager, A Kummer criterion for imaginary quadratic fields , Compositio Math. 47 (1982), no. 1, 31-42. · Zbl 0506.12008
[26] R. I. Yager, On two variable \(p\)-adic \(L\)-functions , Ann. of Math. (2) 115 (1982), no. 2, 411-449. JSTOR: · Zbl 0496.12010
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