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p-adic regulators on curves and special values of p-adic L-functions. (English) Zbl 0655.14010
Let C be a smooth projective curve over the algebraic closure \({\bar {\mathbb{Q}}}_ p\) of \({\mathbb{Q}}_ p\). The main objective of this paper is to define a (higher) p-adic regulator map \(r_{p,C}: K_ 2(C)\to Hom(H^ 0(C,\Omega^ 1_ C),{\bar {\mathbb{Q}}}_ p)\) having properties similar to those of the higher complex regulator maps introduced by A. A. Bejlinson [Funct. Anal. Appl. 14, 116-118 (1980); translation from Funkts. Anal. Prilozh. 14, No.2, 46-47 (1980; Zbl 0475.14015)]. The technique for the construction of \(r_{p,C}\) is a p-adic analytic integration theory on C; this was developed by the first author for \({\mathbb{P}}^ 1\) [Invent. Math. 69, 171-208 (1982; Zbl 0516.12017)] and is extended here to the general case provided that the Jacobian of C has good reduction (only for these curves a regulator is defined in this paper). The key notion for the integration theory is that of a logarithmic F-crystal on a basic wide open (rigid analytic) space.
In the second part of the paper (§§ 4,5), for \(C=E\) an elliptic curve over \({\mathbb{Q}}\) with complex multiplication, under certain conditions a formula is obtained that relates the p-adic regulator \(r_{p,C}\) to a special value of the p-adic L-function of E. It is pointed out that this formula is a precise p-adic analogue of a corresponding result of S. Bloch in the complex case [Proc. Int. Congr. Math., Helsinki 1978, Vol. 2, 511-515 (1980; Zbl 0454.14011)]. The authors expect that their result will support a p-adic version of the Beilinson conjectures, in the same way as Bloch’s theorem stimulated and provided evidence for A. A. Bejlinson’s conjectures [J. Sov. Math. 30, 2036-2070 (1985); translation from Itogi Nauki Tkeh., Ser. Sovrem. Probl. Mat. 24, 181-238 (1984; Zbl 0588.14013)].
Reviewer: F.Herrlich

14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
14H25 Arithmetic ground fields for curves
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
14K22 Complex multiplication and abelian varieties
14G20 Local ground fields in algebraic geometry
Full Text: DOI EuDML
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