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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 $$\overline{\mathbb Q}_p$$ of $$\mathbb Q_p$$. The main objective of this paper is to define a (higher) $$p$$-adic regulator map $$r_{p,C}: K_2(C)\to \operatorname{Hom}(H^0(C,\Omega^1_C),\overline{\mathbb Q}_p)$$ having properties similar to those of the higher complex regulator maps introduced by A. A. Beĭlinson [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. Beĭlinson’s conjectures [J. Sov. Math. 30, 2036–2070 (1985); translation from Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat. 24, 181–238 (1984; Zbl 0588.14013)].

MSC:
 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
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References:
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