\(p\)-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)].


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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