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


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


Zbl 0581.14012
Full Text: DOI


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