## 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

Zbl 0581.14012
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### References:

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