p-adic K-theory of elliptic curves
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 \textit{W. G. Dwyer} and \textit{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.
generalized conjecture of Birch and Swinnerton-Dyer
scientific article; zbMATH DE number 4019207
scientific article