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Surjectivity of \(p\)-adic regulators on \(K_2\) of Tate curves. (English) Zbl 1138.19001

In this paper the author establishes the sujectivity of the \(p\)-adic regulator map
\[ K_2 (E_K) \otimes {\mathbb Q}_p \to H^2_{\text{cont}} (E_K, {\mathbb Q}_p (2)) \]
when \(K\) is a finite extension of \({\mathbb Q}_p\) contained in \({\mathbb Q}_p(\zeta) \) for some root of unity \(\zeta\) and \(E_K\) is an elliptic curve over \(K\), the order of whose \(j\)-invariant with respect to the order on \(K^*\) is negative. \(K_2\) here denotes Quillen \(K\)-theory and \(H^2_{\text{cont}} ({\mathbb Q}_p(2))\) denotes continuous étale cohomology with coefficients in the Tate twist. A consequence of this result is that the torsion subgroup of \(K_1(E_K)\) is finite and if \(E_K\) is a Tate curve \(K^*/q^{\mathbb Z}\) this torsion group is explicitly described as \(\mu_n\oplus \mu_n\oplus \mu_n/(q,K^*)_n\) where \(\mu\) is the group of roots of unity in \(K\), \(n\) is the cardinality of \(\mu\) and \((\quad,\quad)_n\) is the Hilbert symbol.
Note that there is an error in the proof of this theorem (in section 8.1 of the paper) which the author corrects in a subsequently published erratum [Invent. Math. 172, 213–229 (2008; Zbl 1138.19002)]

MSC:

19F27 Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects)
11G20 Curves over finite and local fields
11G55 Polylogarithms and relations with \(K\)-theory

Citations:

Zbl 1138.19002
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References:

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