  I prove the surjectivity of the $p$-adic regulator from Quillen's $K_2$ of
Tate curve to the $p$-adic etale cohomology group when the base field is
contained in a cyclotomic extension of $Q_p$. This implies the finiteness of
torsion part of $K_1$ of Tate curves thanks to Suslin's exact sequence.
