For elliptic curve $E$ defined over $K_v$ we know that $E(K_v)$ = $Z_p^{[K_v:Q_p]} + T$(direct sum) where v is prime of K above p and T is finite abelian group(By prop 6.3 in Silverman's book). In the proof of that proposition, Silverman used Formal Group.

My question is that it is still true for Jacobian of (hyperelliptic) curve of genus g larger than 1?

In other words, $J(C)(K_v) = Z_p^{g[K_v:Q_p]} + T$(direct sum) holds?

Thank you in advance.