Let $E$ is an elliptic curve defined over a number field $F$, $p$ a prime of $\mathbb{Z}$ and $v$ a valuation of $F$ that does not lie over $p$. Call $F_v$ the completion of $F$ with respect to $v$ and $I_v$ the inertia group of $Gal(\bar{F}_v/F_v)$.
I read in Greenberg's article https://arxiv.org/abs/math/9809206 at pag.24 that if $E$ has additive reduction at $v$, then $H^0(I_v,E[p^\infty])$ is finite, where $E[p^\infty]$ denotes the $p$-primary subgroup of $E$.
Could someone explain me why, or give a reference?
I really don't know how to join the notion of additive reduction with the action of the inertia group on $E[p^\infty]$.