I have a little confusion. In an article entitled "Torsion Points of Elliptic Curves" (page 15),it is written that-
From our equation for the line through $P_1$ and $P_2$, we know that $s_1 = αt_1 + β$. Because $s_1 \in p^{v}R_p$, we know that $s_1 \in p^{3v}R_p$. And, because $α \in p^{2v}R_p$ and $t_1 \in p^vR_p$, we have that $αt_1 \in p^{3v}R_p$. Therefore, the equation for the line gives us that $β \in p^{3v}R_p$.
I don't understand -
Because $s_1 \in p^{v}R_p$, we know that $s_1 \in p^{3v}R_p$.
How one concludes $s_1 \in p^{3v}R_p$ from $s_1 \in p^{v}R_p$?
We can say/infer $s_1 \in p^{v}R_p$ from $s_1 \in p^{3v}R_p$ not the other way around, then why author wrote
Because $s_1 \in p^{v}R_p$, we know that $s_1 \in p^{3v}R_p$. ?
I think this is a poor phrasing on the author's part. It would make more sense to say
"Since $(t_1,s_1)\in E(p^\nu)$, then $s_1\in p^{3\nu}R_p$"
They establish at the bottom of page 13 that we must have $t\in p^\nu R_p$ and $s\in p^{3\nu}R_p$ for any $(t,s)$ on the curve, this following from the defining equation of E.