I need help in understanding a passage in a paper by Hindry and Silverman, "The Canonical Height and Integral Points on Elliptic Curves". (re. page 439)
Let $ E(K) $ be an elliptic curve with multiplicative reduction at a valuation $ \nu$, and K a function field of zero characteristic. The torsion points of this curve are supposed to have the form $ E(K)_\mathrm{tors} $ $\cong $ $\dfrac{\mathbb{Z}}{m\mathbb{Z}} \times \dfrac{\mathbb{Z}}{n\mathbb{Z}} $.
They select a point P so that it kills the right side of the product, i.e., $ \dfrac{E(K)_\mathrm{tors}}{\langle P\rangle} \cong \dfrac{\mathbb{Z}}{m\mathbb{Z}} \times \dfrac{\mathbb{Z}}{m\mathbb{Z}} $.
Then, they use the isogeny theorem to construct another elliptic curve $ E'$ such that $ E$ is isogenous with $ E'$.
Next, they construct the Tate models for these curves: $ E(K_\nu) \cong \dfrac{K_\nu^{*}}{\langle q\rangle } $ and $ E'(K_\nu) \cong \dfrac{K_\nu^{*}}{\langle q'\rangle} $, where $ K_\nu $ is the completion at $ \nu $, $ q $ and $ q' $ have the property that $\nu(q) > 1$ and $\nu(q') > 1$.
Up to this point everything is fine. However, in the next line they find that there are elements $q_1$ and $q'_1$ such that $ q = q_1^m$ and $ q' = q'_1^m $; the reason being that there is a subgroup of type $(m,m)$ in both curves.
At first, I thought that one could see this by elementary group theory, but that only gives that $ q^{k} = q_1^m$ for some power $k$ of $q$, and I could not continue from this point. Am I missing something? Can anyone help me understand this?
cheers
$. I changed it to $\langle p\rangle$.
– 2012-08-16