From Silverman's AEC page 332:

I need to understand why the determination of the following local kernel $$ ker \Big( H^1(G_v, E[\phi]) \rightarrow WC(E/K_v)[\phi] \Big) $$

is straightforward. The book says that it is the same as answering the questions whether a curve has a point over a complete local field (which I understand).

This is further reduced by Hensel's Lemma to checking whether the curve has a point in some finite ring $R_v/ \mathcal{M}_v^e$ for some easily computable integer $e$.

Now, as far as I know and understand, Hensel's Lemma says that for a polynomial $f(x) \in R_v[x]$, if it has a root in $R_v/\mathcal{M}$, then it lifts to a unique root in $R_v$ and hence $K_v$. However, this version does not seem to be directly used here. I suspect that maybe some several variable version is being used? And where did that $e$ come from?

I would really appreciate it if someone could explain it to me. Thank you.

Cross-posted here: https://math.stackexchange.com/questions/3590360/clarification-using-hensels-lemma-in-determining-if-a-curve-has-a-k-v-ration

From Silverman's AEC page 332:

I need to understand why the determination of the following local kernel $$ ker \Big( H^1(G_v, E[\phi]) \rightarrow WC(E/K_v)[\phi] \Big) $$

is straightforward. The book says that it is the same as answering the questions whether a curve has a point over a complete local field (which I understand).

This is further reduced by Hensel's Lemma to checking whether the curve has a point in some finite ring $R_v/ \mathcal{M}_v^e$ for some easily computable integer $e$.

Now, as far as I know and understand, Hensel's Lemma says that for a polynomial $f(x) \in R_v[x]$, if it has a root in $R_v/\mathcal{M}$, then it lifts to a unique root in $R_v$ and hence $K_v$. However, this version does not seem to be directly used here. I suspect that maybe some several variable version is being used? And where did that $e$ come from?

I would really appreciate it if someone could explain it to me. Thank you.