# Clarification: Using Hensel's Lemma to determine $K_v$-rational points on a curve

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.

you really shouldn't crosspost. Anyway, you've slightly misstated Hensel's lemma, you left out the assumption that $$f(x)$$ has a simple root in $$R_v/\mathcal{M}_v$$. That's where the $$e$$ is coming from. In general, if $$f(x)$$ has a root of higher multiplicity in $$R_v/\mathcal{M}_v$$, then you need to work in $$R_v/\mathcal{M}_v^e$$. So if the curve is non-singular modulo $$v$$, then you can take $$e=1$$, and it really does reduce to the 1-variable Hensel lemma, since the curve has dimension 1. However, if the curve is singular and it has a singular point defined over $$R_v/\mathcal{M}_v$$, that's not enough to conclude that the point can be lifted, you need to work with a larger value of $$e$$.