# A question on Serre' open image theorem and p-adic Lie group

Recently I read Serre's wonderful book Abelian l-Adic Representations and Elliptic Curves . There is a theorem in section2.2 chapterIV.

Theorem: If $$E$$ has no complex multiplication, then $$g_l=End(V_l)$$, i.e. $$G_l$$ is open in $$Aut(V_l)$$.

Symbols: Let $$E$$ be a elliptic curve over a number field $$K$$. And let $$\rho_l:Gal(\bar{K}/K)\rightarrow Aut(V_l(E))$$ be the representations of Tate-module, where $$V_l(E)=T_l(E)\otimes_{Z_l}Q_l$$. Let $$G_l=Im(\rho_l)$$ and let $$g_l\subseteq End(V_l)$$ be the Lie algebra of $$G_l$$.

Question: We know $$G_l$$ be the closed subgroup of $$Aut(V_l(E))$$, so by the p-adic version closed subgroup theorem, we have $$G_l$$ is the embeded Lie subgroup of $$Aut(V_l(E))$$. But if we have $$g_l=End(V_l)$$, how can we deduce that $$G_l$$ is open in $$Aut(V_l)$$?

In the real Lie groups case, if $$H$$ is the closed Lie subgroup of a n-dim Lie group $$G$$, and assume $$H$$ and $$G$$ have the same dimension, can we deduce that $$H$$ is open in $$G$$?

Thanks for any answers!

I think this is straightforward after noting that for a finite-dimensional vector space $$V$$, $$End(V)$$ is the Lie algebra of $$Aut(V)$$. One has to be a bit careful in the ultrametric case because the exponential map does not converge on the entire Lie algebra, but it does on an open additive subgroup, and its image is an open subgroup (here: of $$Aut(V)$$); since by the theorem, it's contained in $$G_l$$, this group is also open in $$Aut(V)$$.
Added in response to a comment: It is true, and good to know, that $$exp$$ is generally not an open map. However:
2. Then again, all we need is that the image of $$exp$$ (if you want, "the image of its entire domain") is open. This is already shown (by explicitly computing radius of convergence for $$exp$$ and $$log$$ in the ultrametric case, which gives an open ball around $$1$$ to which on open ball around $$0$$ in the Lie algebra exponentiates) in ch. II (!), §8 no. 4, whose setting is that of a complete normed algebra and its unit group, for which $$End(V)$$ and $$Aut(V)$$ is a standard example. (Looking through this proof, annoyingly there are again references to that Fascicule, but those seem to be standard convergence results, which anyone other than Bourbaki would take for granted as known from an analysis course.)
• Thanks for your answer. But how do we know the image of the exponential map acting on an open additive subgroup is an open subgroup of $Aut(V)$? As far as I know. the exp map is not an open map in general. – Eiang Jul 9 at 7:04