I am reading Silverman's book, $\textit{The Arithmetic of Elliptic Curves}$.
Let $E/K$ be an elliptic curve over a local field $K$. Let $\widetilde{E}/k$ be the reduction of $E$. Let $\widetilde{E}_{ns}$ denote the non-singular part of $\widetilde{E}$. Then if $E$ has multiplicative (resp. additive) reduction, then $\widetilde{E}_{ns}(\bar{k})\cong\bar{k}^*$ (resp. $\bar{k}^+$). [Proposition VII.5.1]
The way he used to prove this statement is by constructing a special map related to the tangent line(s) on the singular point. My questions are:
- Are this statement just a special case of a much more general theory in algebraic geometry?
- Are there any other proofs without using the special map mentioned above?