Let $E$ be an elliptic curve defined over a complete local ring K in characteristic greater than 5 (one may assume that E is given by a Weierstrass equation of the form $y^2 = x^3 + Ax + B$). Denote by $\tilde{E}$ the reduction of E modulo the unique maximal ideal $\mathcal{M}$ of the ring of integers of K. Denote by $E_0(K)$ the subgroup of $E(K)$ that consists of all points P of E(K) that reduce to nonsingular points of $\tilde{E}(k)$ ($k$ is the residue field). I am interested in the finiteness of the quotient $E(K)/E_0(K)$. In Silverman’s book, The Arithmetic of Elliptic curves, chapter VII, the finiteness of this quotient is proved by arguments from algebraic geometry (scheme theory). However, Silverman says that an explicit proof working with a minimal Weierstrass Equation and addition formulae is possible (J. Tate writes the same in his article The Arithmetic of Elliptic curves), when the reduction has a cusp singularity.
Does anyone have a reference for such an explicit proof ? Or can anyone provide it here ? I managed the cases (where $v$ is the normalized valuation of K)
1) $v(A)\geq 1$ and $v(B) = 1$. In this case the quotient has cardinality $1$.
2) $v(A)=2$ and $v(B) \geq 2$. In this case the quotient has cardinality $2$.
3) $v(A)=3$ and $v(B) \geq 5$. In this case the quotient has cardinality $2$.
4) $v(A)\geq 4$ and $v(B) = 5$. In this case the quotient has cardinality $1$.
I am missing three cases, that I cannot prove:
1) $v(A)\geq 2$ and $v(B) = 2$. In this case the quotient should have cardinality $3$.
2) $v(A)\geq 3$ and $v(B) = 4$. In this case the quotient should have cardinality $3$.
3) $v(A) = 2$ and $v(B) \geq 3 $. In this case the quotient should have cardinality $4$.
Thank you very much!