= Formal kernel of a minimal Weierstrass equation
{title2=$E_1(K)\cong\widehat E(\mathfrak m_K)$}
For an <elliptic curve> over a <local field> with integral <Minimal Weierstrass equation>, let $E_0(K)$ consist of points with nonsingular reduction and let $E_1(K)$ be the kernel of $E_0(K)\to\widetilde E_{\mathrm{ns}}(k)$. This definition also applies at a prime of <bad reduction of an elliptic curve>. The point at infinity is always smooth. With $t=-x/y$ and $w=-1/y$, the <Weierstrass equation of an elliptic curve> becomes
$$
w=t^3+a_1tw+a_2t^2w+a_3w^2+a_4tw^2+a_6w^3.
$$
The unit linear coefficient in $w$ gives a unique integral <formal power series> $w(t)=t^3+O(t^4)$. Then $x=t/w$ and $y=-1/w$, so the nonidentity points of $E_1(K)$ have $v(x)=-2r$ and $v(y)=-3r$ for $r=v(t)>0$. The <local parameter> $t$ bijects $E_1(K)$ with $\mathfrak m_K$ and turns its addition into the <formal group of an elliptic curve>. At <good reduction> this agrees with the <kernel of reduction of an elliptic curve>.
Back to article page
