The local $p$-parity formula for elliptic curves with a $p$-isogeny

Let ${\mathcal K}$ be a local field of characteristic zero, and let $E/{\mathcal K}$ be an elliptic curve admitting a ${\mathcal K}$-rational isogeny $\phi:E\to E'$ of prime degree $p$. Define

$$\sigma_\phi(E/{\mathcal K})=(-1)^{\operatorname{ord}_p\frac{\#\operatorname{coker}\phi_{\mathcal K}}{\#\ker\phi_{\mathcal K}}},$$

where $\phi_{\mathcal K}$ is the induced map on ${\mathcal K}$-rational points. If $p\ne2$, let ${\mathcal K}(\ker\phi)$ be the field obtained by adjoining the coordinates of points in $\ker\phi$.

Local $p$-parity formula. If $p\ne2$, then

$$w(E/{\mathcal K})=\sigma_\phi(E/{\mathcal K})(-1,{\mathcal K}(\ker\phi)/{\mathcal K}).$$

If $p=2$ and $E$ has a model $y^2=x^3+ax^2+bx$ with $\ker\phi=\{O,(0,0)\}$, then

$$w(E/{\mathcal K})=\sigma_\phi(E/{\mathcal K})\begin{cases}(a,-b)_{\mathcal K}(-2a,a^2-4b)_{\mathcal K},&a\ne0,\\(-2,-b)_{\mathcal K},&a=0.\end{cases}$$

These identities are local formulas used to prove the $p$-parity conjecture in the isogeny cases $p=2$ and $3$, and the paper establishes them in characteristic zero.