The Birch--Swinnerton-Dyer formula for the false Tate representations

Let $p$ be an odd prime, let $K=\mathbb{Q}(\mu_p)$ and let $F$ be the false Tate extension layer used in the paper. Let $E/\mathbb{Q}$ be an elliptic curve with $E(K)$ finite, let $\sigma$ and $\rho$ be the associated Artin representations, and let $\mathcal A_{E/K}$ and $\mathcal A_{E/F}$ be the fractional ideals comparing invariant differentials on the relevant Néron models with the global Néron differential. Write $c_v$ and $c_w$ for Tamagawa factors and $\Sha$ for Tate--Shafarevich groups. Birch--Swinnerton-Dyer consequence. The groups $\Sha(E/K)$ and $\Sha(E/F)$ are finite, and

$$\frac{L(E,\sigma,1)\epsilon(\sigma)}{\Omega_+(E)^{(p-1)/2}(2\Omega_-(E))^{(p-1)/2}}=\frac{N_{K/\mathbb{Q}}(\mathcal A_{E/K})\#\Sha(E/K)\prod_vc_v}{\#E(K)^2},$$

while

$$\frac{L(E,\rho,1)\epsilon(\rho)}{\Omega_+(E)^{(p-1)/2}(2\Omega_-(E))^{(p-1)/2}}=\sqrt[p-1]{\frac{\#E(K)^2N_{F/\mathbb{Q}}(\mathcal A_{E/F})\#\Sha(E/F)\prod_wc_w}{\#E(F)^2N_{K/\mathbb{Q}}(\mathcal A_{E/K})\#\Sha(E/K)\prod_vc_v}}.$$

If the second formula reads $0=0$, it asserts that $L(E,\rho,1)=0$ exactly when $E(F)$ is infinite. This is conditional on the Birch--Swinnerton-Dyer conjecture for $E/K$ and $E/F$.