The norm relations test for positive rank of elliptic curves

Let $E$ be an elliptic curve over $\mathbb{Q}$, let $F / \mathbb{Q}$ be a finite Galois extension with $G = \operatorname{Gal}(F / \mathbb{Q})$, and let $\rho$ be an irreducible representation of $G$. Write $\mathbb{Q}(\rho)$ for the field generated by the character values of $\rho$, and for $\sigma \in \operatorname{Gal}(\mathbb{Q}(\rho) / \mathbb{Q})$ let $\rho^\sigma$ denote the Galois-conjugate representation. Suppose that

$$\left( \bigoplus_{\sigma \in \operatorname{Gal}(\mathbb{Q}(\rho) / \mathbb{Q})} \rho^\sigma \right)^{\oplus m} = \left( \bigoplus_i \mathbb{C}[G / H_i] \right) \ominus \left( \bigoplus_j \mathbb{C}[G / H_j'] \right)$$

for some $m \geq 1$ and subgroups $H_i,H_j' \leq G$. Norm relations test. If either

$$\frac{\prod_i C_{E / F^{H_i}}}{\prod_j C_{E / F^{H_j'}}}$$

is not a norm from some quadratic field $\mathbb{Q}(\sqrt{D}) \subset \mathbb{Q}(\rho)$, or is not a rational square when $m$ is even, then $E$ has a point of infinite order over $F$. The test is based on the conjectural properties of the Birch–Swinnerton-Dyer quotient and is known in the source to follow from the parity conjecture for twists under stated hypotheses, while its unconditional conjectural status is not resolved there.