Diaz 2007, Corollaire 7: if τ or 1/τ is in ℒ̃, then e^{2iπτ} and e^{−2iπ/τ} are not both algebraic

Here $\mathcal{L}$ is the set of logarithms of algebraic numbers, and $\widetilde{\mathcal{L}}$ is the $\overline{\mathbb{Q}}$-vector space spanned by $1$ and $\mathcal{L}$. Roy's strong six exponentials theorem is carried as the hypothesis hSSE: if $x_1, x_2$ are $\overline{\mathbb{Q}}$-linearly independent and so are $y_1, y_2, y_3$, then one of the six products $x_iy_j$ is not in $\widetilde{\mathcal{L}}$. Baker's theorem is carried as the hypothesis hB, in its two-logarithm inhomogeneous form, as in DiazModulus.candidate_one_log_saturation.

Let $\tau$ lie in the upper half-plane, with $\tau \in \widetilde{\mathcal{L}}$ or $1/\tau \in \widetilde{\mathcal{L}}$. Then $e^{2i\pi\tau}$ and $e^{-2i\pi/\tau}$ are not both algebraic.

This is the conclusion of the statement (C4) of Diaz (1997), equivalent there to the injectivity of the modular invariant $J$ on $\overline{\mathbb{Q}} \cap (D \setminus \{0\})$, under the extra hypothesis that $\tau$ or $1/\tau$ lies in $\widetilde{\mathcal{L}}$. Diaz presents it as adding to Proposition 3 of Diaz (1997).

Proof. Théorème 5 (2) (DiazModulus.diaz_2007_th5) with $u = \tau$, $v = 2i\pi/\tau$, or with $u = 1/\tau$, $v = 2i\pi\tau$; the middle number is $2i\pi \in \mathcal{L}$.

Novelty. None: this is Corollaire 7 of Diaz (2007), p. 388, with his proof. The contribution of this node is the formal proof.