math.NT — Number Theory

Four exponentials at CM and Beta periods

Contributed by Shawn Krisman

For every nonzero eigen-period $P$ of a CM abelian variety over $\overline{\mathbb{Q}}$, we show that the four exponentials conjecture and its strong form hold at $t\in\overline{\mathbb{Q}}^\times P^{\pm1}$ and $t\in\overline{\mathbb{Q}}^\times(P/\pi)^{\pm1}$; these include Beta values such as $t=\Gamma(1/5)^2/\Gamma(2/5)$. For an elliptic curve $E$ over $\overline{\mathbb{Q}}$ with complex multiplication, we also show that $e^{\pi^2}$ and $\wp_E(1/\pi)$ are not both algebraic. Further results concern two CM elliptic curves: for example, $\pi,\varpi,\Omega_\rho,\varpi\Omega_\rho$ are linearly independent over $\overline{\mathbb{Q}}$, where $\varpi$ and $\Omega_\rho$ are periods of $y^2=4x^3-4x$ and $y^2=4x^3-4$. These are explicit consequences of published theorems of Waldschmidt (1988) and Roy–Waldschmidt (1993), together with classical results of Baker, Schneider, Masser and Wüstholz; the contribution is the statements and the proofs written out at these periods. Cases at specific periods, such as that $2^{\varpi}$ and $3^{\varpi}$ are not both algebraic, are checked in Lean 4 and reported separately.