HEXAGONbeta

Four exponentials at CM and Beta periods

Primarily AI-generated textHuman understanding: no partsmath.NT — Number Theory

Contributed by Shawn Krisman

Version 2 / Nov 07, 2026 / CC BY 4.0

Abstract

For every nonzero eigen-period PP of a CM abelian variety over Q‾\overline{\mathbb{Q}}, we show that the four exponentials conjecture and its strong form hold at t∈Q‾×P±1t\in\overline{\mathbb{Q}}^\times P^{\pm1} and t∈Q‾×(P/π)±1t\in\overline{\mathbb{Q}}^\times(P/\pi)^{\pm1}; these include Beta values such as t=Γ(1/5)2/Γ(2/5)t=\Gamma(1/5)^2/\Gamma(2/5). For an elliptic curve EE over Q‾\overline{\mathbb{Q}} with complex multiplication, we also show that eπ2e^{\pi^2} and ℘E(1/π)\wp_E(1/\pi) are not both algebraic. Further results concern two CM elliptic curves: for example, π,ϖ,Ωρ,ϖΩρ\pi,\varpi,\Omega_\rho,\varpi\Omega_\rho are linearly independent over Q‾\overline{\mathbb{Q}}, where ϖ\varpi and Ωρ\Omega_\rho are periods of y2=4x3−4xy^2=4x^3-4x and y2=4x3−4y^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 2ϖ2^{\varpi} and 3ϖ3^{\varpi} are not both algebraic, are checked in Lean 4 and reported separately.

References

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Version history

  1. v1 · 26 pagesSubmitted by Shawn KrismanInitial depositSupersededNov 07, 2026
  2. v2 · 26 pagesSubmitted by Shawn KrismanSubmitted as a new versionSupersededNov 07, 2026
  3. v3 · 26 pagesSubmitted by Shawn KrismanSubmitted as a new versionCurrentNov 10, 2026