Conjecture 3.23

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $\Lambda\subset\mathbb C$ be a rank-two lattice, with Weierstrass functions $\wp_\Lambda,\zeta_\Lambda$ normalized by $\zeta_\Lambda^{\prime}=-\wp_\Lambda$ and $\zeta_\Lambda(z)=1/z+O(z^3)$ at zero. Its invariants satisfy $(\wp_\Lambda^{\prime})^2=4\wp_\Lambda^3-g_2\wp_\Lambda-g_3$ and $g_2^3-27g_3^2\ne0$. For $0\ne\omega\in\Lambda$, define the constant $\eta=\zeta_\Lambda(z+\omega)-\zeta_\Lambda(z)$. Transcendence degree counts the largest algebraically independent subfamily over $\mathbb Q$.

Statement recorded in the supplied source.

For every such lattice, nonzero period $\omega$, and $u\in\mathbb C\setminus\Lambda$, is \[\operatorname{trdeg}_{\mathbb Q}\mathbb Q\left(g_2,g_3,\pi/\omega,\wp_\Lambda(u),\zeta_\Lambda(u)-(\eta/\omega)u\right)\ge2?\] [S2]

Short English statement

Must this field of elliptic invariants, one normalized period and two elliptic-function values have at least two algebraically independent elements?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 8700041 (AMR-086-0041). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Michel Waldschmidt, Open Diophantine Problems, arXiv:math/0312440v2 (24 January 2004), Conjecture 3.23. https://arxiv.org/abs/math/0312440.