Conjecture 3.24 — Bertrand

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.

For $0<|q|<1$, put $E_4(q)=1+240\sum_{n\ge1}n^3q^n/(1-q^n)$ and $\Delta(q)=q\prod_{n\ge1}(1-q^n)^{24}$, and define the classical modular invariant $J(q)=E_4(q)^3/\Delta(q)$. Set $D=q\,d/dq$. Two nonzero complex numbers are multiplicatively dependent if $q_1^a q_2^b=1$ for integers $(a,b)\ne(0,0)$.

Statement recorded in the supplied source.

For every $n\ge1$ and algebraic $q_1,\ldots,q_n$ with $0<|q_i|<1$, does algebraic dependence over $\mathbb Q$ of the $3n$ numbers $J(q_i),DJ(q_i),D^2J(q_i)$ imply that some distinct $i,j$ have multiplicatively dependent $q_i,q_j$? [S2]

Short English statement

Must any algebraic relation among modular invariant values and their first two logarithmic derivatives at algebraic arguments force a multiplicative relation between two arguments?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 8700042 (AMR-086-0042). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Michel Waldschmidt, Open Diophantine Problems, arXiv:math/0312440v2 (24 January 2004), Conjecture 3.24. https://arxiv.org/abs/math/0312440.