Conjecture 3.25 — 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 $(a,b)\ne(0,0)$. For an integer $s\ge1$, $\Phi_s(X,Y)\in\mathbb Z[X,Y]$ is the classical modular polynomial characterized by $\Phi_s(j(E),j(E^{\prime}))=0$ when the complex elliptic curves are linked by a cyclic isogeny of degree $s$.

Statement recorded in the supplied source.

For algebraic $q_1,q_2$ with $0<|q_i|<1$, suppose an irreducible polynomial $P\in\mathbb Q[X,Y]$ satisfies $P(J(q_1),J(q_2))=0$. Must $P=c\Phi_s$ for some $s\ge1$ and $c\in\mathbb Q^\times$, and must $q_1,q_2$ be multiplicatively dependent? [S2]

Short English statement

Are all irreducible algebraic relations between two such modular values modular-polynomial relations, with multiplicatively dependent arguments?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 8700043 (AMR-086-0043). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Michel Waldschmidt, Open Diophantine Problems, arXiv:math/0312440v2 (24 January 2004), Conjecture 3.25. https://arxiv.org/abs/math/0312440.