Modular Schanuel conjecture without derivatives

Let z1,,znH+z_1,\ldots,z_n\in\mathbb{H}^{+}, write z=(z1,,zn)\mathbf{z}=(z_1,\ldots,z_n), and let jj be the modular function. Let Σ\Sigma be the set of special points, and let dimG(Q)(zΣ)\dim_{G(\mathbb{Q})}(\mathbf{z}\mid\Sigma) denote the corresponding G(Q)G(\mathbb{Q})-dimension relative to Σ\Sigma. Modular Schanuel conjecture.

tr.deg.QQ(z,j(z))dimG(Q)(zΣ).\mathrm{tr.deg.}_{\mathbb{Q}}\mathbb{Q}\left(\mathbf{z},j(\mathbf{z})\right)\geq \dim_{G(\mathbb{Q})}(\mathbf{z}\mid\Sigma).

The source states that this derivative-free version follows from Modular Schanuel with derivative, so it is a consequence rather than an independent conjectural assumption. Its status is therefore the same unresolved status as the stronger conjecture.

Sources & referencesView supporting material

Primary source

Isaac A. Broudy and Sebastian Eterović, “Schanuel Type Conjectures and Disjointness”, arXiv:2211.09556 (2022).

Additional references

4 papers in this index state this conjecture (2019–2022). The statement above is taken from the most recent of them; the others are arXiv:2209.12192, arXiv:2010.00102, arXiv:1907.09858.

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