Modular Schanuel conjecture without derivatives

About 7 years old · traced to

Let z1,…,zn∈H+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 dim⁡G(Q)(z∣Σ)\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))≥dim⁡G(Q)(z∣Σ).\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.

References

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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