On the CM exception to a generalization of the St\'ephanois theorem

TL;DR

This paper explores exceptions to a generalization of the Stéphaniois theorem in genus 2, revealing positive-dimensional sources of exceptions related to CM points, using Humbert relations and conditional on Schanuel's conjecture.

Contribution

It identifies new positive-dimensional exceptions to the generalized Stéphaniois theorem in genus 2, linking them to CM points and Humbert relations within a transcendental framework.

Findings

Existence of positive-dimensional exceptions in genus 2

Relation of exceptions to CM points

Use of Humbert singular relations in analysis

Abstract

There are two classical theorems related to algebraic values of the j-invariant: Schneider's theorem and the St\'ephanois theorem. Schneider's theorem for the j-invariant states that the transcendence degree $\operatorname{trdeg} \mathbb{Q}(\tau, j(\tau)) \geq 1$ with the sole exception of CM points. In contrast, CM points do not constitute an exception to the St\'ephanois theorem, which states $\operatorname{trdeg} \mathbb{Q}(q,j(q))\geq 1$ for the Fourier expansion ($q$-expansion) of the j-invariant, for any $q$. Schneider's theorem has been generalized to higher dimensions, and in particular holds for the Igusa invariants of a genus 2 curve. These functions have Fourier expansions, but a result of St\'ephanois type is unknown. In this paper, we find that there are positive dimensional sources of exceptions to the generic behaviour expected in genus 2, and we discuss their relation to CM points. We utilize Humbert singular relations, putting them into the transcendental framework. The computations of the transcendence degree for CM points are conditional to Schanuel's conjecture.