On the CM exception to a generalization of the Stéphanois theorem

Abstract

There are two classical theorems related to algebraic values of the j-invariant: Schneider's theorem and the Stéphanois theorem. Schneider's theorem for the j-invariant states that the transcendence degree $\operatorname{trdeg} \mathbb{Q}(τ, j(τ)) \geq 1$ with the sole exception of CM points. In contrast, CM points do not constitute an exception to the Stéphanois 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 Stéphanois 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.