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On the CM exception to a generalization of the Stéphanois theorem

Desirée Gijón Gómez

TL;DR

The paper investigates how CM points on genus 2 moduli interact with the Fourier Igusa invariants j1,j2,j3 through Humbert singular relations to understand transcendence properties of the associated q-expansions. It introduces the lattices \mathcal{L}_{\\boldsymbol{\\tau}} and \mathcal{L}^{lin}_{\\boldsymbol{\\tau}} of Humbert singular and linear relations, and connects their ranks to the endomorphism algebra End_0(A_{\\boldsymbol{\\tau}}). The authors demonstrate that CM points can lie on positive-dimensional special subvarieties of \\mathcal{A}_2, where the expected generic lower bounds for transcendence degrees may fail; Humbert relations yield concrete linear constraints among the period coordinates, producing extra algebraic relations among the q_i and Igusa invariants. Conditional on Schanuel’s conjecture (and related Gelfond–Schneider considerations for certain cases), the paper derives upper bounds for the minimal transcendence degree of the full Igusa-Invariant vector together with q-expansions, and it classifies CM and QM cases via Shimura and modular curves, linking to known moduli descriptions such as X_0^D(N) and K(N). The results illuminate when CM points are isolated versus when they are part of special subvarieties, with implications for functional transcendence questions in higher genus and connections to Humbert surfaces and Siegel modular geometry.

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.

On the CM exception to a generalization of the Stéphanois theorem

TL;DR

The paper investigates how CM points on genus 2 moduli interact with the Fourier Igusa invariants j1,j2,j3 through Humbert singular relations to understand transcendence properties of the associated q-expansions. It introduces the lattices \mathcal{L}_{\\boldsymbol{\\tau}} and \mathcal{L}^{lin}_{\\boldsymbol{\\tau}} of Humbert singular and linear relations, and connects their ranks to the endomorphism algebra End_0(A_{\\boldsymbol{\\tau}}). The authors demonstrate that CM points can lie on positive-dimensional special subvarieties of \\mathcal{A}_2, where the expected generic lower bounds for transcendence degrees may fail; Humbert relations yield concrete linear constraints among the period coordinates, producing extra algebraic relations among the q_i and Igusa invariants. Conditional on Schanuel’s conjecture (and related Gelfond–Schneider considerations for certain cases), the paper derives upper bounds for the minimal transcendence degree of the full Igusa-Invariant vector together with q-expansions, and it classifies CM and QM cases via Shimura and modular curves, linking to known moduli descriptions such as X_0^D(N) and K(N). The results illuminate when CM points are isolated versus when they are part of special subvarieties, with implications for functional transcendence questions in higher genus and connections to Humbert surfaces and Siegel modular geometry.

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 with the sole exception of CM points. In contrast, CM points do not constitute an exception to the Stéphanois theorem, which states for the Fourier expansion (-expansion) of the j-invariant, for any . 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.
Paper Structure (16 sections, 28 theorems, 126 equations, 1 table)

This paper contains 16 sections, 28 theorems, 126 equations, 1 table.

Table of Contents

  1. Introduction
  2. Questions and our main result
  3. Special subvarieties of A2
  4. Organization
  5. On linear dependence relations
  6. First obstructions and the simplest case
  7. Humbert singular relations
  8. Basics on Humbert singular relations
  9. The lattice of singular relations
  10. From Humbert singular relations to linear relations
  11. The case rank L = 1, and proof of Theorem \ref{['TeoBigTeo']} 1), 2)
  12. The case rank L = 2,3
  13. Shimura curves by an Eichler order
  14. A collection of modular curves
  15. End of proof of Theorem \ref{['TeoBigTeo']}
  16. ...and 1 more sections

Key Result

Theorem 1.1

For $\tau \in \mathbb{H}$, $\operatorname{trdeg} \mathbb{Q} (\tau, j(\tau)) \geq 1$ with the sole exception of $\tau$ quadratic imaginary.

Theorems & Definitions (68)

  • Theorem 1.1: Schneider's theorem Schneider
  • Theorem 1.2: The Stéphanois theorem Stephanois
  • Theorem 1.3: Cohen-Shiga-Wolfart ShigaWolfart and CohenHumSurfTransc
  • Definition 1.4
  • Remark 1.5
  • Theorem 1.6
  • Conjecture 2.1
  • Conjecture 2.2: The Gelfond-Schneider conjecture
  • Lemma 2.3: Under Conjecture \ref{['ConjGelfSchne']}
  • Remark 2.4
  • ...and 58 more