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On the $v$-adic values of G-functions III

Georgios Papas

TL;DR

The paper advances the study of $v$-adic values of $G$-functions attached to a $1$-parameter family of QM abelian surfaces by constructing global, nontrivial relations among their values at QM-points across archimedean, ordinary, and supersingular places. Leveraging André's $G$-functions method, it exhibits place-dependent polynomials that vanish under specialization, and proves a height bound for QM-points tied to the count of supersingular proximity places $|\Sigma_{\mathrm{ssing}}(s,s_0)|$. Crucially, the results reveal an independence of the relations from ordinary places, reducing the Zilber-Pink-type question to controlling supersingular proximity, and they connect to broader finiteness results when combined with work of Daw–Orr and related theories. The work thus contributes both to the arithmetic of QM abelian surfaces in $\mathcal{A}_2$ and to the strategy for proving Zilber-Pink in this setting, with explicit computational tools and a detailed discussion of the supersingular proximity phenomenon.

Abstract

In this third part in this series we continue from \cite{papaspadicpart1}, the study of relations among values of G-functions associated to a $1$-parameter family of principally polarized abelian surfaces. In particular, we establish relations among the values of these G-functions, in both the archimedean and $p$-adic setting, at points corresponding to abelian surfaces with Quaternionic multiplication. We also discuss applications to the Zilber-Pink conjecture in $\mathcal{A}_2$ that naturally follow from our discussion.

On the $v$-adic values of G-functions III

TL;DR

The paper advances the study of -adic values of -functions attached to a -parameter family of QM abelian surfaces by constructing global, nontrivial relations among their values at QM-points across archimedean, ordinary, and supersingular places. Leveraging André's -functions method, it exhibits place-dependent polynomials that vanish under specialization, and proves a height bound for QM-points tied to the count of supersingular proximity places . Crucially, the results reveal an independence of the relations from ordinary places, reducing the Zilber-Pink-type question to controlling supersingular proximity, and they connect to broader finiteness results when combined with work of Daw–Orr and related theories. The work thus contributes both to the arithmetic of QM abelian surfaces in and to the strategy for proving Zilber-Pink in this setting, with explicit computational tools and a detailed discussion of the supersingular proximity phenomenon.

Abstract

In this third part in this series we continue from \cite{papaspadicpart1}, the study of relations among values of G-functions associated to a -parameter family of principally polarized abelian surfaces. In particular, we establish relations among the values of these G-functions, in both the archimedean and -adic setting, at points corresponding to abelian surfaces with Quaternionic multiplication. We also discuss applications to the Zilber-Pink conjecture in that naturally follow from our discussion.
Paper Structure (27 sections, 19 theorems, 62 equations)

This paper contains 27 sections, 19 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Summary of main results
  3. Applications to Zilber-Pink
  4. Outline of the paper
  5. Notation
  6. Background on QM abelian surfaces
  7. QM abelian surfaces: A short review
  8. Conventions about QM abelian surfaces
  9. Periods of QM surfaces
  10. The archimedean case
  11. The case of ordinary places
  12. The case of supersingular places
  13. Background G-functions
  14. Relations among values of G-functions
  15. Relations at archimedean places
  16. ...and 12 more sections

Key Result

Proposition 1.4

Let $Y_G\in K[[x]]^{\mu}$ be the G-functions associated to a $1$-parameter family $f:\mathcal{X}\rightarrow S$ of principally polarized abelian surfaces and assume that the center $s_0$ is a $QM$-point. Let $s\in S(\bar{\mathbb{Q}})$ be another $QM$-point and let $v\in \Sigma_{K(s)}$ be a place for

Theorems & Definitions (41)

  • Conjecture 1.2: Zilber-Pink in $\mathcal{A}_2$
  • Proposition 1.4
  • Definition 1.5
  • Definition 1.6
  • Theorem 1.7
  • Corollary 1.9
  • Remark 1.10
  • Lemma 2.1
  • proof
  • Theorem 2.2: patankarthesis, Theorem $2.2.2$
  • ...and 31 more