Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Georgios Papas

TL;DR

The paper develops a uniform G-function framework for 1-parameter families of abelian varieties, focusing on v-adic values arising from splittings in $ ext{A}_2$ and establishing $v$-adic relations encoded by polynomials $R_{s,v}$ that vanish on G-function values when local data align with splitting types. By leveraging the Gauss–Manin connection and comparison isomorphisms, the author constructs a matrix $Y_G(x)$ of $G$-functions and uses André–Bombieri’s Hasse principle to derive height bounds for points $s$ where fibers exhibit “unlikely rich” endomorphism structures, yielding finiteness results compatible with Zilber–Pink-type conjectures. The work treats ordinary and supersingular finite places separately and extends to archimedean settings, with a detailed program for handling bad reduction via Hyodo–Kato cohomology and conjectural relative comparison isomorphisms. Together, these components offer a rigorous path to uniform height bounds and finiteness statements for unlikely intersections in Shimura-type settings, and they lay groundwork for higher-genus generalizations and a fuller treatment of bad reduction. The approach has potential to significantly advance effective finiteness results in Zilber–Pink problems by connecting archimedean and non-archimedean analyses through $G$-functions and period data.

Abstract

This is the first in a series of papers aimed at studying families of G-functions associated to $1$-parameter families of abelian schemes. In particular, the construction of relations, in both the archimedean and non-archimedean settings, at values of specific interest to problems of unlikely intersections. In this first text in this series, we record what we expect to be the theoretical foundations of this series in a uniform way. After this, we study values corresponding to ``splittings'' in $\mathcal{A}_2$ pertinent to the Zilber-Pink conjecture.

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

TL;DR

The paper develops a uniform G-function framework for 1-parameter families of abelian varieties, focusing on v-adic values arising from splittings in and establishing -adic relations encoded by polynomials that vanish on G-function values when local data align with splitting types. By leveraging the Gauss–Manin connection and comparison isomorphisms, the author constructs a matrix of -functions and uses André–Bombieri’s Hasse principle to derive height bounds for points where fibers exhibit “unlikely rich” endomorphism structures, yielding finiteness results compatible with Zilber–Pink-type conjectures. The work treats ordinary and supersingular finite places separately and extends to archimedean settings, with a detailed program for handling bad reduction via Hyodo–Kato cohomology and conjectural relative comparison isomorphisms. Together, these components offer a rigorous path to uniform height bounds and finiteness statements for unlikely intersections in Shimura-type settings, and they lay groundwork for higher-genus generalizations and a fuller treatment of bad reduction. The approach has potential to significantly advance effective finiteness results in Zilber–Pink problems by connecting archimedean and non-archimedean analyses through -functions and period data.

Abstract

This is the first in a series of papers aimed at studying families of G-functions associated to -parameter families of abelian schemes. In particular, the construction of relations, in both the archimedean and non-archimedean settings, at values of specific interest to problems of unlikely intersections. In this first text in this series, we record what we expect to be the theoretical foundations of this series in a uniform way. After this, we study values corresponding to ``splittings'' in pertinent to the Zilber-Pink conjecture.

Paper Structure

This paper contains 47 sections, 30 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. Applications: Galois orbits and height bounds
  3. Main Results
  4. Applications to Zilber-Pink
  5. Places of bad reduction
  6. Outline of the paper
  7. Notation
  8. Period matrices and splittings
  9. Period matrices
  10. Periods and splittings
  11. Bases induced from an isogeny
  12. Background on the G-functions method
  13. Recollections on comparison isomorphisms
  14. Height bounds
  15. Reductions
  16. ...and 32 more sections

Key Result

Theorem 1.3

Let $f:\mathcal{X}\rightarrow S$ be as above and $s_0\in S(\bar{\mathbb{Q}})$ be either an $E^2$-point or an $E\times CM$-point. Let $s\in S(\bar{\mathbb{Q}})$ be another point which is of either of the above types, i.e. $E^2$ or $E\times CM$. Let $v\in\Sigma_{\mathbb{Q}(S,s_0,s)}$ for which $s$ and where $\iota_v:\mathbb{Q}(S,s_0,s)\hookrightarrow \mathbb{C}_v$ is the embedding corresponding to t

Theorems & Definitions (71)

  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Theorem 1.8
  • Corollary 1.9
  • Definition 2.1
  • Remark 2.2: Bases for products of abelian varieties
  • Definition 2.3
  • Remark 2.4
  • ...and 61 more