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The large Galois orbits conjecture under multiplicative degeneration

Christopher Daw, Martin Orr

TL;DR

This work proves the PEL-type Large Galois Orbits conjecture for Hodge-generic curves in the moduli space $\mathcal{A}_g$ that exhibit multiplicative degeneration, enabling a complete Zilber–Pink result in $\mathcal{A}_2$ for such curves and new Zilber–Pink cases for $g\ge3$. The authors implement André's G-functions strategy by constructing period functions via uniformisations at all places and by deriving archimedean and non-archimedean period relations tied to endomorphisms; these relations yield height bounds and, through Masser–Wüstholz isogeny estimates, Galois lower bounds that scale with the discriminant of $\mathrm{End}(A_s)$. A key technical achievement is a cohesive framework connecting formal, rigid, and complex uniformisations to interpret $v$-adic and archimedean evaluations of period G-functions, culminating in explicit polynomial relations among period data and their use to bound unlikely intersections. The results advance the unlikely intersections program in Shimura varieties, provide concrete Zilber–Pink consequences in low dimensions, and supply a robust method to extend LGO-type bounds to broader families with multiplicative degenerations."

Abstract

We establish the PEL type large Galois orbits conjecture for Hodge generic curves in $\mathcal{A}_g$ possessing multiplicative degeneration. Combined with our earlier works, this concludes the proof of the Zilber-Pink conjecture in $\mathcal{A}_2$ for such curves. We also deduce several new cases of Zilber-Pink in $\mathcal{A}_g$ for $g\geq 3$. Our proof uses André's G-functions method, using formal and rigid uniformisation of semiabelian schemes to interpret the $p$-adic evaluations of the period G-functions.

The large Galois orbits conjecture under multiplicative degeneration

TL;DR

This work proves the PEL-type Large Galois Orbits conjecture for Hodge-generic curves in the moduli space that exhibit multiplicative degeneration, enabling a complete Zilber–Pink result in for such curves and new Zilber–Pink cases for . The authors implement André's G-functions strategy by constructing period functions via uniformisations at all places and by deriving archimedean and non-archimedean period relations tied to endomorphisms; these relations yield height bounds and, through Masser–Wüstholz isogeny estimates, Galois lower bounds that scale with the discriminant of . A key technical achievement is a cohesive framework connecting formal, rigid, and complex uniformisations to interpret -adic and archimedean evaluations of period G-functions, culminating in explicit polynomial relations among period data and their use to bound unlikely intersections. The results advance the unlikely intersections program in Shimura varieties, provide concrete Zilber–Pink consequences in low dimensions, and supply a robust method to extend LGO-type bounds to broader families with multiplicative degenerations."

Abstract

We establish the PEL type large Galois orbits conjecture for Hodge generic curves in possessing multiplicative degeneration. Combined with our earlier works, this concludes the proof of the Zilber-Pink conjecture in for such curves. We also deduce several new cases of Zilber-Pink in for . Our proof uses André's G-functions method, using formal and rigid uniformisation of semiabelian schemes to interpret the -adic evaluations of the period G-functions.
Paper Structure (48 sections, 47 theorems, 117 equations)

This paper contains 48 sections, 47 theorems, 117 equations.

Table of Contents

  1. Introduction
  2. Implications for Zilber--Pink
  3. Large Galois orbits conjecture
  4. The strategy
  5. Non-archimedean interpretation of G-functions
  6. Non-archimedean relations
  7. Archimedean relations
  8. Literature review
  9. Structure of the paper
  10. Preliminaries
  11. Notation
  12. Places of a number field
  13. Inverse function theorem for globally bounded power series
  14. Analytification functors
  15. Analytification of schemes
  16. ...and 33 more sections

Key Result

Theorem 1.1

Let $g \geq 2$. Let $C$ be a geometrically irreducible Hodge generic algebraic curve in $\mathcal{A}_g$ defined over $\overline \mathbb{Q}$ such that the Zariski closure of $C$ in the Baily--Borel compactification of $\mathcal{A}_g$ intersects the zero-dimensional stratum of the boundary. Then there

Theorems & Definitions (96)

  • Theorem 1.1
  • Corollary 1.2
  • proof
  • Corollary 1.3
  • proof
  • Conjecture 1.4
  • Remark 1.5
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • ...and 86 more