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A note on unlikely intersections in Shimura varieties

Vahagn Aslanyan, Christopher Daw

Abstract

We discuss the relationships between the André-Oort, André-Pink-Zannier, and Mordell-Lang conjectures for Shimura varieties. We then combine the latter with the geometric Zilber-Pink conjecture to obtain some new results on unlikely intersections in Shimura varieties.

A note on unlikely intersections in Shimura varieties

Abstract

We discuss the relationships between the André-Oort, André-Pink-Zannier, and Mordell-Lang conjectures for Shimura varieties. We then combine the latter with the geometric Zilber-Pink conjecture to obtain some new results on unlikely intersections in Shimura varieties.
Paper Structure (9 sections, 10 theorems, 15 equations)

This paper contains 9 sections, 10 theorems, 15 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Shimura varieties
  4. (Weakly) special subvarieties
  5. Weakly optimal and (weakly) atypical subvarieties
  6. Generalised Hecke orbits
  7. Mordell-Lang in Shimura varieties
  8. Combining Mordell-Lang with geometric Zilber-Pink
  9. Terminology in unlikely intersections

Key Result

Theorem 1.2

Let $S$ be a Shimura variety of abelian type and let $\Sigma\mathop{\mathrm{\subseteq}}\nolimits S$ denote the union of the special points of $S$ and finitely many generalised Hecke orbits. Any subvariety $V$ of $S$ contains only finitely many maximal weakly atypical subvarieties $W$ with the proper

Theorems & Definitions (24)

  • Conjecture 1.1: Zilber-Pink
  • Theorem 1.2
  • Theorem 1.3
  • Definition 3.1
  • Remark 3.2
  • Definition 3.3
  • Lemma 3.4
  • proof
  • Conjecture 3.5: Mordell-Lang for $S$
  • Conjecture 3.6: Mordell-Lang for $S$
  • ...and 14 more