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Special geodesics and atypical intersections

Matteo Tamiozzo

Abstract

Let $C$ be a complex irreducible plane curve that is not the vanishing locus of a modular polynomial. We show that $C$ contains finitely many real algebraic curves whose projection on each coordinate axis is a union of special geodesics.

Special geodesics and atypical intersections

Abstract

Let be a complex irreducible plane curve that is not the vanishing locus of a modular polynomial. We show that contains finitely many real algebraic curves whose projection on each coordinate axis is a union of special geodesics.

Paper Structure

This paper contains 15 sections, 3 theorems, 2 equations.

Table of Contents

  1. Introduction
  2. Special properties of real quadratic geodesics
  3. Detecting strongly special curves
  4. Question
  5. Acknowledgments
  6. Real atypical intersections in $Y(1)\times Y(1)$
  7. The Zilber--Pink conjecture for $Y(1)^n$
  8. Special geodesics in $Y(1)$
  9. Curves with special geodesic projections
  10. Proof of the main result
  11. Setup
  12. Base change
  13. Strongly atypical curves in $V$
  14. Application of Zilber--Pink
  15. Answer to Question \ref{['ssub-mainq']}

Key Result

Theorem 1.2.1

(André and98) An irreducible complex algebraic curve in $Y(1)\times Y(1)$ that is not horizontal nor vertical is strongly special if and only if it contains infinitely many points whose coordinates are Heegner points.

Theorems & Definitions (10)

  • Theorem 1.2.1
  • Remark 1.2.5
  • Remark 2.1.1
  • Theorem 2.1.2
  • Remark 2.1.3
  • Remark 2.2.1
  • Definition 2.3.1
  • Theorem 2.3.3
  • Remark 2.3.4
  • Remark 2.4.7