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Points of low degree on curves over function fields

Sièna van Schaick

Abstract

We show that the geometric classification of smooth projective curves admitting infinitely many points of degree $d\leq 5$ extends from number fields to function fields of characteristic 0. Over number fields, this classification was established by Faltings for $d=1$, Harris--Silverman for $d=2$, Abramovich--Harris for $d=3,4$ and Kadets--Vogt for $d=4,5$. Our approach uses a specialization argument to reduce the problem over function fields to the number field case.

Points of low degree on curves over function fields

Abstract

We show that the geometric classification of smooth projective curves admitting infinitely many points of degree extends from number fields to function fields of characteristic 0. Over number fields, this classification was established by Faltings for , Harris--Silverman for , Abramovich--Harris for and Kadets--Vogt for . Our approach uses a specialization argument to reduce the problem over function fields to the number field case.

Paper Structure

This paper contains 10 sections, 10 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. Conventions
  3. Acknowledgements
  4. Preliminaries
  5. The symmetric product
  6. Constant varieties
  7. Debarre--Fahlaoui curves
  8. Points of degree d points induced by abelian varieties
  9. Specialization lemmas
  10. Proof of the main theorem

Key Result

Theorem 1.1

Let $K$ be a number field, and let $X/K$ be a smooth projective curve. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (24)

  • Theorem 1.1: Classification over number fields Fal83, HS91, AH91, KV25
  • Definition 1.2
  • Theorem A: \ref{['thm:main_result']}
  • Theorem B
  • Theorem C: Lemmas \ref{['lem:specialization_thm_genus_0']}, \ref{['lem:specialization_thm_genus_1']} and \ref{['lem:specialization_thm_DF']}
  • Lemma 2.1
  • proof
  • Example 2.2
  • Definition 2.3
  • Remark 2.4
  • ...and 14 more