Non-Archimedean and Tropical Techniques in Arithmetic Geometry Público

Morrow, Jackson (Spring 2020)

Permanent URL: https://etd.library.emory.edu/concern/etds/76537249n?locale=es
Published

Abstract

Let K be a number field, and let C/K be a curve of genus g ≥ 2. In 1983, Faltings famously proved that the set C(K) of K-rational points is finite. Given this, several questions naturally arise:

1. How does this finite quantity #C(K) varies in families of curves?

2. What is the analogous result for degree d > 1 points on C?

3. What can be said about a higher dimensional variant of Faltings result?

In this thesis, we will prove several results related to the above questions.

In joint with with J. Gunther, we prove, under a technical assumption, that for each positive integer d > 1, there exists a number Bd such that for each g > d, a positive proportion of odd hyperelliptic curves of genus g over Q have at most Bd

“unexpected” points of degree d. Furthermore, we may take B2 = 24 and B3 = 114. Our other results concern the strong Green–Griffiths–Lang–Vojta conjecture, which

is the higher dimensional version of Faltings theorem (nee the Mordell conjecture). More precisely, we prove the strong non-Archimedean Green–Griffiths–Lang–Vojta conjecture for closed subvarieties of semi-abelian varieties and for projective surfaces admitting a dominant morphism to an elliptic curve. 

Table of Contents

1 Introduction . . . . . . . . . . . . . . . . 1

1.1 Organization ............................... 3

2 Background — Rational points on curves ...............................4

2.1 The method of Chabauty–Coleman ................... 4

2.2 Symmetric power Chabauty–Coleman.................. 7

3 Irrational points on hyperelliptic curves 9

3.1 Introduction................................ 9

3.2 Arithmetic and geometry of hyperelliptic Jacobians . . . . . . . . . . 14

3.3 Bounding the number of unexpected degree d points. . . . . . . . . . 21

3.4 Explicit bounds on the number of unexpected quadratic points . . . . 27

3.5 Explicit bounds on the number of cubic points . . . . . . . . . . . . . 31

4 Background — Hyperbolicity ............................... 36

4.1 Varieties of general type ......................... 36

4.2 Hyperbolicity in complex analytic setting . . . . . . . . . . . . . . . . 37

4.3 Hyperbolicity in the algebraic setting .................. 39

4.4 Hyperbolicity in the non-Archimedean analytic setting . . . . . . . . 42

4.5 The conjectures of Green–Griffiths–Lang–Vojta . . . . . . . . . . . . 45 

5 Statement of results on the strong non-Archimedean Green–Griffiths– Lang–Vojta conjecture . . . . . . . . 47

6 The non-Archimedean Green–Griffiths–Lang–Vojta for closed sub-varieties of a semi-abelian variety . . . . . . . . 49

6.1 Introduction................................ 49

6.2 Non-Archimedean entire curves in semi-abelian varieties . . . . . . . . 50

7 The non-Archimedean Green–Griffiths–Lang for projective surfaces dominating an elliptic curve . . . . . . . . 55

7.1 Introduction................................ 55

7.2 Related results .............................. 56

7.3 Surfaces of general type of irregularity one . . . . . . . . . . . . . . . 57

7.4 Semi-coverings............................... 58

7.5 Non-Archimedean entire curves in projective varieties of general type dominating an elliptic curve ....................... 59

7.6 Non-Archimedean entire curves in projective surfaces dominating an ellipticcurve................................ 64

Bibliography ............................... 67 

About this Dissertation

Rights statement
  • Permission granted by the author to include this thesis or dissertation in this repository. All rights reserved by the author. Please contact the author for information regarding the reproduction and use of this thesis or dissertation.
School
Department
Degree
Submission
Language
  • English
Research Field
Palabra Clave
Committee Chair / Thesis Advisor
Última modificación

Primary PDF

Supplemental Files