Non-Archimedean and Tropical Techniques in Arithmetic Geometry 公开

Morrow, Jackson (Spring 2020)

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

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