Non-Archimedean and Tropical Techniques in Arithmetic Geometry 公开
Morrow, Jackson (Spring 2020)
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
| School | |
|---|---|
| Department | |
| Degree | |
| Submission | |
| Language |
|
| Research Field | |
| 关键词 | |
| Committee Chair / Thesis Advisor |
Primary PDF
| Thumbnail | Title | Date Uploaded | Actions |
|---|---|---|---|
|
|
Non-Archimedean and Tropical Techniques in Arithmetic Geometry () | 2020-04-17 12:52:08 -0400 |
|
Supplemental Files
| Thumbnail | Title | Date Uploaded | Actions |
|---|