Cover Page
The handle http://hdl.handle.net/1887/25871 holds various files of this Leiden University
dissertation.
Author: Kosters, Michiel F.
Title: Groups and fields in arithmetic
Issue Date: 2014-06-04
Groups and fields in arithmetic
Proefschrift
ter verkrijging van
de graad van Doctor aan de Universiteit Leiden,
op gezag van Rector Magnificus prof. mr. C.J.J.M. Stolker,
volgens besluit van het College voor Promoties
te verdedigen op woensdag 4 juni 2014
klokke 10:00 uur
door
Michiel Kosters
geboren te Leidschendam
in 1987
Samenstelling van de promotiecommissie:
Promotor:
Prof. dr. H. W. Lenstra
Overige leden:
Prof.
Prof.
Prof.
Prof.
Prof.
Prof.
dr.
dr.
dr.
dr.
dr.
dr.
T. Chinburg (University of Pennsylvania)
R. Cramer (Centrum Wiskunde & Informatica)
B. Edixhoven
B. Moonen (Radboud Universiteit Nijmegen)
P. Stevenhagen
D. Wan (University of California Irvine)
Groups and fields in arithmetic
Michiel Kosters
c Michiel Kosters, Leiden 2014
Typeset using LATEX
Printed by Ridderprint, Ridderkerk
The 63 ‘squares’ on the cover correspond to the units of a finite field of 64 elements.
The placement of the squares is based on the module structure over the subfield of 8
elements. There are 6 types of squares corresponding to the 6 different multiplicative
orders of the elements. Self-similarities have been added for aesthetic reasons.
Contents
Introduction
vii
Chapter 1. The algebraic theory of valued fields
1. Introduction
2. Definition of valuations
3. Main results
4. Preliminaries
5. Extending valuations
6. Normal extensions
7. Algebraic extensions
8. Defects in the discrete case
9. Frobenius formalism
1
1
2
3
10
15
17
24
30
32
Chapter 2. Normal projective curves
1. Introduction
2. Normal projective curves
3. Curves over finite fields
4. Hyperelliptic curves
37
37
37
47
55
Chapter 3. Images of maps between curves
1. Introduction
2. Proof of the first theorem
3. Chebotarev density theorem
4. Density theorem: infinite algebraic over a finite field
5. Proof of second theorem
6. Examples of density calculations and lower bounds
61
61
63
65
68
73
74
Chapter 4. Polynomial maps on vectors spaces over a finite field
1. Introduction
2. Degrees
3. Relations between degrees
4. Proof of main theorem
5. Examples
77
77
77
79
80
82
Chapter 5. Subset sum problem
1. Introduction
2. Proofs of the theorems
83
83
84
v
vi
Contents
Chapter 6. Shape parameter and some applications
1. Introduction
2. Fourier transform
3. Shape parameter
4. Applications of the shape parameter to finite fields
5. Computing the shape parameter
87
87
88
92
96
99
Chapter 7.
1.
2.
3.
4.
5.
6.
7.
Deterministically generating Picard groups of hyperelliptic curves
over finite fields
103
Introduction
103
Realizing Galois groups together with Frobenius elements
105
A generic algorithm
107
Hyperelliptic curves: statements of the results
108
Additive x-coordinate
111
Multiplicative x-coordinate
117
The algorithm
121
Chapter 8. Automorphism groups of fields
1. Introduction
2. Prerequisites
3. Properties of the automorphism groups
4. Degree map of categories
5. Examples of degrees and an application
6. Faithful actions on the set of valuations
123
123
123
125
129
133
139
Bibliography
145
Samenvatting
147
Dankwoord
153
Curriculum Vitae
155
Index
157
Introduction
The title of this thesis is ‘Groups and fields in arithmetic’. This title has been
chosen in such a way that every chapter has to do with at least two of the nouns in the
title. This thesis consists of 8 chapters in which we discuss various topics and every
chapter has its own introduction. In this introduction we will discuss each chapter
very briefly and give only the highlights of this thesis.
Chapter 1 and 2 are preliminary chapters. In Chapter 1 we discuss algebraic
extensions of valued fields. This chapter has been written to fill a gap in the literature.
It does contain some new results. In Chapter 2 we discuss normal projective curves,
especially over finite fields. This chapter does not contain any significant new results.
Chapter 3 and 4 concern polynomial maps between fields. In Chapter 3 we study
the following. A field k is called large if every irreducible k-curve C with a k-rational
smooth point has infinitely many smooth k-points. We prove the following theorem
(Corollary 1.3 from Chapter 3).
Theorem 0.1. Let k be a perfect large field. Let f ∈ k[x]. Consider the induced
evaluation map fk : k → k. Assume that k \ f (k) is not empty. Then k \ f (k) has the
same cardinality as k.
In the case that k is an infinite algebraic extension of a finite field, we prove
density statements about the image (Theorem 1.4 from Chapter 3).
In Chapter 4 we prove the following theorem (Theorem 1.2 from Chapter 4).
Theorem 0.2. Let k be a finite field and put q = #k. Let n be in Z≥1 . Let f1 , . . . , fn ∈
k[x1 , . . . , xn ] not all constant and consider the evaluation map f = (f1 , . . . , fn ) : k n →
k n . Set deg(f ) = maxi deg(fi ). Assume that k n \ f (k n ) is not empty. Then we have
|k n \ f (k n )| ≥
n(q − 1)
.
deg(f )
In Chapter 5 we give an algebraic proof of the following identity (Theorem 1.1
from Chapter 5).
Theorem 0.3. Let G be an abelian group of size n and let g ∈ G, i ∈ Z with 0 ≤ i ≤ n.
Then the number of subsets of G of cardinality i which sum up to g is equal to
s
X
X
1
n/s
N (G, i, g) =
(−1)i+i/s
µ
#G[d],
n
i/s
d
s| gcd(exp(G),i)
d| gcd(e(g),s)
where exp(G) is the exponent of G, e(g) = max{d : d| exp(G), g ∈ dG}, µ is the
M¨
obius function, and G[d] = {g ∈ G : dg = 0}.
vii
viii
Introduction
Chapter 6 is a preliminary chapter for Chapter 7. In Chapter 6 we introduce the
concept of the shape parameter of a non-empty subset of a finite abelian group. We
use this in Chapter 7 to prove the following (Theorem 1.1 from chapter 7).
Theorem 0.4. For any > 0 there is a deterministic algorithm which on input a
hyperelliptic curve C of genus g over a finite field k of cardinality q outputs a set of
generators of Pic0 (C) in time O(g 2+ q 1/2+ ).
In Chapter 8 we study automorphism groups of extensions which are not algebraic.
One of our results is the following (Theorem 5.8 from Chapter 8).
Theorem 0.5. Let Ω be an algebraically closed field and let k be a subfield such
that the transcendence degree of Ω over k is finite but not zero. Endow Ω with the
discrete topology, ΩΩ with the product topology and Autk (Ω) ⊆ ΩΩ with the induced
topology. Then there is a surjective continuous group morphism from Autk (Ω), the
field automorphisms of Ω fixing k, to a non finitely generated free abelian group with
the discrete topology.
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