CONTROLLING RAMIFICATION IN NUMBER FIELDSA DISSERTATIONSUBMITTED TO THE DEPARTMENT OF MATHEMATICS
AND THE COMMITTEE ON GRADUATE STUDIESOF STANFORD UNIVERSITYIN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

Simon Rubinstein-Salzedo
August 2012

c Copyright by Simon Rubinstein-Salzedo 2012
All Rights ReservediiI certify that I have read this dissertation and that, in my opinion, it
is fully adequate in scope and quality as a dissertation for the degreeof Doctor of Philosophy.

(Akshay Venkatesh)Principal Adviser

I certify that I have read this dissertation and that, in my opinion, it
is fully adequate in scope and quality as a dissertation for the degreeof Doctor of Philosophy.

(Brian Conrad)

I certify that I have read this dissertation and that, in my opinion, it
is fully adequate in scope and quality as a dissertation for the degreeof Doctor of Philosophy.

(Persi Diaconis)

Approved for the University Committee on Graduate Studies

iii

iv

Abstract
This thesis focuses on two aspects of limited ramication and is split up into two independent sections. The rst section (which comprises the second and third chapters)
is on the distribution of class groups of cyclic cubic elds. We propose an explanation
for the discrepancy between the observed number of cyclic cubics whose 2-class groupis C2 × C2 and the number predicted by the Cohen-Lenstra heuristics, in terms of aninvariant living in a quotient of the Schur multiplier group. We also show that, insome cases, the denition of the invariant can be simplied greatly, and we compute105 examples.
The second section (which comprises the fourth and fth chapters) discusses
branched covers of algebraic curves, especially covers of elliptic curves with one branch
point. We produce some techniques that allow us to write down explicit equations forsuch maps, and then we give examples of number elds which arise from such covers.
Finally, we present some possibilities for future works that the author hopes topursue.

Acknowledgements
I would like to thank my advisor, Akshay Venkatesh, for teaching me so many things,
suggesting interesting problems that were delightful to work on, helping me makeprogress on the frequent occasions when I encountered diculties, and always being
infectiously enthusiastic about mathematics. I would also like to thank the mathematics department at Stanford University as a whole; it is exciting to be around somany people who also like mathematics and who are likely to tell me exciting newthings at any moment. I am honored to have been able to work with and among suchpeople.
I would like to thank the people who have helped me with my thesis. In addition to Akshay Venkatesh, who helped a tremendous amount, I would like to thankDan Boneh, Brian Conrad, Persi Diaconis, and Ronen Mukamel for serving on mycommittee and reading my thesis. Many other people also gave me valuable feedback and mathematical suggestions about my work, including Rebecca Bellovin, Greg
Brumel, Otis Chodosh, Je Danciger, Jordan Ellenberg, Ilya Grigoriev, Julio Gutiérrez, John Jiang, Silas Johnson, Jürgen Klüners, Hendrik Lenstra, Sam Lichtenstein,Michael Lipnowski, Daniel Litt, Jeremy Miller, Dan Murphy, David Roberts, David
Sher, Soarer Ho Chung Siu, Nisan Stiennon, Frank Thorne, Ravi Vakil, and Melanie
Matchett Wood. Thanks to all of them!
I am also grateful for earlier mathematical opportunities I have had in my life.
Many thanks to the College of Creative Studies at the University of California, Santa
Barbara and the mathematics department there. I could not possibly have had abetter undergraduate experience than I had there. There were many people there
who supported me, but I would like to single out Birge Huisgen-Zimmermann, Mihai
ii

Putinar, and Charles Ryavec from the mathematics department, and Joel Feigin fromthe music department for being particularly helpful to me.
Thanks also to the mathematics contest community in the San Francisco Bay
Area, 19992003. I shudder to think about how I would have escaped high schoolwith my love of mathematics intact were it not for the stimulating environment I
received at contests. Thanks also to Steve Headley, for being the primary teacher at
my high school to take an active role in improving my mathematical education; hewent far above and beyond the call of duty required of any teacher and deserves muchcredit for doing so.
I am also appreciative of all the students I have had over the years. Throughthem I remain convinced that I am helping to improve the world, even in those roughtimes when I sit at my desk day after day making no progress on my research and thus
contributing nothing to the totality of human knowledge. When I teach, I can remind
myself that my years of study can help to make someone else's life better. Thanks
especially to four years of SUMaC students, who have indulged me by allowing me
to tell numerous mathematical stories that fascinate me; I hope some of them weresimilarly interested. Some of them are still willing to talk to me, so I'll take that as
a positive sign.
Finally, I would like to thank my family for supporting me throughout my entire
life. They have always encouraged me to pursue intellectual endeavors and to learn
as much as possible.

iii

Contents
Acknowledgementsii

1 Introduction

1

2 The Cohen-Lenstra Heuristics

6

2.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

2.2

The Cohen-Lenstra Heuristics . . . . . . . . . . . . . . . . . . . . . .

11

2.3

Malle's computations . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

3 Roots of Unity

19

3.1

A Concise Guide to the Chapter . . . . . . . . . . . . . . . . . . . . .

19

3.2

Schur Multipliers and Variants . . . . . . . . . . . . . . . . . . . . . .

20

3.3

Lifting Invariants for Hurwitz Spaces . . . . . . . . . . . . . . . . . .

23

3.4

Invariants for Number Fields . . . . . . . . . . . . . . . . . . . . . . .

27

3.5

Computing the invariant . . . . . . . . . . . . . . . . . . . . . . . . .

31

3.6

A sample invariant computation . . . . . . . . . . . . . . . . . . . . .

37

3.7

The data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

4 Background on Bely maps

41

4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

4.2

Connection to Number Theory . . . . . . . . . . . . . . . . . . . . . .

44

4.3

Generalities on Bely maps . . . . . . . . . . . . . . . . . . . . . . . .

45

4.4

Computations of Bely maps . . . . . . . . . . . . . . . . . . . . . . .

48

4.5

Number Fields and Dessins . . . . . . . . . . . . . . . . . . . . . . . .

51

iv

5 Origamis

54

5.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

54

5.2

A rst computation of an algebraic origami . . . . . . . . . . . . . . .

56

5.3

A non-totally ramied example . . . . . . . . . . . . . . . . . . . . .

62

5.4

Thickening to a family . . . . . . . . . . . . . . . . . . . . . . . . . .

71

5.5

Global structure of the family . . . . . . . . . . . . . . . . . . . . . .

77

5.6

Extracting number elds from origamis . . . . . . . . . . . . . . . . .

80

5.7

Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

82

6 Future possibilities

83

6.1

The Cohen-Lenstra heuristics . . . . . . . . . . . . . . . . . . . . . .

83

6.2

Bely maps and origamis . . . . . . . . . . . . . . . . . . . . . . . . .

84

A Long numbers

85

Bibliography

96v

List of Tables
3.1

Invariant Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vi

39

List of Figures
3.1

Braid group generator . . . . . . . . . . . . . . . . . . . . . . . . . .

26

4.1

A sample dessin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

4.2

A sample dessin, with additional labels . . . . . . . . . . . . . . . . .

47

5.1

An origami diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

5.2

A degenerate origami diagram . . . . . . . . . . . . . . . . . . . . . .

58

5.3

A three-dimensional version of a degenerate genus 2 curve . . . . . .

58

5.4

Swiss cross origami . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62

5.5

An elliptic curve with a typical basis of H1 . . . . . . . . . . . . . . .

65

5.6

A genus 2 curve with a typical basis of H1 . . . . . . . . . . . . . . .

65

5.7

Figure 5.6 in a dierent form . . . . . . . . . . . . . . . . . . . . . . .

67

5.8

Another version of the standard elliptic curve . . . . . . . . . . . . .

77

5.9

A Dehn twist of the Swiss cross . . . . . . . . . . . . . . . . . . . . .

78

5.10 A rearrangement of the previous gure . . . . . . . . . . . . . . . . .

78

vii

viii

Chapter 1
Introduction
This thesis is comprised of work on two projects. The rst project, discussed inchapters 2 and 3, is on the Cohen-Lenstra heuristics. The Cohen-Lenstra heuristics
are concerned with the distribution of class groups in number elds. Papers by Cohen,
Lenstra, and Martinet from the 1980s provide conjectural probability distributions
for the Sylow p-subgroups of class groups of number elds of certain types. For themost part, these conjectured distributions match the available data well, but they do
not match the data at a few primes.
In Chapter 2, we give some background on algebraic number theory and the
Cohen-Lenstra heuristics. There is nothing original in this chapter; it is only intended
to give the reader an overview of previous work in the eld. In Chapter 3, we present
the author's contributions to the subject. In particular, we focus on the case of cyclic
cubic elds for which the Sylow 2-subgroup of their class groups are C2 × C2 , therst case in which the Cohen-Lenstra distribution does not match available data. We
propose an explanation for the failure of the Cohen-Lenstra heuristic in this case,which is described in Sections 3.4 and 3.5.
Briey, what we do is to construct an invariant associated to cyclic cubic eldswith Sylow 2-subgroup of the class group isomorphic to C2 × C2 , valued in C2 . We
conjecture that the invariant is equidistributed in these two classes as the discriminantof the elds goes to innity, and the remainder of Chapter 3 is devoted to giving
numerical evidence to support this conjecture. For the purpose of computation, the1

2

CHAPTER 1. INTRODUCTION

most useful pieces can be found in Theorem 3.5.4 and Algorithm 3.5.6. From there,
we compute invariants for 105 examples of elds of this type, and we nd that theinvariant is roughly equidistributed, but with a small bias in favor of invariant 1 whichappears to drop o gradually.
The second project, discussed in chapters 4 and 5, is on explicit branched coversof algebraic curves. This problem has been carefully studied in the case of maps from

P1 to P1 , especially in the form of dessins d'enfants and Bely maps. We present a
brief overview of known results in this eld in Chapter 4. There is little new material
in this chapter. The author computed the examples independently, but they arepresumably known, in some form, by many other people.
In Chapter 5, we study explicit branched covers of elliptic curves. Surprisingly
little work has been done in this area in the past, even though it is a natural follow-up
to the study of Bely maps and dessins d'enfants. We present several techniques forconstructing these covers.
The most interesting techniques are probably those found in Section 5.3. In thissection, we construct an explicit degree-5 cover of an elliptic curve by a genus-2 curveramied only above one point. While the main result of the section, Theorem 5.3.1,
is easy to verify once it is handed to us, it is interesting to come up with the equationof the curve independently. Hence, the techniques that go into coming up with andconjecturing Theorem 5.3.1 are far more exciting than is the actual statement of thetheorem.
The main theme running through both projects is that of ramication. Our workon the Cohen-Lenstra heuristics involves constructing unramied or minimally ramied extensions of number elds. Our work on covers of algebraic curves involves
constructing covers which are ramied only at a few specied points, and with specied ramication types. From such covers, it is possible to nd number elds which
are ramied only at specied sets of primes.
The number elds we construct give positive answers in certain cases to the following natural extension of the inverse Galois problem: given a nite group G and anite set S of primes, is it possible to nd a Galois extension K/Q with Galois group

G so that K/Q is unramied outside S ?

3The classical inverse Galois problem asks the following:

Question 1.0.1. Let G be a nite group. Is there a Galois extension K/Q so that
Gal(K/Q) ∼= G?
In what follows, I shall present a small sample of the work that has been done on
this problem. Note that it is far from being complete.
This problem goes back to the 19th century, and it still remains an open problem,and one that attracts much attention today. There is no nite group G for which theinverse Galois problem for G is known to have a negative answer, but the question
remains unknown for large classes of groups G.
Hilbert attempted this problem by constructing algebraic curves over Q. Such
curves give function elds K/Q(t). If Gal(K/Q(t)) ∼= G, then Hilbert was able toconstruct number elds over Q with Galois group G. Hilbert was then able to answer
the inverse Galois problem in the armative for all the symmetric and alternatinggroups. This is the approach taken in chapter 4, so we will postpone further discussionof this method until then.
For many other classes of nite groups G, the inverse Galois problem is known tohave a positive solution. It is easy to show that if G is an abelian group, then we cannd a suitable K ; we can construct it as a subeld of some cyclotomic eld.
A much deeper result, due to Shafarevich in [af54], is that the inverse Galois
problem holds for all solvable groups G. Once the solvable groups were taken care of,focus shifted toward simple groups.
Since Hilbert had already answered the inverse Galois problem for alternatinggroups, the next case to look at was that of PSL2 (Fp ). Shih in [Shi74] was able toprove the following result:

Theorem 1.0.2 (Shih). Let p be an odd prime for which at least one of 2, 3, and 7
is a quadratic non-residue modulo p. Then PSL2 (Fp ) occurs as a Galois group overQ.
Starting with the surprising discovery in 1965 of a new sporadic simple group,
the race was on to nd more of them; by 1982, the 21 new sporadic groups had

4

CHAPTER 1. INTRODUCTION

all been found. Number theorists naturally turned to the sporadic groups as a newsource of special cases of the inverse Galois problem. In the mid-1980s, several papers
including [MZM86], came out constructing the small Mathieu groups M11 and M12
as Galois groups.
At around the same time, Thompson developed a new technique, known as rigidity,
to tackle the inverse Galois problem. In [Tho84], he was able to prove the following
result:Theorem 1.0.3. Let G be a nite group, C = (C1 , . . . , Cs ) a sequence of conjugacy
classes of G, andAG (C) = {(g1 , . . . , gs ) ∈ C1 × · · · × Cs : g1 · · · gs = 1}.
Suppose that AG (C) is nonempty.

Then G acts on AG (C) by g(g1 , . . . , gs ) =(gg1 g −1 , . . . , ggs g −1 ). If G acts transitively on AG (C), and one (hence every) element of AG (C) generates G, then G occurs as a Galois group over Q.
Using this theorem, Thompson showed that the Monster group occurs as a Galois
group over Q.
After much work on sporadic groups, the inverse Galois problem is almost completein this case: of the 26 sporadic groups, it remains open only for M23 .
More recently, several people have become interested in the question of number
elds with limited ramication (and perhaps given Galois group). This question has
a geometric interpretation in terms of the étale fundamental group. If S is a scheme,
then there is a pronite group π1ét (S) which serves a similar function to the (pronitecompletion of the) topological fundamental group; that is, it classies nite connectedét (S) denote the set of abstract nite groups
covers (or covering spaces) of S . If we let πAét measures the Galois groups (or groups ofwhich occur as quotients of π1ét (S), then πA

deck transformations) which can occur for nite covers.
To use this machinery in the case of Galois groups over Q, we let S =Spec(Z[1/N ]), where N is the product of the primes at which we allow ramication. The question of whether G occurs as a Galois group over Q of a eld unramied

5

ét (Spec(Z[1/N ])).
at primes not dividing N is then equivalent to asking whether G ∈ πA

An interesting discussion along these lines can be found in [Har94].
One concrete question, posed by Gross in [Gro98], is the following:

Question 1.0.4. Let p be a prime. Is there a number eld K , Galois over Q, unramied away from p and ∞, so that Gal(K/Q) is nonsolvable?
This question has now been solved, in the armative. For primes p ≥ 11, thisquestion was answered by Serre in [Ser73], by Dembélé for p = 2 in [Dem09], by
Dembélé, Greenberg, and Voight for p = 3 and 5 in [DGV11] and by Roberts for

p = 5 with an explicit polynomial in [Rob11]. Very recently, Dieulefait in [Die12]
solved this question for p = 7, nally answering the question completely. Thesepapers used modular forms to construct the relevant elds.
The general question, of which pairs (G, S) of a Galois group and a set of ramied
primes, can occur in a number eld, is still very much open. It is this question thatinspires the work done in this thesis, and to which I hope to be able to contribute.
But at the moment, there is still much to be done before I will be able to construct
many new examples.

Chapter 2
An Introduction to the
Cohen-Lenstra Heuristics
2.1

Introduction

Before we begin our discussion of the Cohen-Lenstra heuristics, it will be helpful torecall some basic denitions and results from algebraic number theory and class eldtheory.

Denition 2.1.1. Let K be a number eld with ring of integers oK . A fractional
ideal of K is a nonzero nitely generated oK -submodule of K . Let I(K) denote theset of fractional ideals of K .
The fractional ideals of K form a group under multiplication. The identity element
is the fractional ideal oK itself.

Denition 2.1.2. A principal fractional ideal of K is a fractional ideal of the form
aoK , for some a ∈ K × . Let P (K) denote the set of principal fractional ideals.
The principal fractional ideals form a subgroup of the group of fractional ideals.

Denition 2.1.3. The class group of K is the quotient Cl(K) = I(K)/P (K).
One of the most important theorems in basic algebraic number theory is thefollowing:62.1. INTRODUCTION

7

Theorem 2.1.4. If K is a number eld, then Cl(K) is a nite group.
See 17, page 71, of [CF86] for a proof, or the discussion in IV.3 of [FT93] for aneective version, providing a bound on its size.

Denition 2.1.5. The class number h(K) is dened to be the order of Cl(K).
The class group was rst studied in connection with Fermat's Last Theorem, byErnst Kummer in the middle of the 19th century, based on a failed attempt by GabrielLamé to give a proof of this celebrated then-conjecture. The idea was as follows. Let

p ≥ 3 be a prime. We wish to show that xp + y p = z p has no solutions in positive
integers. So, we try to factor the left side as
p−1
Y(x + ζ i y),
i=0where ζ is a primitive pth root of unity. Since the right side is a pth power, if all thefactors in the product are pairwise relatively prime, then each one of them must bea pth power as well. One can then attempt to show that this is impossible. A careful
exposition of this approach can be found in [Was97].
However, this approach implicitly assumes that Z[ζ] is a unique factorization domain, and this is false in general. (In fact, it is false for all primes p ≥ 23.)
It it not too dicult to show that K has class number 1 exactly when oK is aunique factorization domain. Hence, the class number is a measure of the failure ofoK to be a unique factorization domain. More concretely, if we have some elementx ∈ oK with nonunique factorization, we can construct a nonprincipal ideal.
√√
Example. Let K = Q( −5), so that oK = Z[ −5]. Then K has class number 2.
We can give an explicit element of oK which has nonunique factorization:

6 = 2 × 3 = (1 +√−5)(1 −√

−5).

We can use this factorization to give us an example of a nonprincipal ideal, namely√(2, 1 + −5). (Any other similar pair of factors would also do. Any two such pairs
provide ideals in the same ideal class, however.)8

CHAPTER 2. THE COHEN-LENSTRA HEURISTICS

It will also be helpful for this chapter to recall various facts about ramicationin number elds. We begin by recalling the following theorem, known as the ef g
Theorem:

Theorem 2.1.6 (ef g Theorem). Let L/K be an extension of number elds of degreen, and let p ⊂ oK be a prime ideal. Suppose poL factors in oL as
poL =

g
Y

Pei i ,

i=1

where the Pi are distinct primes in oL . Let fi = [oL /Pi : oK /p]. Then

n=

g
X

ei fi .

i=1

If, furthermore, L/K is a Galois extension, then all the ei 's and fi 's are equal, so wewrite these common values as e and f , respectively. We then haven = ef g.
This is Theorem 20 in [FT93]. A proof can be found there.
An important invariant associated to a number eld is its discriminant, dened
as follows:Denition 2.1.7. Let K be a number eld with ring of integers oK . Let {αi } be an
integral basis for K , i.e.,

oK =

M

Zαi

i

as a Z-module.
1. If α ∈ K , then we dene the trace of α to be the trace of the multiplication by

α map with respect to the basis {αi } of K . (Of course, this is independent ofchoice of basis.) We denote this number by Tr(α).

2.1. INTRODUCTION

9

2. The discriminant of K is dened to bedisc(K) = det(Tr(αi αj ))i,j .
We can also dene the discriminant of an extension of number elds L/K , but
the situation is a little bit more subtle, since oL need not be a free oK -module. To
remedy this, we dene the discriminant of L/K to be the ideal of oK dened asfollows: suppose [L : K] = n, and suppose that A = {α1 , . . . , αn } ⊂ oL generates anite-index free oK -subalgebra of oL . Let discA (L/K) = det(Tr(αi αj ))i,j . Then thediscriminant ideal of L/K is the ideal of oK generated by all the discA (L/K)'s.

Denition 2.1.8. Let L/K be an extension of number elds, and let p ⊂ oK be aprime ideal. We say that p is ramied in L if, in the factorization given by the ef g
Theorem, some ei > 1.
It turns out that, for any extension, there are only nitely many ramied primes.
These are exactly the primes dividing the discriminant of L/K .
It is also possible to talk about a eld being ramied or unramied at ∞.

Denition 2.1.9. An extension L/K of number elds of degree n is said to be unramied at ∞ if, whenever ι : K → R is an embedding, then there exist n embeddingsι̃ : L → R whose restriction to K is equal to ι. Otherwise, the extension is said to beramied at ∞.
In particular, if K is totally real, then L/K is unramied at ∞ if and only if L is
also totally real.
One of the most remarkable achievements of twentieth century mathematics was
the development of class eld theory. Global class eld theory relates abelian extensions of number elds (and function elds) to certain generalizations of class groups.
We briey describe some of the key results in class eld theory now.

Theorem 2.1.10. Let K be a number eld. Then there exists a unique (up to isomorphism) maximal Galois extension L/K which is unramied at all prime ideals

p as well as at ∞, and so that Gal(L/K) is abelian. Furthermore, in this case,
Gal(L/K) ∼= Cl(K).

10

CHAPTER 2. THE COHEN-LENSTRA HEURISTICS

See page 61 of [Chi09] for more information and generalizations of this theorem.

Example.

1. If K = Q, then its class group is trivial, so there are no unramied

abelian extensions of Q.

√
2. If K = Q( −5), then Cl(K) = C2 , so there is a unique unramied quadratic√√
extension of K . This extension is Q( −5, −1).
√
3. If K = Q( −23), then Cl(K) = C3 , so there is a unique unramied cyclic cubic√
extension of K . This extension is Q( −23, α), where α is a root of x3 − x − 1.
This extension L is known as the Hilbert class eld of K and is frequently denotedby H(K). It is perhaps worth noting that the isomorphism given in the theorem is
not merely an abstract isomorphism; rather, by using some powerful theorems in classeld theory such as (a very weak version of) the Chebotarev density theorem, we candescribe the map quite explicitly. The map in question is known as the Artin map.
We can also construct variants of the Hilbert class eld. For instance, the maximal
Galois unramied abelian extension of K for which the Galois group is abelian andfor which [L : K] is a pth power corresponds via the Artin map in class eld theory(or simply by Galois theory) to the Sylow p-subgroup of Cl(K).
There is also a variant of the class group known as the narrow class group, and it
will be used later in this section. Let P + (K) denote the group of principal fractional
ideals (a) of K for which a is totally positive, i.e., so that ι(a) > 0 for every embeddingι : K ,→ R. Let Cl+ (K) = I(K)/P + (K). We call Cl+ (K) the narrow class group ofK.
As we saw above, there is a eld (the Hilbert class eld) associated to the classgroup of a number eld. Similarly, there is a eld, called the narrow class eld,
associated to the narrow class group. The narrow class eld of K is the maximalabelian extension of K that is unramied at all nite primes.

√

Example. Let K = Q( 3). Then Cl(K) = 1, and Cl+ (K) = C2 . Hence, there exists
a quadratic extension L/K which is unramied at all nite places, but is ramied at√ √innity. This eld is L = Q( 3, −1).

2.2. THE COHEN-LENSTRA HEURISTICS

2.2

11

The Cohen-Lenstra Heuristics

Cohen and Lenstra in [CL84] were interested in studying the distribution of classgroups of quadratic elds, and perhaps, abelian extensions of Q more generally. These
heuristics were extended by Cohen and Martinet in [CM87] and [CM90] in 1987 and1990 to elds of more general type.
Since class groups are nite abelian groups, we can attempt to understand themby understanding their p-Sylow subgroups. Assuming that the distributions of the

p-Sylow subgroups are independent for dierent p, we can then patch together the
distribution of class groups from the distribution of the p-Sylow subgroups for allp. (The independence of distributions at each prime is conjectural, but it is wellsupported by numerical data.)
Therefore, we will restrict ourselves to looking at the distribution of the p-Sylow
subgroups of the class groups of number elds. Let K be a number eld. Then we
let Clp (K) denote the p-Sylow subgroup of Cl(K), and we let hp (K) denote its order.
In the case of quadratic elds, Cohen and Lenstra made the following conjecture:

Conjecture 2.2.1 (Cohen-Lenstra). Let p be an odd prime. Let D± (X) denote theset of real (respectively imaginary) quadratic elds K with | disc(K)| < X . Let A bea nite abelian p-group. Then#{K ∈ D± (X) : Clp (K) ∼= A}
λ (A) = lim
X→∞#D± (X)±

exists, and we haveλ+ (A) = c+ | Aut(A)|−1 × |A|−1 ,λ− (A) = c− | Aut(A)|−1

for certain explicit constants c+ and c− , which are independent of A.
The statement of this conjecture suggests many further questions. One such question is why we need to restrict to the case of of an odd prime p. The reason is that the
2-torsion in the class group is controlled by a dierent phenomenon, namely genus
theory. In the general case, genus theory is quite involved (see [Frö83] for a complete

12

CHAPTER 2. THE COHEN-LENSTRA HEURISTICS

treatment), but in the case of 2-torsion in a quadratic eld, it is readily understood:
if K is a quadratic eld and r is the number of primes dividing disc(K), then the
2-torsion of Cl(K) is isomorphic to C2r−1 or C2r−2 (see [FT93], Corollary 1 to Theorem
39, for a more precise statement and a proof). In particular, the 2-torsion in quadraticelds is rarely equal to 0 and can easily become arbitrarily large.
Another question that is likely to arise is what sort of heuristic evidence we havethat suggests such a result might be true, or even a reasonable guess. One answer,
supported by analogies in other areas of mathematics (especially the theory of stacks
in algebraic geometry) is that objects in general are best and most uniformly treated
when we take a quotient by the automorphism group of each object. An instance of
this phenomenon that will become increasingly relevant in later chapters is that ofelliptic curves. Most elliptic curves (over the complex numbers) have automorphismgroup C2 , but those with j -invariant 0 or 1728 have larger automorphism groups. The
moduli stack of elliptic curves is then the quotient of the coarse moduli space A1 ofelliptic curves by automorphism groups, and this is the best object to use to study
families of elliptic curves, since there are unexpected phenomena that occur for thesej -invariants.
It is also worth at this point giving a plausibility argument for this conjecture,
which comes from a result of Friedman and Washington in [FW89]. Suppose wehave a (large) collection of ideals I1 , . . . , In in a number eld K . Then their images in

Cl(K) satisfy various relations. In fact, we expect them to satisfy n or n + 1 relations,depending on whether K is imaginary quadratic or real quadratic, respectively. Assuming we have chosen enough ideals to generate the class group, the class group willthen be the cokernel of this matrix, considered as a map of free Z-modules. Hence,the class group should be well-modeled by the cokernel of a random matrix of theright size.
Unfortunately, this approach does not make any sense as stated, since there is no
obvious measure to put on these matrices. To x this problem, we restrict to the

p-power torsion and consider the matrices to have entries in Zp . Now, if we assume
that the entries are distributed identically and independently according to the Haar
measure on Zp , then we have a well-dened distribution on their cokernels. Friedman

2.2. THE COHEN-LENSTRA HEURISTICS

13

and Washington showed that, as n → ∞, a limiting distribution exists and is equal
to the distribution in the above conjecture.
Here, it is necessary to try to justify the dierence between the imaginary
quadratic and real quadratic cases, but it is clear from even the most casual glance ata table of class numbers that imaginary and real quadratic elds behave very dierently. Indeed, there are exactly nine imaginary quadratic elds with class number one,while there appear to be innitely many (and in fact, around 75% of) real quadraticelds with class number one. It is hard to give very compelling reasons to believethat the rank of the unit group should be a determining factor in the distributionof class groups, but let us try to demonstrate that at least there ought to be someconnection.
One place in which it becomes clear that there is a serious dierence between realand imaginary quadratic elds is in the analytic class number formula. If K is any
number eld, and ζK is the Dedekind ζ function for K , then we have the formulaζK (0) = −hR,w

where h is the class number, R is the regulator, and w is the number of roots of unity
in K . For all but nitely many quadratic elds, w = 2, so that term is not so critical.
However, in the case of an imaginary quadratic eld, R = 1, while for real quadraticelds, R depends on the fundamental unit of K . We should think of hR as being one
block from an analytic point of view; it is hard to write down an analytic expressionthat separates h from R. If we were to study the distribution of hR in place of h, thenwe should expect much greater similarities between the real quadratic and imaginary
quadratic situations.
Another place the unit group comes into consideration in the study of class groups
is in the construction of class groups of quadratic elds in terms of binary quadratic
forms. For imaginary quadratic elds K with discriminant −∆, the class group isexactly the group of reduced binary quadratic forms with discriminant −∆. For real
quadratic elds, the analogous construction only gives us the narrow class group,
which is either equal to the class group or else twice as large (in the case of real

14

CHAPTER 2. THE COHEN-LENSTRA HEURISTICS

quadratic elds).
In order to remedy this, we might think of the class group of a real quadratic eldas being the narrow class group modulo the cyclic subgroup generated by a uniformly
chosen element. (This is far from being actually true, but it is reasonable intuition.)
Quotienting out by one element of a group modeled after the narrow class groupshould correspond to allowing one extra relation in the matrix. This should be ourintuition for why the class group of a real quadratic eld should be modeled by ngenerators and n + 1 relations.
Of course, we need not lose interest in class groups as soon as we step beyond
quadratic elds: we could ask the same question for elds of other types. Furthermore,in this case, the 2-power torsion will not necessarily be governed by genus theory, so
we might also allow p to be 2. So, we could make the following guess, by attempting toapply the Cohen-Lenstra heuristics to situations for which we have no a priori reason
for believing that they are appropriate, apart from a random matrix-type argument
or a fuzzy argument of the type given just above.

Heuristic 2.2.2 (Proto-Cohen-Lenstra Heuristics). Let n be a positive integer, and
let G be a transitive permutation group on a set of size n. Furthermore, let (r1 , r2 ) bea signature, with r1 + 2r2 = n. Let D(X) be the set of number elds K the absolutevalue of whose discriminant is less than X , and so that the Galois group Gal(K ] /Q)of the Galois closure of K is isomorphic to G, and K has r1 real embeddings and r2
pairs of complex conjugate embeddings. Finally, let p be a prime not dividing |G| andlet A be a nite abelian p-group. Then#{K ∈ D(X) : Clp (K) ∼
= A}
X→∞
#D(X)
lim

exists, and is inversely proportional to | Aut(A)| × |A|r1 +r2 −1 .
However, the proto-Cohen-Lenstra heuristics sometimes fail for silly reasons.
Here's an example:

Lemma 2.2.3. Let n = 3 and G = C3 in the above heuristics. If p ≡ 2 (mod 3),
then the p-rank of K , rp (K) := dimFp (Cl(K)/p Cl(K)), is even for all such K .

2.2. THE COHEN-LENSTRA HEURISTICS

15

Proof. The Galois group C3 = {1, σ, σ 2 } of K over Q acts on Clp (K). The numberof ideal classes of order p is prp (K) − 1, which is congruent to 0 (mod 3) if and only
if rp (K) is even. Hence, if rp (K) is odd, then there must be a nontrivial p-torsion
ideal class C xed by the Galois action. In particular, there's a nonprincipal ideal2a ∈ C of order p so that a, aσ , and aσ are all in the same ideal class. We now
2show that their product a1+σ+σ is equal to the principal ideal (N a)oK . For any
22a ∈ a, N a = a1+σ+σ ∈ a1+σ+σ , and the N a generate (N a)oK as an oK -module, so22(N a)oK ⊂ a1+σ+σ . Now, the norms of both ideals a1+σ+σ and (N a)oK are (N a)3 .
Hence, they are equal. Furthermore, (N a)oK is principal, since it is generated by theelement N a. Hence C 3 = 1 in Clp (K). But this is impossible, as 3 - p.



We can patch the proto-Cohen-Lenstra heuristics by excluding those A that are
ruled out by this Lemma and related ones. Furthermore, as the proof of the lemma
hints, for those A that are allowable, we need the automorphisms to be compatible
with the Galois action, in the case that K is actually a Galois number eld. Moreprecisely, if G = C` , then we need A to be a Z[ζ` ]-module. This suggests the following
renement of the proto-Cohen-Lenstra heuristics, at least in the case where n = ` isa prime, and G = C` :Heuristic 2.2.4 (Rened Cohen-Lenstra Heuristics). Let ` be an odd prime, and let
G = C` . Let D(X) be the set of C` number elds with absolute discriminant less thanX . (Such elds are necessarily totally real.) Also, let p be a prime dierent from `and A an abelian p-group with the structure of a Z[ζ` ]-module. Then#{K ∈ D(X) : Clp (K) ∼
= A}
X→∞
#D(X)
lim

exists, and is inversely proportional to | AutZ[ζ` ] (A)| × |A|`−1 .
Another, more modern and sometimes cleaner, way to interpret the rened CohenLenstra heuristics is based on the following idea from probability theory. Let µ be aprobability distribution on R. Dene the k th moment of µ to be

Z ∞
ak =
−∞

xk dµ.

16

CHAPTER 2. THE COHEN-LENSTRA HEURISTICS

From knowing the sequence of moments a1 , a2 , . . ., it is possible to reconstruct µ under
fairly mild hypotheses. To be more precise, dene the moment generating function
to be the power series

∞
X

xk
ak .
A(x) =
k!
k=1
Then assuming that A(x) has positive radius of convergence, µ is the only probability
distribution having moment generating function A(x). (See [Bil95], Chapter 30, fora proof.)In the context at hand, we can dene an analogue of a moment for a probability
distribution f of nite abelian p-groups as follows. Fix a nite abelian p-group A, andlook at the expected number of surjections (in whichever category is appropriate) froman f -random nite abelian p-group X to A. This number behaves as the  Ath momentof X . Just as in the situation for classical moments of probability distributions, theseAth moments of X determine f , assuming that f is fairly well-behaved.
We now put this in proper context. If A is a nite abelian p-group that also has
the structure of a Z[ζ` ]-module, then we would like to understand the number

E(# SurjZ[ζ` ] (Cl(K), A)).
Here Surj is the set of surjective maps, and E denotes the expected value. Let's lookat the case of n = 3 and G = C3 . Then the rened Cohen-Lenstra heuristics areequivalent to

1
X→∞ #D(X)lim

XK∈D(X)

prp (K) =2
 1+ 1pp≡1(mod 3),

1 + 1p2

p≡2(mod 3).

(2.2.1)In the case when A = Cp (if p ≡ 1 (mod 3)) or A = Cp × Cp (if p ≡ 2 (mod 3)),
then

X
1
(prp (K) − 1).
X→∞ #D(X)E(# SurjZ[ζ3 ] (Cl(K), A)) = lim

K∈D(X)

So, in particular, if p = 2, we'd expect the number of surjections from the class groupof a random C3 eld to C2 × C2 to be 1/4. As we shall see shortly, however, this2.3. MALLE'S COMPUTATIONS

17

appears not to be the case.

2.3

Malle's computations

Notation. We use the following notation: for q, k ∈ N, letkY(1 − q −1 ),(q)k =∞
Y(q)∞ =(1 − q −1 ).

i=1

i=1

After performing many tests, Malle proposed a list of cases in which the CohenLenstra heuristics are expected to fail. In particular, when p = 2, they should always
fail. In the case of C3 elds, the Cohen-Lenstra heuristics predict that the Sylow
2-subgroup of the class group should be isomorphic to C2 × C2 with probability

1 (4)∞≈ .0765.
12 (4)1
Instead, in his sample of over 16 million elds, he nds that the actual probability
is closer to .13, nearly twice as large as expected. Similarly, equation (2.2.1) does
not seem to hold when p = 2: equation (2.2.1) predicts that the average size of the
maximal elementary abelian 2-subgroup of Cl(K) be 45 , but Malle's computations
suggest that the correct number is 32 .
In terms of expected number of surjections, it appears thatE(# SurjZ[ζ3 ] (Cl(K), C2 × C2 )) = 1/2,

(2.3.1)rather than 1/4, as mentioned in the previous section.
In general, Malle expects the Cohen-Lenstra heuristics to fail at the prime p when
the ground eld contains pth roots of unity. In this case, he expects that if A isa nontrivial abelian p-group, then Clp (K) ∼= A more often than the Cohen-Lenstra
heuristics predict.

Remark 2.3.1. For C3 elds, the Cohen-Lenstra prediction also fails for p = 3, sincethe 3-torsion in the class group is governed by genus theory, just as in the case of

18

CHAPTER 2. THE COHEN-LENSTRA HEURISTICS

p = 2 for quadratic elds. More generally, the Cohen-Lenstra predictions at a prime p
do not hold for elds with Galois group G if p divides |G| because of genus theory. In
this thesis, we are not especially interested in the failure for that reason, since genus
theory is well-understood. Thus, we will only be concerned with deviations due tothe existence of pth roots of unity.
In the case of quadratic elds, it is still interesting to look at the 4-ranks ofclass groups; that is, the maximum value of r so that Cl(K) contains a subgroupisomorphic to C4r . These are no longer controlled by genus theory, except that wehave an obvious bound for the 4-rank in that it cannot exceed the 2-rank. In [Ger89],
Gerth investigates the following situation. Let F be an imaginary quadratic eldwith odd class group. He then wishes to determine the distribution of 4-ranks of classgroups of quadratic extensions K of F , ordered by the absolute norm of the relative
discriminant of K over F . He proves the following theorem:

Theorem 2.3.2 (Gerth, [Ger89]).

√
1. If F 6= Q( −1), then the probability thatthe 4-rank of Cl(K) is equal to r is

(2)∞.
2j(j+1) (2)j (2)j+1√
2. If F = Q( −1), then the probability that the 4-rank of Cl(K) is equal to r is
3
2(j+1)(j+2)/2 (4).
∞Of particular importance is the dierence in behavior when F contains a fourthroot of unity and when F does not contain a fourth root of unity. Inspired by this andrelated results, Malle conjectured that deviations from the Cohen-Lenstra heuristics
occur when the ground eld contains roots of unity.

Chapter 3
The Cohen-Lenstra Heuristics andRoots of Unity
3.1

A Concise Guide to the Chapter

The goal of this chapter is to explain the discrepancy in (2.3.1): the class group of aC3 eld tends to contain more copies of C2 × C2 than the Cohen-Lenstra heuristics
predict. In this chapter, we will not work directly with C3 elds, but rather withquartic A4 elds, which will be slightly more convenient. However, this is only aminor distinction as there is a natural bijection between totally real A4 elds and C3
elds K with a Galois-equivariant surjection from Cl(K) to C2 × C2 .
The chapter begins with some preliminary material in 3.2 on the Schur multiplierand a variant of it called the reduced Schur multiplier. After that, we present somebackground material on lifting invariants in 3.3. This material is motivational: we
do not use the results of this section in our new results or computations. However,much of what we do is heavily inspired by the results of 3.3. Thus, we include thissection in order to make future constructions not seem like a bolt from the blue.
The new material begins in 3.4. Here we present an invariant associated to an

A4 eld which is expected to explain the discrepancy in (2.3.1). In 3.5, we present
an algorithm to compute the invariant, and we show that in certain circumstances,
the invariant has a simple interpretation as the parity of the class group of a certain
19

20

CHAPTER 3. ROOTS OF UNITY

eld. In 3.6, we perform an explicit computation of the invariant for the smallest

A4 eld. Finally, we end this chapter with Table 3.1, which summarizes the datacollected from 105 elds.

3.2

Schur Multipliers and Variants

Our proposed correction to the Cohen-Lenstra heuristics in the presence of roots ofunity can be described in terms of the reduced Schur multiplier. We rst recall thedenition of the Schur multiplier, then move on to the reduced Schur multiplier.

Denition 3.2.1. Let G be a group.
1. The Schur multiplier group of G is dened to be the second homology group

H2 (G, Z).
2. A central extension of G is a short exact sequencee → G → 1,
0→A→Ge.
where A is an abelian group, and A is contained in the center of G
3. A stem extension of G is a central extension

e→G→1
0→A→G
e and the derived
so that A is contained in the intersection of the center of Ge.
subgroup of G
e of maximal order; in fact, there may
If G is nite, there is a stem extension Ge need not be isomorphic. However,be more than one of maximal order, and such Ge varies over maximal stem extensions, the corresponding A are all isomorphic,
as G
and they are isomorphic to the Schur multiplier group H2 (G, Z). If, in addition, G
is a perfect group (i.e., G = [G, G] is its own commutator subgroup), then there is ae.
unique such group G

3.2. SCHUR MULTIPLIERS AND VARIANTS

21

Suppose G is a nite group. Then H2 (G, Z) is a nite group all of whose elements
have order dividing the order of G. Also, for a prime p, the Sylow p-subgroup of

H2 (G, Z) is trivial if the Sylow p-subgroup of G is cyclic. For convenience, we provide
a few examples of Schur multiplier groups.

Proposition 3.2.2.

1. (Schur 1907, also Corollary 2.2.12 in [Kar87]) Let G bethe nite abelian groupG∼= Cn1 × Cn2 × · · · × Cnk ,(m)with ni+1 | ni for 1 ≤ i ≤ k − 1. Let Cndenote the direct product of m copies

of Cn . Then

H2 (G, Z) ∼× · · · × Cn(k−1).
= Cn2 × Cn(2)3k
2. Let G = C2 × C2 . Then H2 (G, Z) ∼= C2 . There are two stem extensions of G:

0 → C2 → D8 → C2 × C2 → 0
and

0 → C2 → Q8 → C2 × C2 → 0.

(3.2.1)

3. Let G = An be the alternating group on n letters. Then



1



H2 (G, Z) = C2



C
6

n ≤ 3,n ≥ 4 and n 6= 6, 7,
n = 6, 7.

If n = 4, then we have A4 ∼= PSL2 (F3 ), and the unique stem extension of A4 is

0 → C2 → SL2 (F3 ) → A4 → 1.

(3.2.2)Furthermore, the sequence (3.2.1) is the sequence of Sylow 2-subgroups of(3.2.2). If n = 5, we have A5 ∼= PSL2 (F5 ), and the unique stem extension

22

CHAPTER 3. ROOTS OF UNITY

of A5 is0 → C2 → SL2 (F5 ) → A5 → 1.
en for the maximal stem extension of An .
We write A
4. Let G = Sn be the symmetric group on n letters. Then
1
H2 (G, Z) =C2n ≤ 3,n ≥ 4.

For n ≥ 4, there are two nonisomorphic double covers of Sn .
In fact, what we really need is not the full Schur multiplier group, but a certain
quotient of it, associated to a certain union of conjugacy classes of G. To this end,
x a union of conjugacy classes c ⊂ G. Let

e→G→1
0 → H2 (G, Z) → G
be a Schur cover. Suppose x ∈ c and y ∈ G commute. Lift x and y to xe and ye,e. (This can be done in multiple ways; choose one arbitrarily.) Then
respectively, in G
the commutator [ex, ye]Ge lies in H2 (G, Z), and this element is independent of the choiceof lifts. Call this element hx, yiGe . Let Qc denote the subgroup of H2 (G, Z) generated
by all the hx, yiGe 's.

Denition 3.2.3. The reduced Schur multiplier of a pair (G, c) is the quotient
H2 (G, c, Z) = H2 (G, Z)/Qc .
ec = G/Qe c.
A reduced Schur cover of (G, c) is the quotient G
A reduced Schur cover is a largest stem extension of G so that c lifts bijectivelyec .
to a union of conjugacy classes ec⊂GRemark 3.2.4. We will tend to be slightly sloppy with our terminology when referring to reduced Schur covers. In the future, when we refer to a reduced Schur cover3.3. LIFTING INVARIANTS FOR HURWITZ SPACES

23

ec , we shall assume that it comes packaged with a union of conjugacy classes eec
Gc⊂Gwhich bijects onto c, even when no explicit choice of ec is provided.

Example. Suppose G = A5 . If c is the conjugacy class of 3-cycles, then H2 (G, c, Z) ∼=C2 , and the corresponding extension is
e5 → A5 → 1.
0 → C2 → A
However, if c is the conjugacy class of (12)(34), then H2 (G, c, Z) is trivial. To see this,e5 are conjugate
it suces to show that the two lifts of an element of order 2 in A5 to A

e5 . Since A5 ∼
e5 ∼
in A
= PSL2 F5 and A= SL2 F5 , it suces to work with matrices. Let
g ∈ PSL2 F5 be the image of the matrix
2

0

!
.

0 −2
Its two lifts to SL2 F5 are

ge1 =

2

0

!

0 −2

,

ge2 =

!
−2 0
0

2

.

In SL2 F5 , ge1 and ge2 are conjugate, since if

s=

!
0 −1
1

0

,

then

s−1 ge1 s = ge2 .

3.3

Lifting Invariants for Hurwitz Spaces

This section is provided for motivation, as 3.4 would not seem to be very natural
without the knowledge of the material in this section. However, it is not logicallynecessary to read this section in order to understand the new results.

24

CHAPTER 3. ROOTS OF UNITY

In this section, we describe lifting invariants for branched covers of the disk. Thatis, we describe a combinatorial algorithm to determine whether or not two covers arein the same connected component of the space of all such covers. Before we do that,
we need to recall the denition of a monodromy group of a nite cover.

Denition 3.3.1. Let X be a connected topological space, and let f : Y → X be a(not necessarily connected) nite cover of X , of degree d. Let x ∈ X be a basepoint,
and let f −1 (x) = {y1 , . . . , yd } ⊂ Y . Suppose γ is a loop in X based at x. For each

i with 1 ≤ i ≤ d, there is a unique lift γ
ei of γ with γ
ei (0) = yi . We now dene a
permutation πγ ∈ Sd to be the permutation which sends i to j , where γ
ei (1) = yj . The
map γ 7→ πγ induces a map π1 (X, x) → Sd , called the monodromy representation. Its
image is called the monodromy group of f .
We recall some information about Hurwitz spaces, all of which can be found ingreater detail in [EVW12].
Pick a basepoint x on the boundary of the disk, and let Conf n denote the conguration space of n distinct unlabeled points in the interior of the disk. We areinterested in branched covers of the disk with n branch points. We say two suchcovers f1 : Y1 → D and f2 : Y2 → D are isomorphic if there exists a homeomorphism

g : Y1 → Y2 so that the triangle
Y1 @

g

@@
@@
f1 @@

D

/Y
2
~
~
~
~~
~~~ f2

commutes. In particular, f1 and f2 must have the same branch points.
Fix a positive integer d and a nite group G ≤ Sd , and let HurG,n denote themoduli space of branched covers of the disk with n branch points and monodromy
group a subgroup of G. Equivalently, a point in HurG,n is a point {P1 , . . . , Pn } ∈Conf n , together with a homomorphismρ : π1 (D2 − {P1 , . . . , Pn }, x) → G.

(3.3.1)

3.3. LIFTING INVARIANTS FOR HURWITZ SPACES

25

There are some variants of HurG,n which will also be important for us.

• Let CHurG,n be the subspace of HurG,n consisting of the connected covers.
Equivalently, a cover is connected if and only if the map ρ in (3.3.1) is surjective.

• If c ⊂ G is a union of conjugacy classes of G, let HurcG,n be the subspace ofHurG,n consisting of covers for which all the local monodromies around the Pi
lie in c. Dene CHurcG,n similarly.

• Let Zc/G be the free abelian group on the conjugacy classes inside c. If m ∈ Zc/G
is a multi-index, we let HurcG,m be the subspace of HurcG,|m| consisting of covers
for which there are mi branch points with local monodromy inside ci for eachconjugacy class ci ⊂ c.

• We dene other variants similarly.
In this section, we will be interested in understanding the connected components of

HurcG,n . Note that, since a cover is determined up to isomorphism by its monodromy
at the branch points, we can express π0 HurG,n and π0 HurcG,n in terms of quotients ofGn and cn , respectively. To do this, we need to dene the braid group and its action
on Gn (and cn ).

Denition 3.3.2.

• Let n ≥ 2 be an integer. The braid group Brn on n strands isthe group generated by elements σ1 , . . . , σn−1 , subject to the relations [σi , σj ] =1 if |i − j| ≥ 2, andσi σi+1 σi = σi+1 σi σi+1
for 1 ≤ i ≤ n − 2.

• There is an action of the braid group Brn on Gn and cn . If σi is the ith generator
in the presentation above, then its action on (g1 , . . . , gn ) is given byσi (g1 , . . . , gn ) = (g1 , . . . , gi−1 , gi gi+1 gi−1 , gi , . . . , gn ).

26

CHAPTER 3. ROOTS OF UNITY

Figure 3.1: This denotes σ1 ∈ Br4 , where the strands are labeled 1 through 4, frombottom to top, and are considered as going from left to right.
The term braid group comes from making braids with n strands. We show asample braid on four strands in Figure 3.1.
With this denition, we can make a precise statement equating connected components of Hurwitz space to n-tuples of elements of G: we have

π0 HurG,n ∼= Gn / Brn ,π0 HurcG,n ∼= cn / Brn .

The maps π0 HurG,n → Gn / Brn and π0 HurcG,n → cn / Brn send a cover to the n-tupleof local monodromies at the branch points.
The key result is the following, based on unpublished work of Conway and Parker.

Theorem 3.3.3 (Conway-Parker). There exists a constant N , depending on G, c,and n, so that if mi ≥ N for all i, then∼ e
π0 CHurcG,m → Gc.

The map in the theorem has a very concrete, combinatorial description. Let(g1 , . . . , gn ) represent a class in π0 CHurcG,m . Lift g1 , . . . , gn ∈ c to ge1 , . . . , gen ∈ e
c.
Q
ec .
Then the lifting invariant associated to (g1 , . . . , gn ) is the product gei ∈ G
We can give a similar description of the connected components of the space ofbranched covers of P1 . The only dierence is that the product of all the gi 's must be
equal to 1, so we instead get a map from the connected components to H2 (G, c, Z).

Example. Let G = PSL2 F11 ≤ S12 , and let c be the set of elements of order 3, 5,
ec = SL2 F11 , and e
and 11. Then H2 (G, c, Z) = C2 = {±1}, G
c is the set of elements of

3.4. INVARIANTS FOR NUMBER FIELDS

27

order 3, 5, and 11 inside SL2 F11 . Consider two covers given by the following triples
of elements of c: (g1 , g2 , g3 ) and (h1 , h2 , h3 ), where

g1 =

h1 =

!
−1 4
−1 3
!
−1 4
−1 3

g2 =

h2 =

!
−1 1
−1 0
!
−1 3
−4 0

g3 =

h3 =

!
−3 −1
−2 −1
!
−4 −3
0

−3

Then the invariant for the triple (g1 , g2 , g3 ) is 1 ∈ H2 (G, c, Z), whereas the invariant
for the triple (h1 , h2 , h3 ) is −1 ∈ H2 (G, c, Z). Hence (g1 , g2 , g3 ) and (h1 , h2 , h3 ) do notlie in the same connected component of Hurwitz space.

3.4

Invariants for Number Fields

We now introduce the invariant of Ellenberg and Venkatesh as described in [EV10].
This will be an attempt to explain (2.3.1), as follows: Ordinarily, we would expect
the right-hand side of (2.3.1) to be 1/4, but in this case, it is twice as large as we
anticipate. To each surjection ϕ : Cl(K)  C2 × C2 of Z[ζ3 ]-modules, we associate
an invariant z(ϕ) ∈ {0, 1}. We then hope thatE(#{ϕ ∈ SurjZ[ζ3 ] (Cl(K), C2 × C2 ) : z(ϕ) = 0}) =
and

1
4

1
E(#{ϕ ∈ SurjZ[ζ3 ] (Cl(K), C2 × C2 ) : z(ϕ) = 1}) = .
4(For convenience, we'll overuse the notation z a little bit: sometimes, we'll write z(ϕ)
to denote the invariant of a surjection, and sometimes we'll write z(ρ) to denote theinvariant of a eld corresponding to a representation ρ : GK → C2 × C2 .)The motivation for this comes from the case of function elds, which was studied
by Ellenberg, Venkatesh, and Westerland in [EVW09]. In this case, the extensions areparametrized by a Hurwitz space, which may have several connected components. On
each connected component, the number of extensions agrees (asymptotically) with

28

CHAPTER 3. ROOTS OF UNITY

the Cohen-Lenstra predictions, but there is a discrepancy when there are multiple
connected components. This is discussed at the end of [EVW12]. The invariant
used there for classifying connected components of Hurwitz space is essentially the
Conway-Parker invariant of 3.3.
In the number eld case, we have no Hurwitz space to parametrize the extensions,but we are left with a vestige of the connected components, which are given in termsof the Schur multiplier. These vestiges of connected components are parametrized ina manner quite similar to the way connected components were parametrized in 3.3.
We now consider the following scenario, which is a modication of that consideredby Ellenberg and Venkatesh in [EV10]. Let K be a number eld or function eld, let

G be a nite group, and let c = c1 ∪ · · · ∪ cr ⊂ G be a union of conjugacy classes.
Then, we assume that the following conditions hold:Conditions 3.4.1.

1. G has trivial center.

2. c generates G.
3. If n is prime to the order of an element g ∈ c, then g n ∈ c.

Example. These conditions hold if G = A4 and c is the union of the two conjugacy
classes of 3-cycles. They also hold if G = A5 and c either the conjugacy class of
3-cycles or the conjugacy class of a product of two disjoint 2-cycles.

Lemma 3.4.2 (Ellenberg-Venkatesh). Let K be a totally real number eld, let G bea nite group, and let c be a union of conjugacy classes of G, satisfying Conditions
3.4.1 above, and let ρ : GK → G be a homomorphism so that
1. ρ is trivial at all innite places.
2. ρ is tamely ramied, and the image of each inertia group IP in GK is a cyclic
subgroup contained in c ∪ {1}.
Furthermore, we assume that 2H2 (G, c, Z) = 0. Then ρ lifts to an extension ρe : GK →ec which is trivial at all innite places and tamely ramied.
G

3.4. INVARIANTS FOR NUMBER FIELDS

29

Remark 3.4.3. The conclusion of the Lemma is sometimes still valid even when
the hypotheses are not all satised. In particular, if G = C2 × C2 and c consists ofjust the identity, the conclusion still holds, even though c does not generate G. Thisparticular case will show up again shortly.
We may now dene the invariant z(ρ) ∈ H2 (G, c, Z). If H2 (G, c, Z) = 0, setz(ρ) = 0. Otherwise, assume that H2 (G, c, Z) 6= 0. For each nite place v of K , let
pv be the corresponding prime and kv the residue eld at v , and let qv be the size of
kv . Let Iv be the inertia group of GK at v , and x an element π ∈ pv − p2v . We havea map Iv → kv× , given by σ 7→ σ(π)/π mod pv . Let gv be any inverse image of −1
so that gv topologically generates a subgroup of Ivtame of index qv2−1 . Now, each x ∈ cis the image of a unique x∗ ∈ ec.

Denition 3.4.4. The invariant z(ρ) is dened to bez(ρ) =

Yv niteρe(gv )(ρ(gv )∗ )−1 ∈ H2 (G, c, Z).

(3.4.1)This invariant is independent of choice of ρe, gv , and Iv .

Remark 3.4.5. Denition 3.4.4 is heavily inspired by the invariant of Theorem 3.3.3.
The ramied places v of K , i.e., those for which the image of inertia at v is nontrivial,
are analogous to the branch points in the disk. In the Conway-Parker theorem, wedene the invariant to be the product of the lifts of the local monodromies. In
Denition 3.4.4, ρe(gv ) is analogous to the local monodromy, and ρ(gv )∗ is analogous
ec .
to a lift to ec⊂GDenition 3.4.6. If L/K is a Galois extension with group G, we say that all ramication of L/K is of type c if L/K is tamely ramied, and for each prime P of L,
either P is unramied, or else a generator of the inertia group at P is contained in c.
We consider Galois extensions L/K with Galois group G with the following properties:

Conditions 3.4.7.

1. G and c satisfy Conditions 3.4.1 above.

30

CHAPTER 3. ROOTS OF UNITY

2. All ramication of L/K is of type c.
3. K and L are totally real number elds.
In the case where G = A5 and K = Q, if c is the conjugacy class of 3-cycles sothat H2 (G, c, Z) ∼= C2 , we can dene the invariant in more down-to-earth terms. In

ec = Ae5 and eec .
this case, G
c is the conjugacy class of elements of order 3 in GeClaim 3.4.8. If L/Q is the A5 -eld which is the xed eld of the kernel of ρ and L
is the xed eld of the kernel of ρe, then the invariant z(ρ) is the number of primese, modulo 2.
p ≡ 3 (mod 4) with even ramication degree in L

e is unramied above
Proof. We check the contribution to (3.4.1) at each prime. If L/Lev , then the contribution to the product is 1 ∈ {±1}. If v ≡ 3 (mod 4) and L/Lisec , and ρ(gv )∗ has odd order, so theramied above v , then ρe(gv ) has even order in Gecontribution to the product is −1. If v ≡ 1 (mod 4) and L/Lis ramied above v ,then ρe(gv ) has odd order, as does ρ(gv )∗ , so they are equal. Hence in this case, thecontribution to the product is 1. Thus, the product is −1 to the number of primes
e . Since all ramied primes in L have odd
congruent to 3 mod 4 which ramify in L/Leramication degree and all ramied primes in L/Lhave ramication degree 2, theclaim is valid.



e. Suppose we have
Remark 3.4.9. The invariant is independent of the choice of L
e0 . Then Le and Le0 dier by a totally real quadratic twist GQ → C2 unanother lift L
ramied at 2, and in any real quadratic eld, the number of ramied primes congruent
to 3 (mod 4) is even.
In the case where G = A4 and K = Q, we can take c to be the union of two
conjugacy classes consisting of all 3-cycles of G. Then H2 (G, c, Z) ∼= C2 , and we takeec = Ae4 and eec . The invariant is dened
Gc the collection of elements of order 3 in G
just like in the case of G = A5 above: if L/Q is the A4 -eld over Q which is the xede4 -eld L
e which is tamely
eld of a homomorphism ρ : GQ → A4 , we can lift to an A
ramied and totally real. The invariant z(ρ) is again the number of primes p ≡ 3e, modulo 2.
(mod 4) with even ramication degree in L
Much of the value of the invariant rests on our belief in the following conjecture:3.5. COMPUTING THE INVARIANT

31

Conjecture 3.4.10. Assume Conditions 3.4.7 hold. As we vary L by discriminant,z(ρ) is equidistributed over H2 (G, c, Z).

Remark 3.4.11. We can think about A4 invariants in one of two ways. First, ofcourse, they are invariants of A4 elds. But an A4 eld also corresponds to a Galois
cubic eld K together with a Galois-equivariant surjection ϕ : Cl(K)  C2 × C2 :given such a eld K and a surjection ϕ, we construct an unramied C2 × C2 cover
H of K , so that H is Galois over Q (and hence K ), with Gal(H/K) ∼= C2 × C2 andGal(H/Q) ∼= A4 . (This construction is described in section 3.5, and an example isgiven in detail in section 3.6.) Now, let c be the trivial conjugacy class in C2 × C2 .
Although c does not generate C2 ×C2 , the conclusion of Lemma 3.4.2 still holds. In thise 1 and H
e 2 , with Gal(He 1 /K) ∼case, we have H2 (G, c, Z) ∼= C2 , so H lifts to elds H= D8

e 2 /K) ∼and Gal(H= Q8 which are tamely ramied and totally real. Since the Sylow 2e4 is isomorphic to Q8 , H
e 2 is an A
e4 -eld. If, furthermore, all ramication
subgroup of Ae 2 is a lift of H of the type described above. Hence,in H is of type 3-cycle, then H
we can also think of the invariant associated to an A4 eld H as being the invariant
associated to a pair (K, ϕ), where K is a C3 eld an ϕ a surjection from Cl(K) to

C2 × C2 .

3.5

Computing the invariant

In this section, we present an algorithm that takes an A4 eld ramied at one prime
e4 lift of it. We then prove that the invariant associated to an A4
and produces an A
eld is closely related to the class group; this will help us compute tables of invariants
e4 eld in every case.
much more quickly than if we had to construct the A
We are now in a position to calculate the invariant associated to a totally real A4
eld H . Let c be set of all 3-cycles in A4 ; this is a union of two conjugacy classes in

A4 . If H is ramied at exactly one prime, we will prove below that all ramicationin H is of type c, so the Lemma in the previous section tells us that we can lift H toe with Galois group Ae4 ∼a tamely ramied and totally real extension H= SL2 (F3 ). If

H is ramied at more than one prime, all we can say is that the ramication of someprime is of type c.

32

CHAPTER 3. ROOTS OF UNITY

Proposition 3.5.1. If E/Q is an A4 eld ramied at exactly one rational prime,then all ramication is of type 3-cycle.
This follows quickly from the following more general Lemma:

Lemma 3.5.2. If E/Q is a nite Galois extension with Galois group G, then theinertia groups at the ramied nite places of E generate G.
Proof. Let E0 be the intersection of the xed elds of all the inertia groups. Then

E0 /Q is a nite Galois extension unramied at all nite places. Hence E0 = Q, andso the inertia groups generate G.

Proof of Proposition. There are no wildly ramied A4 extensions of Q ramied atexactly one rational prime (see [Jon]), so E must be tamely ramied. Hence, its
ramication type must either be that of 3-cycles, or that of products of two disjoint
2-cycles. The latter case cannot happen by the Lemma, because the products of two
disjoint 2-cycles do not generate A4 .



Now, we shall see how to lift a totally real A4 eld H ramied at exactly onee4 eld H
e . Note that, sincerational prime to a tamely ramied and totally real AeH/Hwill be a quadratic extension, tamely ramied is equivalent to being unramied
above 2.

Algorithm 3.5.3
Input: A quartic polynomial f dening a quartic eld L with Galois closure a totallyreal A4 eld H ramied at exactly one prime.
√e , so that He hasOutput: An element α ∈ L so that L( α) with Galois closure H
e4 , and so that H
e is totally real and tamely ramied.
Galois group A
1.

×2
Let {αi } be a set of representatives of o×
L /oL which includes 1.

2.

Let {Cj }j∈J be the 2-torsion ideal classes of L.

3.

For j ∈ J , let Ij denote an integral ideal in Cj . Let the ideal (1) be the representative of the trivial ideal class.

4.

Each Ij2 is a principal ideal; let βj be a generator for Ij2 .

5.

Let γ1 = 1.

3.5. COMPUTING THE INVARIANT

6.

33

Let p be the rational prime at which L is ramied. Then poL splits as p1 p32 , forsome prime ideals p1 , p2 ⊂ oL . Let J = p1 p2 . Suppose that the order of J in theclass group is r. Let γ2 be a generator for the principal ideal J r .

7.
8.

Let ∆ = {αi βj γk }.

√e4 . Stop once we have found one
For δ ∈ ∆, check if L( δ) has Galois group A

that does, and call this element δ .
√9. If L( δ) is tamely ramied and totally real, let α = δ .
√10. If L( δ) is tamely ramied and totally complex, let q be a rational prime with√q ≡ 3 (mod 4) so that L( δ) is unramied at q . Let α = −qδ .
√11. If L( δ) is wildly ramied and totally real, let q be a rational prime with q ≡ 3√(mod 4) so that L( δ) is unramied at q . Let α = qδ .
√12. If L( δ) is wildly ramied and totally complex, let α = −δ .
13. Return α.

Proof of Algorithm 3.5.3. By Lemma 3.4.2, we know that there is an α ∈ H so that√
e4 . Suppose we have such an α. Let q beH( α) is Galois over Q with Galois group A√
a rational prime dierent from p, and let qoH = q1 · · · qg . In order for H( α) to be
Galois over Q it is necessary and sucient that the class of α in H × /H ×2 is stable
under the action of Gal(H/Q). If the class of α in H × /H ×2 is Gal(H/Q)-stable, thenthe parity of vqi (α) must be the same for all i. If vqi (α) ≡ 1 (mod 2) for all i and theclass of α is Gal(H/Q)-stable, then vqi (α/q) ≡ 0 (mod 2) for all i, and the class of

α/q is still Gal(H/Q)-stable. Hence, we may assume that vq (α) ≡ 0 (mod 2) for all
primes q of H lying over a rational prime dierent from p. Furthermore, the parities
of vpi (α) are equal for all primes pi of H lying over p. Let Ξ be the set of squareclasses of H with even valuation at all primes not lying over p, and with all valuationsat primes over p having the same parity.
e4  A4 . Then A
e4 acts transitively and faithfully
Let B be the kernel of the map A
on a set of 8 objects partitioned into blocks of size 2 so that B xes the blocks. The
e4 /B acts on the blocks in the usual way that A4 acts on 4 objects.
quotient A4 ∼=Ae4 eld is the Galois closure of an octic eld obtained by adjoining theHence any A
square root of some square class in a quartic eld. Hence, we may restrict our listof square classes to check still further by letting ∆ be the set of square classes in Ξ

34

CHAPTER 3. ROOTS OF UNITY

which contain a representative in L. This shows that we can nd a δ in Step 8 so√e4 .
that L( δ) has Galois group A
The remaining steps explain how we can twist by a quadratic character in ore4 be theder to remove wild ramication and ramication at ∞. Let ρe : GQ → A√
Galois representation corresponding to the number eld H( δ). We can nd somequadratic character χ : GQ → C2 so that the representation χeρ is tamely ramied
and unramied at ∞. This completes the proof.

e, we can simply count the number of ramied
Once we have found the desired L
e in order to determine the invariant z(ρ).
primes congruent to 3 (mod 4) in L
Frequently, it is possible to compute the invariant without constructing an explicit
e4 eld. (Still, as a matter of good discipline and for the sake of generality,
lift to an A
it is good to know how to perform the explicit construction.) We recall that Cl2 (K)denotes the Sylow 2-subgroup of Cl(K). We dene variants such as Cl+
2 (K) to mean
the Sylow 2-subgroup of Cl+ (K), and in general, a subscript of 2 in any sort of classgroup will denote the Sylow 2-subgroup of that class group. In the situation at hand,
we have the following characterization of the invariant:

Theorem 3.5.4. Let K be a C3 eld with prime conductor so that Cl2 (K) ∼= C2 ×C2 ,and let H be the everywhere unramied C2 × C2 extension of K , so that H is an A4
eld. Let L ⊂ H be the xed eld of any 3-cycle in Gal(H/Q), so that L is a nonGalois quartic eld over Q with Galois closure H . Suppose furthermore that Cl+
2 (L)is cyclic. Then the invariant associated to H is equal to # Cl(L) (mod 2).

e4 eld H
e containing H so that H
e is totally real and
Proof. By the above, we have an A
e is not unique, any two such He 's diertamely ramied. While the construction of H
e4 be the Galois representation
only by a quadratic twist χ : GQ → C2 . Let ρe : GQ → A
e . Suppose that H/Heassociated to one such H
were ramied at two primes of H above
two distinct primes p1 and p2 of Q, so that p1 ≡ p2 ≡ 3 (mod 4). Let χ : GQ → C2 be√
the quadratic character associated to the number eld Q( p1 p2 ). Then the number
e4 eld which is still tamely
eld associated to the representation χeρ is again an A
ramied, totally real, and contains H , and the corresponding quadratic extension ofe ramies, with the exception of the
H is ramied at exactly the primes at which H

3.5. COMPUTING THE INVARIANT

35

eprimes above p1 and p2 , where it is now unramied. Similarly, if H/H
were ramied
at a prime in H above some rational prime q ≡ 1 (mod 4), then if χ is the quadratic√
character associated to Q( q), then the number eld associated to the representationρeχ is now unramied at the primes above q . Hence, we may assume that there are
no primes congruent to 1 (mod 4) for which the primes in H above p are ramied
e , and there is at most one such prime congruent to 3 (mod 4). If we havein H/He , let χ bea prime q ≡ 3 (mod 4) for which the primes above q are ramied in H/H√
the quadratic character associated to Q( −q). Then the eld associated to χeρ isunramied at the primes above q in H .

e
The above paragraph shows us how to produce a quadratic extension H/Hun∼ee4 . By the argument in the proof oframied at all nite places, so that Gal(H/Q)=A
e descends to a quadratic extension L
e of L, unramied at all nite
Algorithm 3.5.3, He over Q is H
e . Hence, Cl+ (L) is nontrivial. If
places, so that the Galois closure of L2+
Cl2 (L) is cyclic, then there is a unique nontrivial quadratic extension of L unramiede. In this case, Le and hence H
e are toat all nite places, so this extension must be Letally real if and only if Cl2 (L) is nontrivial. If this happens, then H/H
is everywhereeunramied (including at innity), so the invariant is 0. If Cl2 (L) is trivial, then H/H
is ramied only at innity, so we can twist by some character associated to a eld
√e ramied only at p. Hence,Q( −p) for some p ≡ 3 (mod 4) to obtain a totally real H
the invariant in this case is 1. In either case, the invariant is # Cl2 (L) (mod 2).

Remark 3.5.5. The hypothesis that Cl+2 (L) is cyclic holds very frequently. In fact,
there are no exceptions in the 105 elds we tested for inclusion in the data given in
Table 3.1.
In 3.6, we will need to start with a C3 eld K with Cl2 (K) ∼= C2 × C2 andconstruct an A4 eld H containing K so that H/K is everywhere unramied. We
now explain how that is done.

Algorithm 3.5.6
Input: A cubic polynomial f dening a Galois cubic eld K .
Output: A quartic polynomial g so that the Galois closure of g is an A4 eld H
containing K , with H/K everywhere unramied.

36

CHAPTER 3. ROOTS OF UNITY

1.

×2
Let {αi } be a set of representatives of o×
K /oK .

2.

Let {Cj }j∈J be the 2-torsion ideal classes of K .

3.

For j ∈ J , let Ij denote an integral ideal in Cj . Let the ideal (1) be the representative of the trivial ideal class.

4.

Each Ij2 is a principal ideal; let βj be a generator for Ij2 .

5.

Let ∆ = {αi βj }.

6.

√For δ ∈ ∆, let Kδ be the Galois closure over Q of K( δ). If Kδ has Galois group
A4 and is totally real and unramied at 2, let α = δ and stop.

7.
8.

Let x3 − a2 x2 + a1 x − a0 be the minimal polynomial of α over Q.
√
Let b2 = −2a2 , b1 = −8 a0 , b0 = a22 − 4a1 .

9.

Let h(x) = x4 + b2 x2 + b1 x + b0 .

10. (Optional.) Using the LLL algorithm, nd a polynomial g with smaller coecients than h so that g and h generate the same eld; this is implemented in
PARI/GP [The08] as polredabs.
11. Return g .

Proof of Algorithm 3.5.6. We rst show that there is some α ∈ ∆ so that the Galois√
closure H of K( α) has Galois group A4 over Q and so that H/K is everywhereunramied. By class eld theory, we know that there is some such α ∈ K × , so it√
suces to show that if α 6∈ ∆K ×2 , then K( α)/K is ramied somewhere. Observe√that for K( α)/K to be unramied, it is necessary (but not sucient) that α haveeven valuation at all places of K . Those elements of K × which have even valuation
at all places of K are precisely the elements of ∆K ×2 , so this shows that we can ndsuch an α ∈ ∆.
Now we explain the construction of h(x). The A4 eld H is the Galois closure of√a quartic eld L over Q. Since α has degree 3 over Q and is not a square, α has√√√degree 6. Let us call its Galois conjugates ± α, ± β, ± γ . Now, L is generated by
√ √ √
√ √ √
√ √ √
r = α+ β+ γ . The conjugates of r are r2 = α− β− γ , r3 = − α+ β− γ ,√√√and r4 = − α − β + γ , and so we can check explicitly that h as constructed in
Algorithm 3.5.6 is the minimal polynomial of r.

Suppose now that K is a C3 eld ramied at exactly one rational prime p. Then H

3.6. A SAMPLE INVARIANT COMPUTATION

37

as constructed in Algorithm 3.5.6 is also ramied at p and nowhere else. Furthermore,

H is totally real.

3.6

A sample invariant computation

Let us compute the invariant for the smallest A4 eld ramied at one prime. In orderto build this A4 eld, we start with the smallest C3 eld with prime conductor andclass group C2 × C2 . A polynomial generating this eld is p(x) = x3 − x2 − 54x + 169.
Let K denote this eld, and let α be a root of p in K . The unit group isZ2Zo×K = (α − 4) × (α + 4α − 33) × {±1}.

Two ideals whose ideal classes generate the class group are (5, α − 2) and (5, α − 1),and the squares of these ideals are (α2 + 4α − 32) and (α2 + 4α − 35), respectively.
√
To nd the Hilbert class eld of K , it suces to look at K( β), where β is the
product of elements of some subset of {−1, α−4, α2 +4α−33, α2 +4α−32, α2 +4α−35}.
We nd that, if β is one of {γ1 , γ2 , γ3 }, where γ1 = α2 + 4α − 32, γ2 = (α − 4)(α2 +4α − 33)(α2 + 4α − 35) = 12α2 + 48α − 395, and γ3 = (α − 4)(α2 + 4α − 33)(α2 + 4α −√32)(α2 + 4α − 35) = 169α2 + 688α − 5612, then K( β) is an unramied extension of
K . Hence, if β is one of these elements, then the Hilbert class eld H(K) of K is the√
Galois closure of K( β).
Now, we nd an equation for a quartic eld L whose Galois closure is H(K).
√
Take β as in the above paragraph, so that H(K) is the Galois closure of K( β).
Suppose the Galois conjugates of β in K are β1 = β, β2 , β3 . Then the minimal√√√√
polynomial of β over Q has roots ± β1 , ± β2 , and ± β3 . An example of a quartic
√
√
√ √
√
√
polynomial with the same Galois closure has roots β1 + β2 + β3 , β1 − β2 − β3 ,
√
√
√
√
√
√
− β1 + β2 − β3 , and − β1 − β2 + β3 . A polynomial with these roots is

x4 − 34x2 − 40x + 121. Using the PARI/GP function polredabs, we nd that anotherpolynomial that generates the same eld is f (x) = x4 − x3 − 7x2 + 2x + 9. Let L bethe eld Q[x]/(f (x)), and let γ be a root of f in L.

38

CHAPTER 3. ROOTS OF UNITY

Now, we must construct a degree-8 extension of L whose Galois group is isoe4 . First, we must nd generators for the unit group of L. The unit
morphic to A
group of L is isomorphic to Z3 × {±1}, and a basis for the torsion-free part is{γ 2 − 2, γ + 2, γ 2 − 2γ − 4}. The class group of L is trivial, so we get no contribution
from 2-torsion ideal classes. Finally, L is ramied exactly at the prime 163, and (163)
factors as p1 p32 , where p1 = (γ 3 − 4γ − 4) and p2 = (−4γ 3 + 9γ 2 + 16γ − 26). Hence,√p1 p2 = (6γ 3 − 11γ 2 − 23γ + 23). Thus, some eld of the form L( δ), where δ is the
product of elements of some subset of {γ 2 −2, γ +2, γ 2 −2γ −4, 6γ 3 −11γ 2 −23γ +23},√
e4 . We nd that, if δ = γ +2, then the Galois closure of L1 = L( δ)has Galois group A

e4 over Q. Now, L1 is totally real, but it is ramied at 2, and hence
has Galois group A
not tamely ramied. Thus, we need to twist by a character that is ramied at one√
e4 , isprime congruent to 3 (mod 4). In particular, L2 = L( 3δ) has Galois group A
totally real, and is tamely ramied, so it is a lift of the desired form. Now, in order
to compute z(ρ), we need to determine the number of primes congruent to 3 (mod 4)that are ramied to even order. There are two ramied primes of L2 , namely 3 and163. Only 3 is ramied to even order, so z(ρ) = 1.
This computation, and all others in this paper, were done using Sage [S+ 10] andPARI/GP [The08].

3.7

The data

We collected data from the rst 105 C3 elds K ramied at exactly one prime suchthat the Sylow 2-subgroup of the class group is K is isomorphic to C2 × C2 andconstructed the associated A4 elds. Of these, 53891 have invariant 1 and 46109
have invariant 0. So, while we might be forgiven for expecting that the invariant
equidistributes among the two classes, the data seems to exhibit a slight bias that
gradually goes away. Table 3.1 gives incremental data for the invariants.

The rst column denotes the number of elds, the second denotes the number
with invariant 1, and the third denotes the proportion with invariant 1. Hence, we3.7. THE DATA

N
100
200
300
400
500
1000
2000
4000
6000
8000
10000
20000
30000
40000
50000
60000
70000
80000
90000
100000Invariant 1
55
104
160
212
266
536
1063
2183
3279
4372
5456
10862
16267
21638
27064
32400
37768
43176
48578
53891

39

Table 3.1: Invariant Data
Proportion with invariant 1
.5500
.5200
.5333
.5300
.5320
.5360
.5315
.5458
.5465
.5465
.5456
.5431
.5422
.5410
.5413
.5400
.5395
.5397
.5398
.5389

40

CHAPTER 3. ROOTS OF UNITY

suspect, somewhat hesitantly, that the two classes do equidistribute, but that there isa secondary term of slightly lower order that leads to an apparent bias that persists
for a long time. Based on the numerical evidence, and the fact that the number of
cubic elds with absolute value of the discriminant at most x is of the formax + bx5/6 + o(x5/6 )for certain explicitly known constants a and b (see [BST10] and [TT11]), we might
conjecture that the proportion of these C3 elds with invariant 1 among the rst x
by discriminant is

1/2 + cx−1/6 + o(x−1/6 ),where c ≈ 0.27, perhaps with some logarithms thrown in because we are parametrizing
elds in a slightly dierent manner from [BST10] and [TT11]. Still, there is not yet
enough data to be able to distinguish between an error term of the form cx−1/6 and,perhaps, c0 x−1/8 , so this conjecture ought to be taken with more than a grain of salt.

Chapter 4
Background on Bely maps
4.1

Introduction

In this section, we construct explicit maps of smooth projective complex algebraic
curves f : X → Y that are unramied away from a few specied points. Recall thata map f : X → Y of algebraic curves of degree n is said to be unramied at y ∈ Y if

f −1 (y) consists of exactly n distinct points, and that f is said to be unramied if it
is unramied at all y ∈ Y .
Let y ∈ Y be a point, and let x1 , . . . , xr be the distinct preimages of y under f .
(So, if f is unramied at y , then r = n; otherwise r < n.) There exists a neighborhood

U ⊂ Y of y on which f is unramied except perhaps at y . If U is suciently small
in the analytic topology, then the xi 's are in pairwise disjoint components of f −1 (U );write ei for the degree (as a map of topological spaces) of the map from the component
containing xi in f −1 (U ) to U . The ei 's are called the ramication indices.
Ramication indices can be dened more precisely, and more generally, in termsof local rings. As before, let f : X → Y be a degree n map of algebraic curves(now over an arbitrary algebraically closed eld k ), and let y ∈ Y be a point withf −1 (y) = {x1 , . . . , xr }. Fix some xi ∈ f −1 (y). Let A = Oy,Y be the ring of rational
functions on Y regular at y ; similarly, let B = Oxi ,X . Both A and B are local rings,
so they have unique maximal ideals p and P, respectively. The map f induces a map

f ∗ : A → B . Now, since f ∗ is a local homomorphism, there is a unique number ei so41

CHAPTER 4. BACKGROUND ON BELYI MAPS

42

that

f ∗ (p)B = Pei .
This number ei is the ramication index; when k = C, this denition agrees with thepreceding one.
In algebraic number theory and algebraic geometry, we tend to think of ramication as a menace. While there are only nitely many ramied points (in the caseof a map of algebraic curves) or ramied primes (in the case of number elds), it istypically quite dicult to control which primes are ramied. For instance, we might
try to construct number elds of degree n by writing down a random polynomial

f (x) of degree n over Q (which will almost certainly be irreducible, assuming ourrandomized selection is reasonable) and dening a number eld F = Q[x]/(f (x)). In
this case, the ramied primes will be a subset of the primes dividing the discriminantof f . But trying to control the discriminant of f based on its coecients requires us
to solve a very dicult Diophantine equation.
Instead, we look for other methods to control ramication. Here, we are mostlyinterested in the case of algebraic curves, and the story begins with a theorem ofRiemann, which we will state shortly after a brief discussion on monodromy.

Denition 4.1.1. Let f : X → Y be a covering map of degree n of topologicalspaces, and let y ∈ Y be a base point. Let f −1 (y) = {x1 , . . . , xn }. Then we obtain ahomomorphism ρ : π1 (Y, y) → Sn as follows: let γ be a loop in Y based at y . Then,for each 1 ≤ i ≤ n, γ lifts uniquely to a path γ
ei in X with γ
ei (0) = xi . Supposeγei (1) = xj . Then we set ρ([γ])(i) = j . This is independent of the choice of γ in its
homotopy class, and relabeling the points in f −1 (y) is conjugation in Sn . The mapρ, dened up to conjugation in Sn , is called the monodromy representation of f , andthe image of ρ is called the monodromy group of f .
In the case of a branched cover of Riemann surfaces, we also have a notion of local
monodromy.

Denition 4.1.2. Let f : X → Y to be a branched cover of Riemann surfacesof degree n. Let y1 , . . . , yr be the branch points, and pick a base point v ∈ Y

4.1. INTRODUCTION

43

distinct from the branch points. For each branch point yi ∈ Y , x a loop γi ∈π1 (Y − {y1 , . . . , yr }, v) which has winding number 1 around yi and winding number
0 around yj for j 6= i. Let x1 , . . . , xn be the preimages of v under f . Then each

γi induces a permutation σi ∈ Sn , dened as follows. For each j = 1, . . . , n, lift γi
to γ
eij in such a way that γeij (0) = xj . Dene σi (j) by the rule γeij (1) = xσi (j) . The
permutation σi is called the local monodromy around yi with respect to γi .
Note that the local monodromies do depend on the choice of γi .
Riemann's Existence Theorem tells us that if we start with a (possibly punctured)
Riemann surface Y and a homomorphism ρ : π1 (Y, y) → Sn , then we can build an X
and a map f : X → Y of degree n so that the monodromy representation of f is ρ.
More precisely:

Theorem 4.1.3 (Riemann's Existence Theorem). Suppose Y is a connected Riemannsurface (possibly with several punctures). If n ≥ 1 and ρ : π1 (Y ) → Sn is a homomorphism whose image is a transitive permutation group, then there exists a connected
Riemann surface X (possibly with several punctures) and a map f : X → Y realizing

ρ as its monodromy. Furthermore, X and f are unique up to isomorphism.
We can also ll in the punctures to obtain a branched covering X 0 → Y 0 , so that X 0
and Y 0 are compact Riemann surfaces. The relevance of this result for our purposesis that not only do X 0 and Y 0 have the structure of compact Riemann surfaces, butthey actually have the structure of algebraic curves. More precisely:

Theorem 4.1.4. There is an equivalence of categories between the category of compact Riemann surfaces and the category of smooth projective algebraic curves over thecomplex numbers.
Hence, we are free to study either compact Riemann surfaces or smooth projective
complex algebraic curves, whichever happens to be more convenient.
There are many other versions of Riemann's Existence Theorem. The version
quoted in Theorem 4.1.3 can be found in [Don11], 4.2.2. Theorem 4.1.4 is a deep
result. See [Har77], page 441, and the references found there for a discussion of thistheorem.

CHAPTER 4. BACKGROUND ON BELYI MAPS

44

4.2

Connection to Number Theory

In 1979, Bely proved the following remarkable theorem:

Theorem 4.2.1 (Bely, [Bel79]). An algebraic curve C , dened initially over a eld of
characteristic zero, can be dened over Q if and only if there exists a map f : C → P1
which is ramied above three points.
In fact, such a curve C with a map f : C → P1 ramied above three points isn't
just dened over Q; rather, both C and f can be dened over some number eld,which we call a eld of denition for C . However, the eld of denition is not quite
the correct object to study, since a curve and map may be denable over many elds.
To dene the correct object, we need to put a Galois action on the set of equivalenceclasses of Bely pairs.
For i ∈ {0, 1, ∞}, x loops γi around i ∈ P1 so that γ0 γ1 γ∞ is the trivial elementof π1 (P1 − {0, 1, ∞}). Let f : C → P1 and g : D → P1 be two Bely pairs. We
say that (C, f ) and (D, g) are equivalent if f and g have the same degree n, and the0
local monodromies σ0 , σ1 , σ∞ and σ00 , σ10 , σ∞around 0, 1, and ∞ with respect to theγi of f and g , respectively, are simultaneously conjugate, i.e., there is some τ ∈ Sn sothat τ −1 σi τ = σi0 for i = 0, 1, ∞. There is an action of Γ = Gal(Q/Q) on the set ofBely pairs, given by acting on all coecients of the curve and the map; this action
descends to an action on the set of equivalence classes of Bely pairs.

Denition 4.2.2. Let (C, f ) be a Bely pair. Then the eld of moduli of (C, f ) isthe xed eld of the stabilizer of the equivalence class of (C, f ).
The eld of moduli, which can also be described as the intersection of all the eldsof denition for C inside a xed algebraic closure of Q, is a better-behaved object
than a eld of denition. Such elds are of considerable arithmetic interest, thanks
to the following result of Beckmann:

Theorem 4.2.3 (Beckmann, [Bec89]). Let f : X → P1 be a Bely map, and let M bethe eld of moduli for X and f . Let G be the monodromy group of f . Then if p is aprime not dividing the order of G, then X and f have good reduction at p, and p isunramied in M .

4.3. GENERALITIES ON BELYI MAPS

45

The converse fails, although somewhat infrequently: if p divides the order of G,
then p might still be unramied in M . This turns out to be an important point:
if we wish to construct elds which are unramied outside a nite set of primes, we
may wish to allow monodromy groups whose orders have more prime factors, with thehope that some of these extra primes will accidentally turn out to be unramied. Thissituation is analogous to the dierence between eld discriminants and polynomial
discriminants: the primes dividing the eld discriminant are exactly the ramied
primes. The eld discriminant divides the polynomial discriminant for any dening
polynomial of the eld, but there may be extraneous primes dividing the polynomial
discriminant which are nevertheless unramied in the eld.
In fact, Beckmann's Theorem also works for more general branched covers ofcurves over number elds.

Theorem 4.2.4 (Beckmann, [Bec89]). Let Y be an algebraic curve over a number
eld K and f : X → Y a branched cover with monodromy group G. Then if p is aprime of K lying outside the union of
1. the set of primes q dividing the order of G,
2. the set of primes of bad reduction of Y ,3. the set of primes q for which the branch locus becomes singular modulo q,
then X and f have good reduction at p, and p is unramied in the eld of moduli of

X and f .
Roberts in [Rob04] has used this theorem to great eect to construct several eldsof moderately large degree which are unramied outside {2, 3}. The point, as we shallsoon see, is that writing down explicit Bely maps of moderately large degree can bedone relatively quickly with a computer. We shall work out some examples later.

4.3

Generalities on Bely maps

We quickly give just enough background for what we'll need regarding Bely maps; adelightful and more leisurely exposition of this material can be found in [LZ04], andCHAPTER 4. BACKGROUND ON BELYI MAPS

46

Figure 4.1: A sample dessin on P1 . The black dots are points in f −1 (1), and theelements of f −1 (0) are in bijection with the edges between two black dots. We willsoon see how to compute f from looking at the diagram.
reading it comes with this author's highest recommendation.
Following Bely, Grothendieck noted in his famous Esquisse d'un Programme
[Gro97] that, corresponding to an algebraic curve C (dened over the complex numbers initially, and over Q by Bely's Theorem) together with a Bely map f : C → P1
ramied above three points (which we can allow to be 0, 1, and ∞), we can associatea diagram called a dessin d'enfant, or simply a dessin. The dessin of f is dened
to be f −1 ([0, 1]) ⊂ C(C). Typically, we think of a dessin as being a (fat) graph on
the Riemann surface C(C), and it is tempting to draw a picture of it, as we have in
Figure 4.1.

Remark 4.3.1. When we draw such a picture, we typically do not wish to think of
its edges and vertices as consisting of specic points in C(C); rather, we consider it
only as a representative of an isomorphism class of fat graphs on C(C). Hence, thepicture of the actual points in C(C) may look dierent; for example, they may berotated, and some parts of the diagram may be scaled dierently.
We can read o a fair bit of combinatorial data from looking at a dessin. Most
importantly, we can determine the local monodromies around 0, 1, and ∞ of thetarget P1 .
As an example, let us compute the local monodromies around 0, 1, and ∞ of thedessin in Figure 4.1. In gure 4.2, we have drawn white dots to represent the elementsof f −1 (0), and we have arbitrarily labeled the edges and placed the number to the rightof the corresponding edge when we think of the edge as being directed from the black
dot to the white dot. To obtain the local monodromy around 0, consider traversing a

4.3. GENERALITIES ON BELYI MAPS

47

Figure 4.2: The same dessin as in Figure 4.1, but now with the white dots representing
the elements of f −1 (0), and with the edges labeled (in an arbitrary manner).

small counterclockwise circle around each white dot. Each circle will intersect several
edges; the edges it intersects will form a cycle in the cycle decomposition of the local
monodromy. Multiply these cycles together for all such white points to obtain thefull local monodromy around 0. In this case, the local monodromy around 0 is

σ0 = (12)(34)(56)(78).
Similarly, to obtain the local monodromy around 1, we traverse small counterclockwise
circles around each black point to obtain the cycle decomposition. Hence, the local
monodromy around 1 isσ1 = (123)(475)(6)(8).
One way to obtain the local monodromy around ∞ would be to use the relation

σ0 σ1 σ∞ = 1 to see that
σ∞ = (1465873)(2).
Alternatively, we could look at each connected component of the complement of thedessin and travel around the boundary of that component in the clockwise direction
and record the cycle consisting of the edge numbers we encounter in order. (This is
why we have insisted on placing the edge numbers on the right!) This of course also
tells us that

σ∞ = (1465873)(2).

CHAPTER 4. BACKGROUND ON BELYI MAPS

48

From here, we can determine the monodromy group of our dessin: it is the
subgroup of the symmetric group (in this case, S8 ) generated by σ0 and σ1 . In
this case, the monodromy group is the unique simple group of order 168, namely
GL3 (F2 ) ∼= PSL2 (F7 ).
Conversely, if we are given three permutations σ0 , σ1 , σ∞ ∈ Sn with the property
that σ0 σ1 σ∞ = 1, we can nd some algebraic curve C together with a Bely function

f : C → P1 realizing these three permutations. For i ∈ {0, 1, ∞}, let pi denote thecycle type of σi ; we think of pi as a partition of n. The curve C will be connected if
and only if hσ0 , σ1 , σ∞ i ≤ Sn is a transitive permutation group. In the case that C isconnected, the Riemann-Hurwitz formula tells us that if there are a total of r partsin p0 , p1 , p∞ , then C has genus 12 (n + 2 − r).
The following denitions will be useful in the remainder of this section.

Denition 4.3.2. The triple (σ0 , σ1 , σ∞ ) of permutations is called the constellationof the dessin (or the Bely function). The triple (p0 , p1 , p∞ ) of partitions of n is called
the passport.
Note that the monodromy group is the group generated by σ0 , σ1 , and σ∞ .

Denition 4.3.3. Let f : C → X and g : D → X be two branched coverings ofalgebraic curves. We say that f and g are isomorphic if there exists an isomorphism

u : C → D of algebraic curves so that the diagram
C@

u

@@
@@
f @@

X

/D
}
}}
}}g
}
}~

is commutative.

4.4

Computations of Bely maps

In the case when C = P1 , it is frequently possible to compute Bely maps explicitly
in a rather short amount of time. The process is perhaps best illustrated with an
example.

4.4. COMPUTATIONS OF BELYI MAPS

49

Example. Let us suppose that we wish to nd a Bely map f of degree 6 on an
algebraic curve C realizing the passport p0 = (4 + 1 + 1), p1 = (3 + 2 + 1), andp∞ = (4 + 2). By the genus formula given in the introduction, we know that C
has genus 0. Hence, we can write f as a rational function in one variable. Such afunction f has a quadruple zero at one point together with two simple zeros, and also
has a quadruple pole at one point, and a double pole at another. After applying asuitable Möbius transformation, we may assume that the quadruple zero is at 0 andthe quadruple pole is at ∞. Furthermore, if we let g(x) = f (x) − 1, then we knowthat g must have a triple zero at some point, a double zero at another point, and asimple zero at yet another one. We may also assume that the triple zero of g is at
1. Hence, there are eight points of interest to us, of which we have xed three. So,
there are ve points left to locate, as well as a scaling constant.
Putting all this information together tells us that we can write

f (x) = Kx4 (x2 + ax + b)
(x − e)2

andg(x) = f (x) − 1 = K(x − 1)3 (x − c)2 (x − d)(x − e)2

for suitable constants a, b, c, d, e, K . These constraints yield a system of equations

e2 + c2 dK = 0−2e − c2 K − 2cdK − 3c2 dK = 0
1 + 2cK + 3c2 K + dK + 6cdK + 3c2 dK = 0−K − 6cK − 3c2 K − 3dK − 6cdK − c2 dK = 0
3K + 6cK + c2 K + 3dK + 2cdK = bK
−3K − 2cK − dK = aK.
This system of equations is suciently straightforward that Mathematica can solve
it in a matter of seconds, but the resulting algebraic numbers ought not be displayedin full in polite society. However, we can write down the minimal polynomials in

CHAPTER 4. BACKGROUND ON BELYI MAPS

50

concise form, and all the coecients are dened over the quartic eld dened byx4 − 2x3 + 6x2 − 6, which has discriminant −26 · 33 · 5.

Let p = (p0 , p1 , p∞ ) be a passport, and let Σ be the set of constellations (σ0 , σ1 , σ∞ )in Sn . We dene an equivalence relation ∼ on Σ by saying that (σ0 , σ1 , σ∞ ) ∼(τ0 , τ1 , τ∞ ) if there is some ρ ∈ Sn so that ρ−1 σi ρ = τi for all i ∈ {0, 1, ∞}. One canverify that two dessins are isomorphic if and only if they have equivalent constellations, as is done in [LZ04], Proposition 1.2.16.

Example. Let's return to the case of Figure 4.1. To determine the equation here,
we must rst work out the passport. In a neighborhood of each point in f −1 (0),the image of f contains two copies of [0, ε], for ε suciently small. Since there are
four such points, p0 = (2 + 2 + 2 + 2). Similarly, in a neighborhood of each point inf −1 (1), the image of f contains several copies of [1 − ε, 1] for ε suciently small; thenumber of copies is equal to the number of edges emanating from that vertex. Hence

p1 = (3 + 3 + 1 + 1). Finally, the elements of f −1 (∞) are in bijection with the faces ofthe diagram, and the degrees are equal to the number of vertices we reach travelingaround the face (counted with multiplicity). Hence p∞ = (7 + 1).
Since the automorphism group of P1 is 3-transitive, we are allowed three normalization conditions for the function, so we can assume that the pole of order 7 is at∞, the simple pole is at 1, and that the sum of the four double zeros is 0. Hence, we
can write

f (x) = K
andf (x) − 1 = K(x4 + ax2 + bx + c)2x−1(x2 + dx + e)3 (x2 + f x + g).
x−1

4.5. NUMBER FIELDS AND DESSINS

51

Mathematica solves the system of equations easily, giving us

6
a=− ,
7
32
b=− ,
49
159
c=
,
343
4
d=− ,
7
17
e=− ,
49
12
f= ,
7
9
g= ,
7
823543
K=−.
221184(We could, of course, choose dierent normalization conditions, for instance, by
putting the simple pole somewhere else, which would give us dierent Bely maps,
but they would all be isomorphic.)

4.5

Extracting Number Fields with Limited Ramication from Dessins

As we mentioned in section 4.2, it is possible to construct number elds unramied
outside primes dividing |G| and ∞ with Galois group G from a Bely map f : P1 → P1of degree n and monodromy group G. Let us demonstrate this technique with an
example in Figure 4.1, as the example will show us how to perform this construction
in full generality. From the previous section, we know that the Bely map is given by

f (x) = −7(343x4 − 294x2 − 224x + 159)2.
221184(x − 1)

CHAPTER 4. BACKGROUND ON BELYI MAPS

52

Now, consider the polynomial

g(x, t) = −7(343x4 − 294x2 − 224x + 159)2 + t(221184(x − 1)).
The polynomial discriminant of g(x, t) is−2104 324 749 t4 (t + 1)4 .
Hence, if t is chosen so that g(x, t) is irreducible, and Kt is a root eld of g(x, t), thenthe only primes that can ramify (or, equivalently, divide the discriminant of Kt ) are
2, 3, 7, and the primes dividing t and t + 1. Choosing t = −2, for instance, then gives
us a eld ramied only at 2, 3, and 7, with Galois group PGL2 (F7 ).
Note that the monodromy group PSL2 (F7 ) and the Galois group PGL2 (F7 ) are
quite closely related. The monodromy group is in fact the Galois group of a slightlydierent extension, of function elds of curves, and it coincides with the monodromy
group of the corresponding cover of curves. If we delete the ramied points from thesource and target P1 's for f , we obtain a covering space f : X1 → Y1 of punctured
genus-0 curves. However, this cover is not Galois; let X2 be the Galois closure of f .
Now, let X and Y be the obvious compactications of X2 and Y1 , respectively. Thenf induces a map f 0 : X → Y = P1 ramied above three points. Since the monodromy
group of the dessin has order 168, we know that f 0 has degree 168. Now, Q(X)/Q(Y )is a Galois eld extension with Galois group equal to the monodromy group of thedessin. We have a natural map

Gal(Q(X)/Q(Y )) ,→ Gal(Q(X)/Q(Y )),(4.5.1)

which is induced from taking the composita of Q(X) and Q(Y ) with Q. Furthermore,
the group on the left of (4.5.1) is normal in the group on the right.
Now, by Hilbert's Irreducibility Theorem (see [Völ96], Theorem 1.23), the Galois
groups of the specializations of Q(X)/Q(Y ) are equal to Gal(Q(X)/Q(Y )) for all but
a thin set of specializations. Hence, even before computing any Galois groups, we
knew that the Galois groups of g(x, t) would contain PSL2 (F7 ) as a normal subgroup4.5. NUMBER FIELDS AND DESSINS

53√
for almost all values of t. In this case, the quotient is C2 ∼= Gal(Q( −7)/Q). Indeed,√
the Galois group of g(x, t) over Q( −7) is isomorphic to PSL2 (F7 ) for almost all t;√
it is PGL2 (F7 ) over any number eld not containing Q( −7).

Chapter 5
Origamis
5.1

Introduction

In addition to studying Bely maps, it is also interesting to consider branched coversof other algebraic curves. Moving up a genus, then, we can attempt to constructbranched covers of elliptic curves. This is a more dicult problem, but it is still
possible to give concrete covers in at least a few cases.
By the Riemann-Hurwitz formula, an unramied cover of a genus-1 curve mustagain be a genus-1 curve, which means that such a map is simply a composition of
an isogeny of elliptic curves and a translation. Since these maps are well-understood,
we will not be concerned with them here.
However, if we allow one branched point on our elliptic curve, then there arecovers by higher-genus curves. These one-point covers admit a pictorial interpretation
analogous to that of dessins for Bely maps. Such maps are called origamis.
Over C, any elliptic curve E can be written as C/Λ, for some lattice Λ ⊂ C. We
will nd it most helpful to think of E as a fundamental parallelogram for Λ. The
choice of lattice Λ or fundamental parallelogram determines the complex structure
on E . Many of our arguments in this section do not depend on the choice of complex
structure; when this happens, we choose to work with the square lattice Λ = Z[i], andour fundamental parallelogram of choice will be the square S with vertices 0, 1, 1 + i,
and i. (The only reason we prefer this parallelogram is that it is easier to draw than54

5.1. INTRODUCTION

55

Figure 5.1: This diagram represents a genus-2 curve with a degree-3 map to theelliptic curve y 2 = x3 − x. Here we identify opposite edges, meaning that edge a is
identied with edge b, edge c with edge d, edge e with edge f , and edge g with edgeh.
are other parallelograms. It should not generally be assumed that we are interested in
the special properties of the elliptic curve C/Z[i] not enjoyed by other elliptic curves.)
Our elliptic curve will then be the square, with opposite edges identied.
Now, consider a disjoint union of n translates of S , and identify various edges toform an orientable surface X subject to the following requirements:
1. X is connected.
2. Every left edge is identied with a unique right edge, and vice versa.
3. Every top edge is identied with a bottom edge, and vice versa.
If we remove all the vertices of the n squares, the resulting gure carries the structureof a Riemann surface, obtaining its complex structure whose charts are (slightly
enlarged versions of) the original n squares minus the vertices. The resulting Riemann
e is then a compact Riemann surface with several punctures. There is asurface X

e so that its compactication is a compact Riemannunique way of compactifying X
surface; we call this Riemann surface X . Furthermore, X admits a map to the elliptic
curve C/Z[i] by mapping a point in any translate of S to the corresponding pointin S . This map is branched only above the vertex of S . An example can be seen in
Figure 5.1. In this diagram, we have explained the edge identication; in the future,
if there are no markings on the edges, we take this to mean that opposite edges are
identied.

56

CHAPTER 5. ORIGAMIS

Had we chosen to distinguish a dierent elliptic curve with a dierent fundamental
parallelogram P , the corresponding origami would simply consist of of a disjoint unionof n translates of P with similar edge identications.
The origami diagram, though apparently extremely simple, turns out to carry awealth of combinatorial information in readily available form. We will see later how
to read o some of this information at various points throughout this chapter.

5.2

A rst computation of an algebraic origami

In this section, we construct a family of examples of explicit origamis, one for eachgenus g .

Denition 5.2.1. We say that an origami is totally ramied if the preimage of the
branch point is a single point.
The origamis we construct here will all be totally ramied. Later, we constructorigamis which are not totally ramied.

Theorem 5.2.2. For each g ≥ 1 and t 6= 0, −1, the genus-g curve
Ct : y 2 = x(x + 1)(x2g−1 + tj(x)2 ),wherej(x) =

g−1 
X2g − 1i=0

2i(x + 1)i ,admits a degree 2g − 1 map to the elliptic curve

Et : y 2 = x(x + 1)(x + t),
totally ramied above (0, 0) and unramied everywhere else. The map is given by(x, y) 7→ (f1 (x), f2 (x)y), where
f1 (x) =

x2g−1
j(x)2

5.2. A FIRST COMPUTATION OF AN ALGEBRAIC ORIGAMIandf2 (x) =xg−1

Pg−1 2g−1
i=0

2i+1
j(x)3

(x + 1)i57.

Proof. We rst check that (f1 (x), f2 (x)y) actually gives a map from Ct to Et . This
amounts to checking thatf2 (x)2 (x(x + 1)(x2g−1 + tj(x)2 )) = f1 (x)(f1 (x) + 1)(f1 (x) + t)
is a formal identity. This does happen to be the case; hence (f1 (x), f2 (x)y) does denea map from Ct to Et .
Now, we check that the ramication type is as claimed. To do this, we checkthat if f (x, y) = (f1 (x), f2 (x)y) is the map above, and ω = dx
∈ Ω1Et is an invariantydierential on Et , thenf ∗ ω = (2g − 1)

xg−1 dx.
yHence, f is totally ramied at (0, 0) and unramied everywhere else.



Remark 5.2.3. It is worth noting that the map (f1 (x), f2 (x)y) is independent of t
and hence denes a map F : P2C → P2C . If we x an elliptic curve Et in the target P2 ,then F −1 (Et ) is a union of several irreducible components, one of which is Ct . It isalso worth noting that if we take t = 0 or t = −1, then Et is a nodal cubic, and Ct isa singular quintic of arithmetic genus 0. This will be relevant later in this section.
The proof given above thoroughly fails to capture the motivation that went into
the discovery of this result. In fact, the story of nding these examples is much moreinteresting than is the given proof. Therefore, we now discuss how the reader could(and the author did) discover such an example. To do this, we carefully work withthe lowest-degree example: that of a degree-3 origami from a genus-2 curve to an
elliptic curve. Such an origami must necessarily be totally ramied.
In the remainder of this section, as well as in future sections, we perform someeducated guesswork; it will not be clear whether our guesses will turn out to be
successful until we present a proof in the style of that of Theorem 5.2.2.
We will construct a family of genus-2 curves mapping to a family of elliptic curves,parametrized (essentially) by their Legendre form. Hence, for any j -invariant other

58

CHAPTER 5. ORIGAMIS

Figure 5.2: Here, we shrink the dotted edges to a point. The resulting surface has
geometric genus 0.

Figure 5.3: A three-dimensional version of Figure 5.2.
than 0 or 1728, we will actually construct six genus-2 curves mapping to an elliptic
curve with this j -invariant. These six genus-2 curves are generically all distinct.
To do this, we start by constructing a cover C 0 of the nodal cubic

E 0 : y 2 = x3 + x2 ,which we expect to arise as a degeneration of covers of elliptic curves which limit toE 0 . One possibility is that the degenerate origami diagram will look like Figure 5.2,
with the dotted edges collapsed to a point. The curve represented by this origami has
geometric genus 0, since it is a double torus with three pinched loops, as in Figure5.3. Furthermore, since the origamis are totally ramied, the family of covers mustdegenerate to a curve with only one preimage of the branch point in E 0 . Finally,a map C 0 → E 0 can be described as a map from the normalization of C 0 to thenormalization of E 0 .
The next thing to do is to construct explicit equations for C 0 , as well as theirnormalization maps. While in general this is a notoriously dicult problem, it is easyin this case. By the picture, we can see that C 0 has one nodal point and has geometric5.2. A FIRST COMPUTATION OF AN ALGEBRAIC ORIGAMI

59

genus 0; hence it has a Weierstraÿ equation of the form y 2 = (x − a)4 (x − b). We
choose to take a = 0 and b = −1 so that we obtain the curve

C 0 : y 2 = x5 + x4 .
To compute the normalization of E 0 , we note that the map E 0 → P1 given by(x, y) → x + 1 has a square root y/x in C(E 0 ). Letting u = y/x, we have x = u2 − 1and y = u3 − u, so C(E 0 ) = C(u), and the normalization map is P1 → E 0 , given by

u 7→ (u2 − 1, u3 − u). A similar computation shows that the normalization of C 0 is
P1 → C 0 , given by t 7→ (t2 − 1, t(t2 − 1)2 ). Note that, in the normalizations of both
C 0 and E 0 , the preimage of the nodal point is {±1} ⊂ P1 .
Earlier, we worked out that the map on normalizations must have degree 3 andexactly two branch points. Furthermore, the branch points must be {±1}, and theirpreimages must be {±1}. Fortunately, there are very few maps P1 → P1 with only
two branch points: they are simply conjugates of z 7→ z n , where n is the degree ofthe map. In this case, the map on normalizations is

z 7→

z 3 + 3z.
3z 2 + 1

Now, in order to compute the map f : C 0 → E 0 , note that we have the following
commutative diagram:
/ P1

P1


C0

f


/ E0

Furthermore, the vertical maps have near-inverses; the inverse of the vertical arrowon the left is given by (x, y) 7→ y/x2 . Hence, f is the composition of the other three
arrows; putting this together, we havef (x, y) =x3
xy(x + 4),
2(3x + 4) (3x + 4)3

.

We now proceed to prolong f to a map from a family of genus-2 curves to the

60

CHAPTER 5. ORIGAMIS

Legendre family of elliptic curves by means of deformations.
In order to gure out the map from a family of nonsingular genus-2 curves to a
family of elliptic curves, we deform the dening equation for the nodal quintic andfor the map. We let the dening equation of the genus-2 curve be

Ct : y 2 = x5 + (1 + at)x4 + btx3 + ctx2 + dtx,
where a, b, c, d ∈ C[[t]]. The dening equation of the elliptic curve will be

Et : y 2 = x(x + 1)(x + t).
The map will be
(x, y) 7→(x2 + (4 + gt)x)yx3,((3 + et)x + (4 + f t))2 ((3 + et)x + (4 + f t))2.

A priori, a, b, c, d, e, f, g are power series in t; for now, we are only interested in theirconstant terms. Expanding everything out and equating the txi terms for various
values of i gives us a system of linear equations; we then nd thata=9
b = 33c = 40d = 16e=f =g=0
is a solution. Miraculously, these values of a, b, c, d, e, f, g are not merely the constantterms of power series; they are in fact the entire power series. Hence, if we let

Ct : y 2 = x5 + (1 + 9t)x4 + 33tx3 + 40tx2 + 16tx
andEt : y 2 = x(x + 1)(x + t),

5.2. A FIRST COMPUTATION OF AN ALGEBRAIC ORIGAMIthenf (x, y) =x3
xy(x + 4),
2(3x + 4) (3x + 4)361



is a map f : Ct → Et . Indeed, this map is only branched over (0, 0), with its
preimage being (0, 0); we can check this directly, or we can verify that the pullbackof the invariant dierential ω = dx
∈ Ω1Et (which has no zeros or poles) is 3x dx∈ Ω1Ct ,yywhich has a double zero at (0, 0) and no other zeros or poles.
The same method allows us to construct totally ramied origamis in every genus.
For instance, in genus 3, we let

Ct : y 2 = x7 + (1 + 25t)x6 + 225tx5 + 760tx4 + 1200tx3 + 896tx2 + 256tx
andEt : y 2 = x(x + 1)(x + t).
Thenf (x, y) =x5x3 (x2 + 12x + 16)y,(5x2 + 20x + 16)2 (5x2 + 20x + 16)2



is a totally ramied origami f : Ct → Et .
It is worth noticing that, in these cases, we need only change the equation of thegenus-g curve as t varies; in particular, the map does not change. Hence, we havea map f : P2 → P2 so that the inverse images of elliptic curves in a certain family
are all genus-g curves, so that the map is an origami. The proof above explains thisphenomenon.
It would be interesting to see this method generalize to cases where the base neednot be an elliptic curve. In particular, we would like to know to what extent is it
possible to construct branched covers of a curve C in P2 by constructing a suitablemap f : P2 → P2 , chosen so that its branch locus is consistent with the desired
branching properties of the cover of C , and restricting to the map f |D : D → C ,where D is some irreducible component inside f −1 (C) for which f |D : D → C is at.
The author has used this method to construct several examples of branched covers ofhigher-genus curves, but a detailed study of this method may be the topic of future
work.

62

CHAPTER 5. ORIGAMIS

Figure 5.4: This is an origami diagram for a genus-2 curve with a degree-5 map to an
elliptic curve. Above the branch point, we have one triple point and two unramied
points. We call this origami a Swiss cross.

5.3

A non-totally ramied example

The method described in the last section works well for totally ramied origamis,
mainly because we can work out an equation for the degenerate cover. In othercases, it is not clear how the degenerate cover ought to look, in which case we won't
even be able to get started with the previous method. Fortunately, there are otherapproaches. Here we outline one which relies on some numerical magic that looks
daunting in general, but which is possible to make work in at least a few instances.
Here, we work out an equation for origami f : C → E of degree 5 and genus 2,with one triple point (so that the monodromy type is a 3-cycle). An origami diagram
for such a curve can be found in Figure 5.4.

Theorem 5.3.1. The genus-2 curve√
C : − 5y 2 = x(x − 1)(x − α)(x − 2α + 1)(x − 2α),√α = 81 + 36 5,

admits a degree-5 map f to the elliptic curve

E : y 2 = x3 − xbranched only above ∞ with a triple point above ∞. The map is given byf (x, y) = (g(x), h(x)y),5.3. A NON-TOTALLY RAMIFIED EXAMPLE

63

where

x5 + a4 x4 + a3 x3 + a2 x2 + a1 x + a0,
b2 x 2 + b1 x
x6 + c5 x5 + c4 x4 + c3 x3 + c2 x2 + c1 x + c0,
h(x) =
dx2 (x − 2α)2g(x) =

and√a4 = −45(9 + 4 5)√
a3 = 660(161 + 72 5)√
a2 = −3240(2889 + 1292 5)√
a1 = 1980(51841 + 23184 5)√
a0 = −324(930249 + 416020 5)√√
b2 = −100 5(2889 + 1292 5)√√
b1 = 1800 5(51841 + 23184 5)√
c5 = −54(9 + 4 5)√c4 = 1030(161 + 72 5)√
c3 = −7920(2889 + 1292 5)√
c2 = 18780(51841 + 23184 5)√
c1 = 216(930249 + 416020 5)√
c0 = −1944(16692641 + 7465176 5)√√
d = 1000 5(219602 + 98209 5).

Proof. In order to show that f (x, y) gives a map E → C , we need only check thatwhenever (x, y) is a point on C , then (g(x), h(x)y) is a point on E . This amounts tochecking that if√− 5y 2 = x(x − 1)(x − α)(x − 2α + 1)(x − 2α),

64

CHAPTER 5. ORIGAMIS

then

1− √ h(x)2 y 2 = g(x)3 − g(x),5or

h(x)2 x(x − 1)(x − α)(x − 2α + 1)(x − 2α) = g(x)3 − g(x).
This now amounts to verifying a formal identity of rational functions.
on E .
To show that the map is only branched above ∞, pick a dierential ω = dxyThis dierential has no zeros or poles. The pullback f ∗ ω ∈ Ω1C will therefore havezeros or poles exactly at the branch points of f , and the order of each zero or poleis one less than the ramication index at that zero or pole. We compute f ∗ ω to bea constant multiple of dx, which has a double zero at ∞ and no other zeros or poles.
yHence f is branched only above ∞, with a triple point above ∞.

As in the previous section, the proof sheds no light on how one could actually
come up with the equations of E and f without rst being told the answer. We now
describe how to compute it from scratch, once again using some educated guesswork.
All genus-2 curves are hyperelliptic and hence can be written in the form

y 2 = a quintic in x (or a sextic in x, if we prefer, but that will be less convenienthere). Let us decide that f will be branched above the point O = (0 : 1 : 0) of E . Let
us also decide that, if ω = dx
is the invariant dierential on E , then f ∗ ω = a dx
∈ Ω1C ,yyfor some a ∈ C× . Now, f ∗ ω has a double zero at P = (0 : 1 : 0) ∈ C , and no otherzeros or poles. Hence, such a map will be a cover of the desired form.
Now, suppose we had such a map f . Let γ be a loop in C (or an element of

H1 (C, Z), if we prefer). Then we have
Z

∗

Z

f ω=
γ

ω.
f∗ γ

These integrals are known as periods. Now, since H1 (E, Z) is Z2 , the periods of E
form a lattice in C. But since H1 (C, Z) is 4-dimensional, the periods of dx∈ Ω1C areytypically dense in C. In order to nd a curve C that admits such a map, it is necessarythat its periods form a lattice, and in fact a lattice homothetic to that formed by E .

5.3. A NON-TOTALLY RAMIFIED EXAMPLE

65

Figure 5.5: An elliptic curve with a typical basis of H1 shown.

Figure 5.6: A genus 2 curve with a typical basis of H1 shown.

It is now important to be able to determine the degree of the map from understanding its action on loops. That is, suppose we have a map f : C → E , andthat we know what its induced map f∗ : H1 (C, Z) → H1 (E, Z) (or equivalentlyf ∗ : H 1 (E, Z) → H 1 (C, Z)) is. We need to be able to use this information to determine the degree of f ; in other words, the induced map f ∗ : H 2 (E, Z) → H 2 (C, Z).
Let x and y be the duals (in H 1 ) of the loops in E shown in Figure 5.5, oriented
in such a way that x and y intersect positively. Then the fundamental class [E] isx ^ y . Also, let α, β, γ, δ be the duals of the loops in C shown in 5.6, oriented insuch a way that α and β intersect positively, as do γ and δ . Suppose now that

66

CHAPTER 5. ORIGAMISf∗ (α∨ ) = m1 x∨f∗ (β ∨ ) = n1 y ∨f∗ (γ ∨ ) = n2 y ∨f∗ (δ ∨ ) = m2 x∨ .
The induced map on H 2 is then given byf ∗ ([E]) = f ∗ x ^ f ∗ y = (m1 α + m2 δ) ^ (n1 β + n2 γ) = (m1 n1 − m2 n2 )[C].
Hence (the absolute value of) the degree is |m1 n1 − m2 n2 |.
Since we are interested in degree-5 maps, we choose m1 = 1, n1 = 3, n2 = −1,
and m2 = 2. Then we have the following map on H1 :α∨ 7→ x∨ ,β ∨ 7→ 3y ∨ ,γ ∨ 7→ −y ∨ ,(5.3.1)

δ ∨ 7→ 2x∨ .
Now, we specialize to the elliptic curve E : y 2 = x3 − x, which has j -invariant
1728. Computations with this elliptic curve are somewhat nicer than they are with
arbitrary elliptic curves because its period lattice is a square lattice. Hence, we cantake its fundamental periods to be $ and $i, where $ ≈ 5.2441151.
If we write C as y 2 = a quintic in x, let the roots of the quintic be r1 , r2 , r3 , r4 , r5 .
We identify α∨ , β ∨ , γ ∨ , δ ∨ with representatives of their respective homology classes
which pass through two of the Weierstraÿ points of C , i.e., the preimages of r1 , r2 , r3 ,r4 , r5 , and ∞ under the natural hyperelliptic map C → P1 . We can therefore draw
the images of α∨ , β ∨ , γ ∨ , δ ∨ under the hyperelliptic map in P1 ; this picture is given in
Figure 5.7.
Figure 5.7 shows us how the loops in Figure 5.6 are related to periods of dx∈ Ω1C .
y
R r2 dx 1 R dx
The period r1 y is 2 α∨ y , for example. Note that the sum of the three green loops

5.3. A NON-TOTALLY RAMIFIED EXAMPLE

67

Figure 5.7: The data of Figure 5.6 has been rearranged. For example, the integral ofa form over the dual of α is twice the integral from r1 to r2 . This gure and Figure
5.6 have the same symplectic intersection matrix.
is 0 ∈ H1 (C, Z), so we have

Z r2
r1

or

Z r4
r3

since

dx
+
y

dx
=−
y

Z r4

Z r2
r1

Z ∞
r5

r3

dx
+
y

dx
−
y

Z ∞
r5

Z ∞

dx
=2
y

r5

dx
= −3
y

Z r2
r1

dx
= 0,
y
Z r2
r1

dx
y

dx
y

by (5.3.1).
Now, we wish to nd a genus-2 curve whose periods with respect to the dierentialdx
are proportional to $i, 3$, 3$i, and $. That is, if we let C be the curvey

C : y 2 = (x − r1 )(x − r2 )(x − r3 )(x − r4 )(x − r5 ),then we have

Z rj+12
rjdxp(x − r1 )(x − r2 )(x − r3 )(x − r4 )(x − r5 )= Kmj $

for some constant K and for j = 1, 2, 3, 4, where m1 = i, m2 = 3, m3 = 3i, andm4 = 1. Notice now that this holds if we let r1 + r5 = r2 + r4 = 2r3 . Let us also

68

CHAPTER 5. ORIGAMIS

normalize C , by setting r1 = 0 and r2 = 1, which we can do because PGL2 (C) acts
3-transitively on P1C .
Now, it is not a priori clear that we can nd a curve C whose underlying quintichas ve real roots. Still, it is reasonable to begin the search by looking for such a C .
After this simplication, we see that C has the form

C : y 2 = x(x − 1)(x − α)(x − 2α + 1)(x − 2α).
The task of nding α remains. Furthermore, α is an algebraic number.
Using a Newton's method algorithm, we can compute α to high precision, assuming that we can compute periods accurately. Fortunately, this can be done quite
rapidly by using tanh-sinh quadrature, as described in [BB11]. Using this method,
we can compute α to hundreds of digits. The task remaining is to determine which
algebraic number α is, i.e. to nd a nonzero polynomial in Z[x] having α as a root.
This can be done by using continued fractions if α is rational or quadratic, and it canbe done using LLL (see [KZ01]) in higher degrees.
This computation yields a number α, whose rst few digits of α areα ≈ 161.49844718999242907073,
and the continued fraction of this number is roughly[161; 2, 160, 2, 160, 2, 160, 2, 160, 2, 7, . . .].
√A good guess, then, is that α = [161; 2, 160] = 81 + 36 5. Indeed, computation of
more digits helps to increase condence in this guess.

√
It's worth mentioning here that this method does not prove that α = 81 + 36 5

works, since this number might merely be exceptionally close to the true value of α.
However, once we have the equation for the map as well, we will be able to check that
this number is actually correct if we desire to do so. In the opinion of the author,
this should not be seen as a defect of the method any more than having to verify thata clever approach to solving any other problem works by justifying our approach a5.3. A NON-TOTALLY RAMIFIED EXAMPLE

69

posteriori by presenting a complete solution as we eventually do in the above proof.
√
Now, working on the assumption that α = 81 + 36 5, we compute the dening
equation of the map f : C → E . Analytically, E = C/Λ0 , where Λ0 is the period
lattice for C (not for E ; they are only homothetic lattices), and the map f : C → C/Λ0
is given by

Z P
f (P ) =(−∞,0)dxpx(x − 1)(x − α)(x − 2α + 1)(x − 2α).

We can then convert this to a point on E by using the elliptic exponential function,
which is given in terms of the Weierstraÿ ℘-function and its derivative for the lattice

Λ0 : the map C/Λ → E is given by z 7→ (℘(z), ℘0 (z)). If we know the form of the map
f and values of f (P ) at enough points, then we can compose with the map C/Λ → E
and solve a system of linear equations to determine the map, with coecients being
decimal expansions. Then, just as before, we can convert the decimal expansions into
concrete algebraic numbers.
Before we compute the map from C to E , we change variables slightly, letting ourgenus-2 curve be√− 5y 2 = x(x − 1)(x − α)(x − 2α + 1)(x − 2α).
This does not change the isomorphism class of the curve, but it does make the coecients (slightly) cleaner. But since this change of variables also makes the equation of
the curve slightly less pleasant, we will revert to the old form once we have computedit.
The map f : C → E can be written as f (x, y) = (g(x, y), h(x, y)), where g, h :C → P1 . Furthermore, the map x : E → P1 which remembers the x-coordinate has
degree 2, and the map y : E → P1 which remembers the y -coordinate has degree 3.
Hence, g has degree 10, and h has degree 15. Also, from the origami diagram of Figure
5.4, we can see that f respects the hyperelliptic involution: the elliptic involution on
the elliptic curve corresponds to a rotation by π of the fundamental parallelogram;similarly, the hyperelliptic involution on the genus-2 curve corresponds to a rotation

70

CHAPTER 5. ORIGAMIS

by π of the origami diagram. Hence, f (x, −y) = (g(x, y), −h(x, y)), so g(x, y) only
depends on x, and h(x, y)/y only depends on x. Finally, g has two simple poles. This
allows us to determine that we can writeg(x, y) =x5 + a4 x4 + a3 x3 + a2 x2 + a1 x + a0.
b2 x 2 + b1 x + b0

Using the method as outlined above, we nd that√a4 = −45(9 + 4 5)√
a3 = 660(161 + 72 5)√
a2 = −3240(2889 + 1292 5)√
a1 = 1980(51841 + 23184 5)√
a0 = −324(930249 + 416020 5)√√
b2 = −100 5(2889 + 1292 5)√√
b1 = 1800 5(51841 + 23184 5)
b0 = 0.
√Remark 5.3.2. The coecients in the map are surprisingly nice. Let ε = 1+2 5 ; this√is a fundamental unit in Q( 5). Then the coecients in the map are (essentially)
integral multiples of powers of ε. We have√
ε6 = 9 + 4 5,√ε12 = 161 + 72 5√
ε18 = 2889 + 1292 5√
ε24 = 51841 + 23184 5√
ε30 = 930249 + 416020 5√
ε36 = 16692641 + 7465176 5.
Hence, there is some homogeneity in the map: if we replace x by ε6 x, then all the√coecients are either integers or integral multiples of 5. This seems unlikely to be5.4. THICKENING TO A FAMILY

71

a coincidence, but at the moment, we lack an explanation for this phenomenon.

√
To compute h(x, y), we recall that g and h satisfy the relation − 5h2 = g 3 − g .
Hence, we compute g 3 − g . Also, we know that h(x, y) can be written as h1 (x)y , sog(x)3 − g(x)h1 (x) =.
x(x − 1)(x − α)(x − 2α + 1)(x − 2α)2We therefore haveh(x, y) =(x6 + c5 x5 + c4 x4 + c3 x3 + c2 x2 + c1 x + c0 )y,dx2 (x − 2α)2

where

√
c5 = −54(9 + 4 5)√c4 = 1030(161 + 72 5)√
c3 = −7920(2889 + 1292 5)√
c2 = 18780(51841 + 23184 5)√
c1 = 216(930249 + 416020 5)√
c0 = −1944(16692641 + 7465176 5)√√
d = 1000 5(219602 + 98209 5)

With everything now so concrete, we can verify that all our numerics were correct:
that f does in fact dene a map C → E , branched only above ∞, and with a triple
point in C . This completes the construction of a curve C which admits a map to E .

5.4

Thickening to a family

Our goal ought to be larger than simply constructing one curve that admits a map of
the desired type to one elliptic curve; we would like to nd, for each elliptic curve E ,a curve C which admits such a map to E . We would like to perform a degeneration
similar to the one we did in the totally ramied case. Unfortunately, the process here

72

CHAPTER 5. ORIGAMIS

is much more complicated, and we have only the following partial result so far:

Theorem 5.4.1. Let R be the ring C[t]/(t2 ). Let Ct be the genus-2 curve given by
y 2 = x(x − 1)(x − (a0 + a1 t))(x − (b0 + b1 t))(x − (c0 + c1 t)),where√
a0 = 81 + 36 5√
b0 = 161 + 72 5√
c0 = 162 + 72 5

7 47 √+
5
2 30
161 8 √
b1 =
5
+
27
3
161
c1 = 6 + √ ,
12 5
a1 =

and let Et be the elliptic curve over R with j -invariant 1728 given by

y 2 = x3 − x +

t
.
108

Then there is a degree-5 map ft : Ct → Et ramied only above ∞ so that there is one
triple point above ∞, and the other two points are unramied.

Proof. The proof is as in the proof of Theorem 5.3.1. The equation for the map
involves coecients that take much space to write. Therefore, we have relegatedthem to Appendix A.

In fact, we know how to extend the genus-2 curve Ct in Theorem 5.4.1 to C[t]/(t3 ):Conjecture 5.4.2. Let R be the ring C[t]/(t3 ). Let Ct be the genus-2 curve given by
y 2 = x(x − 1)(x − (a0 + a1 t + a2 t2 ))(x − (b0 + b1 t + b2 t2 ))(x − (c0 + c1 t + c2 t2 )),5.4. THICKENING TO A FAMILY

73

where

√
a0 = 81 + 36 5√
b0 = 161 + 72 5√
c0 = 162 + 72 5

7 47 √
5+
2 30
161 8 √
b1 =
+
5
27
3
161
c1 = 6 + √
12 5

371241 √
5+
432 32400
967
961 √
b2 =
+
5
7290 16200
193
971 √
c2 =
+
5,
1440 16200

a1 =

a2 =

and let Et be the elliptic curve over R with j -invariant t2 + 1728 given by

y 2 = x3 − x +

t
.
108

Then there is a degree-5 map ft : Ct → Et ramied only above ∞ so that there is one
triple point above ∞, and the other two points are unramied.
This is only a conjecture, as we have not yet performed the computation analogous
to that of Theorem 5.4.1. The computation is signicantly more tedious, but we
expect to encounter no new diculties in performing it.
The choice of elliptic curve in Theorem 5.4.1 may be surprising, since the elliptic
curve y 2 = x3 − x also has j -invariant 1728. However, we would like to extend thiselliptic curve to a family parametrized by t with j -invariant 1728 + t2 over C[[t]]. One
such elliptic curve over C[[t]] is y 2 = x3 − x + h(t), for some h(t) ∈ C[[t]]. The imageof one such h(t) in C[[t]]/(t2 ) ∼= C[t]/(t2 ) (or even C[t]/(t3 ), as in Conjecture 5.4.2) is

t/108.
We now explain the process by which we found these coecients. We now revert
to a dierent parametrization of the family of elliptic curves and use the elliptic curve

Et : y 2 = x3 − 3

t2 + 1728
t2 + 1728
x
+
2
t2
t2

over C(t), which has j -invariant t2 + 1728.
We look for curves of the form

y 2 = x(x − 1)(x − a(t))(x − b(t))(x − c(t))

74

CHAPTER 5. ORIGAMIS

mapping to the elliptic curvet2 + 1728
t2 + 1728
y =x −3
x+2
.
t2
t2
2

3

√
We let a(t), b(t), and c(t) be power series in t, with a(0) = 81 + 36 5, b(0) =√√161 + 72 5, and c(0) = 162 + 72 5. To do this, we let t0 be some 10−n , for adecently-sized n, so that some consecutive string of digits in a(t0 ) − a(0), b(t0 ) − b(0),and c(t0 ) − c(0) hand us the coecient of t1 in these power series. Letting t = 10−30
is sucient for recognizing the next terms of the power series; the coecients of thelinear terms of a(t), b(t), and c(t) are then seen to be

7 47 √+
5,
2 30
161 8 √
b1 =
+
5,
27
3
161
c1 = 6 + √ .
12 5

a1 =

From here, it is now possible to compute as many terms as desired from the series
expansions of a(t), b(t), and c(t). To demonstrate the algorithm, we compute thequadratic terms. First, it is necessary to compute the power series expansions in t up
to the quadratic term of the roots of the cubic polynomial

x3 − 3

t2 + 1728
t2 + 1728
x
+
2
;
t2
t2

we nd the roots to be

t
t2
72 1
− −
−
+ O(t3 ),t
3 54 34992
2t2
r2 (t) = ++ O(t3 ),3 1749672 1t
t2
r3 (t) =− +−+ O(t3 ).
t
3 54 34992
r1 (t) = −5.4. THICKENING TO A FAMILY

75

Now, we compute the power series expansions of the periods. Letdx

ωt = q.

22x3 − 3 t +1728
x + 2 t +1728
t2
t2

Then the power series expansion for

Z r2 (t)ωt
r1 (t)

isαt1/2 + βt3/2 + O(t5/2 ),and the power series expansion for

Z r3 (t)ωt
r2 (t)

is−αit1/2 + βit3/2 + O(t5/2 ),where

α = 0.30901244621689531973897505786587667 . . . ,β = 0.00098057069999303135023900361703941 . . . .
Call these periods I1 and I2 .

Similarly, we compute power series for the four periods Q1 , Q2 , Q3 , Q4 of the genus2 curvey 2 = x(x − 1)(x − a(t))(x − b(t))(x − c(t))
up to the quadratic term, leaving a2 , b2 , and c2
with respect to the dierential dxy76

CHAPTER 5. ORIGAMIS

terms in the expansion. Now, by the hypotheses we made in (5.3.1), we know that

3Q1 = Q3 ,3Q4 = Q2 ,I1 Q1 = −I2 Q4 .
Substituting in the appropriate power series and truncating as appropriate, we obtaina system of three linear equations in the three variables a2 , b2 , c2 . Solving theseequations gives us

a2 = 0.17129507284189009300135137351845182667 . . . ,b2 = 0.265293223164579431304821281930288591407 . . . ,c2 = 0.268053827540265172544031334191681165555 . . . .
Using a similar method as before, we nd that37
1241 √+
5,
432 32400
967
961 √
b2 =
+
5,
7290 16200
193
971 √
c2 =
+
5.
1440 16200

a2 =While this process is tedious to perform, it is clear how one could carry it out ingeneral to obtain further coecients of the power series expansions of a(t), b(t), andc(t).
The eventual goal of this process is to write a(t), b(t), and c(t) as explicit algebraic
functions of t, not just elements of C[[t]]. Once we have enough terms of the power
series expansions for a(t), b(t), and c(t), we can hope to algebraize the expressions.
However, before we can do that, we need a bound on the degree of a(t), b(t), and c(t)are, as algebraic functions over C(t). We can derive such a bound by looking at the
global structure of the family of genus-2 curves that admit covers of the desired typeto elliptic curves.

5.5. GLOBAL STRUCTURE OF THE FAMILY

77

Figure 5.8: The red part of the gure is a fundamental parallelogram for the same
elliptic curve as that of the square. However, by sliding the top edge of the parallelogram one unit over, we have performed a Dehn twist on the elliptic curve.

5.5

Global structure of the family

What we want to do is to nd an irreducible closed substack Z of the moduli stackof genus-2 curves with a dierential, so that Z maps to the j -line (or A1 ) of elliptic
curves by a map of the given type. In this section, we show how to compute its
degree, which in this case is 9. We can do this directly with the origami diagram.
To do this, consider starting with an elliptic curve represented by a square as its
fundamental parallelogram, and transforming the parallelogram through various otherones until we reach another parallelogram which represents the same elliptic curve(see Figure 5.8). This amounts to performing a Dehn twist, or equivalently an elementof the mapping class group, on the elliptic curve. As we perform such an action on
the elliptic curve, we drag the genus-2 curve above it as well, possibly sending it to agenuinely dierent genus-2 curve, as we see in Figure 5.9. The orientation-preserving
part of the mapping class group of an elliptic curve is PSL2 (Z). Hence, we must see
what PSL2 (Z) does to the genus-2 curves in the ber.
At this point, it is easier to work not with the origami diagram directly, but withan equivalent description. We instead describe the origami as a pair of permutations(g, h) ∈ S5 × S5 , as follows. Number the squares in the origami diagram from 1 to 5.
(There are many ways of doing this; choose one at random, since the choice won't be
relevant at the end.) Let g be the permutation given by moving one square to the rightin the numbered diagram, and let h be the permutation given by moving one square
up. Now, g and h generate a transitive permutation group, and their commutator[g, h] is a 3-cycle. We can easily convert between the origami diagram and the pair

78

CHAPTER 5. ORIGAMIS

Figure 5.9: This is the genus-2 curve we obtain by applying the Dehn twist as shownin Figure 5.8 to the genus-2 curve in Figure 5.4. To turn the gure back into an
origami diagram with squares, we can cut and paste some of the identied edges. We
have labeled the resulting squares by color. This is shown in Figure 5.10.

Figure 5.10: This is a rearrangement of Figure 5.9 back into squares.

5.5. GLOBAL STRUCTURE OF THE FAMILY

79

of permutations, with an ambiguity of drawing an origami diagram with a dierent
choice of edge cuts, and of choosing pairs of permutations which are simultaneously
conjugate: (g, h) and (g 0 , h0 ) are simultaneously conjugate if there is some σ ∈ S5 sothat σ −1 gσ = g 0 and σ −1 hσ = h0 . In this case, we get isomorphic origamis.
Now, we pick two generators of PSL2 (Z) and observe their actions on permutations. We pick the generators

S=

!
1 1
0 1

,

T =

!
1 0
1 1

,

which act on (equivalence classes of) permutations by S(g, h) = (g, gh) and T (g, h) =(hg, h). A computation shows that S and T have action on equivalence classes asgiven in the following diagram:Tt(234), (153)

T

8

T

n
S

S
S

/ (15342), (153)S(234)(15342)
T
S



T

(234), (15324)fT
/ (15342), (13425)
S(15342), (23)(45)
U

T

S

.

(13425), (153)hS/ (23)(45), (13425)
ST/ (15342), (14352)

T
T

Each equivalence class of pairs of permutations in the diagram above corresponds
to one preimage of an (arbitrary) elliptic curve with dierential. Hence, the map

Z → A1 has degree 9.
In fact, it is perhaps worth pausing here to note that we can get a Bely map outof this: if we compactify Z and A1 , we get a map Z → P1 which is ramied only
above three points: 0, 1728, and ∞. A change of variables then gives us a Bely map.
An unfortunate consequence of the fact that Z → A1 has degree 9 is that wecannot hope to get a reasonable parametrization for a family of curves C over the80

CHAPTER 5. ORIGAMIS

Legendre family of elliptic curves y 2 = x(x − 1)(x + 1 − t), since the map from the
Legendre family to A1 has degree 6. Instead, we parametrize our family of elliptic√
curves directly by j -invariant. If j 6= 0, 1728 and we let t = j − 1728, then theelliptic curvey 2 = x3 − 3
has j -invariant j . We use√

t2 + 1728
t2 + 1728
x
+
2
t2
t2

j − 1728 rather than j − 1728 because j − 1728 has a√double zero on the j -stack at j = 1728; hence j − 1728 is a local parameter on thej -stack at j = 1728.

5.6

Extracting number elds from origamis

Just as in the case of Bely maps, we can extract number elds with limited ramication from origamis. Although the examples coming from the origami families in
the previous section are solvable and thus not so exciting from the perspective ofconstructing elds, the techniques are worth presenting here.
Let us start with the degree-3 origami, where the equation of the genus-2 curve is

y 2 = x5 + (1 + 9t)x4 + 33tx3 + 40tx2 + 16tx =: ft (x).
We can factor ft as

ft (x) = x(x + 1)gt (x),gt (x) = x3 + 9tx2 + 24tx + 16t.

The polynomial discriminant of gt is 28 33 t2 (t−1); hence if t = 4 (for example), then thepolynomial discriminant is 212 34 , and the eld discriminant of the eld Q[x]/(g4 (x))
is 34 . If t = 9, then the polynomial discriminant of Q[x]/(g9 (x)) is 211 37 , and the elddiscriminant is 23 35 . Hence, we have constructed some elds ramied only at 2 and3. (In this case, of course, there are much quicker ways to construct cubic elds ofsmall discriminant.)
We can do the same thing with the degree-5 origami, where the equation of the5.6. EXTRACTING NUMBER FIELDS FROM ORIGAMIS

81

genus-3 curve is

y 2 = x(x + 1)gt (x),gt (x) = x5 + 25tx4 + 200tx3 + 560tx2 + 640tx + 256t.

The polynomial discriminant of gt is 232 55 t4 (t − 1)2 . Letting t = 5, we have a eld

Q[x]/(g5 (x)) with polynomial discriminant 236 59 and eld discriminant 24 59 . Its Galois group is F5 = (Z/5Z) o (Z/5Z)× .

There are other number elds that can be extracted from origamis. Let us returnto the degree-3 origami, and consider the elliptic curve E4 corresponding to t = 4,
namely y 2 = x(x + 1)(x + 4). This elliptic curve has conductor 24, and E4 (Q) ∼=C2 × C4 . One of its torsion points is P = (2, 6). The x-coordinates of the preimagesof P under the origami map satisfyx3= 2,(3x + 4)2
or x3 − 2(3x + 4)2 = 0. The number eld corresponding to this cubic polynomial has
discriminant −22 34 . We can also look at the preimage under the map for the degree-5
origami; the x-coordinates of these preimages are roots of the quintic polynomial x5 −2(5x2 + 20x + 16)2 = 0; this polynomial generates a eld of discriminant (−1)2 24 32 55
and Galois group F5 .

We can do something similar with points of innite order. Let t = 10 so that E10
is the elliptic curve y 2 = x(x + 1)(x + 10), which has rank 1 and conductor 25 × 3 × 5.
A generator of E10 (Q) modulo torsion is (−5, 10). The x-coordinates of the preimages
under the degree-3 origami satisfy x3 +5(3x+4)2 = 0, and the corresponding eld has
discriminant 22 33 52 . Under the degree-5 origami, the preimages satisfy x5 + 5(5x2 +20x + 16)2 = 0, and the corresponding eld has discriminant 24 59 and Galois group
F5 .

82

5.7

CHAPTER 5. ORIGAMIS

Computations

The computations described in this work were done using [J+ 10], [S+ 10], [Sym12],[The08], and [Wol07].

Chapter 6
Future possibilities
6.1

The Cohen-Lenstra heuristics

My work brings up many questions that would be interesting to investigate in the
future. Regarding the Cohen-Lenstra heuristics and roots of unity, it would be interesting to gather numerical evidence for equidistribution of the invariant in a widerclass of number elds. This is a tractable problem in theory; we can write down
algorithms to compute the invariants associated to more general number elds. Unfortunately, actually running them takes far longer, as their complexity grows rapidly.
However, it may be possible to do a limited amount of verication, perhaps for nonGalois cubic elds or for A5 elds.
Another possibility to investigate is the dependence on the number of primes
dividing the conductor. My numerical work here is only for the prime conductor
case. While we do not suspect that the behavior of prime-conductor C3 elds diers
from that of arbitrary C3 elds, it would still be nice to have numerical evidence
to suggest that the behaviors are identical. Doing these numerics is well within our
computational ability.
Finally, we noted that we expect the invariant to be asymptotically equidistributed, but that there is a small but denite bias in favor of invariant 1 for relatively
small elds. At the moment, we have no explanation for the origin of this bias; any
explanation for this behavior would be very exciting and would be expected to change83

84

CHAPTER 6. FUTURE POSSIBILITIES

the way we think about the Cohen-Lenstra heuristics.
Finally, a proof of any nontrivial case of the Cohen-Lenstra heuristics in the number eld case would be most welcome. At the moment, this seems to be outside of
the realm of possibility, but we can hope that someone will have a new idea that will
open up the Cohen-Lenstra heuristics to attack.

6.2

Bely maps and origamis

We would like to be able to construct many more explicit examples of origamis. At the
moment, we can construct many examples of totally ramied origamis, but the nontotally ramied case is much more dicult. We would like to complete the family of
3-1-1 origamis which was started in this work; I am condent that I can compute theentire family, but it is quite a time-consuming process that I hope to nish eventually.
Explicit examples of higher-degree families would also be interesting.
Continuing on with branched covers of curves, perhaps the next place to go after
origamis is to unramied covers of higher-genus curves. We expect that some of thetechniques used here will be applicable in this case as well, and indeed, we have some
explicit examples written down.
Motivating this desire for explicit covers of curves is the connection to number
elds with limited ramication. Our totally ramied origamis do provide us with
examples of number elds with limited ramication, but these elds are all solvableand can thus probably be constructed in a less involved manner through class eldtheory. By contrast, the non-totally ramied origamis generally give us non-solvableelds, and we would be more excited to nd these and to understand how many ofthem exist.

Appendix A
Numbers involved in the Proof of
Theorem 5.4.1
In this appendix, we write down the map needed to prove Theorem 5.4.1. The reader
is strongly advised not to look at the numbers in this appendix, but only to beappeased by their presence.
The map from Ct to Et in Theorem 5.4.1 is of the form (x, y) 7→ (f1 (x), f2 (x)y),where

f1 (x) =andf2 (x) =x5 + a4 x4 + a3 x3 + a2 x2 + a1 x + a0
(1 + bt)
b2 x 2 + b1 x

x6 + c5 x5 + c4 x4 + c3 x3 + c2 x2 + c1 x + c0
(1 + dt).
(b2 x2 + b1 x)2Now, the ai 's, bi 's, and ci 's are elements of C[t]/(t2 ), and the constant terms of theai , bi , ci are the same as in Theorem 5.3.1. We denote the linear term of ai by a1i , andsimilarly for the bi and ci . Then we havea1
0 = {−(75(751878161950148022139896155348384918364792135486613076100908884583524065631543785049910794356809061528
4438749279449494854184492327443389410559288599134586937069215555073711077655615884457345219000212841218820709629783929
6872671781194389839478454142449271932087742562297108073974426325192615591431170025801913083643086339687596310037045656
0288503552959787545959172905309042899060560905411008281739575836795460157192149864055430087593490378570977327795652261
1695377725463306452665385118495564945961797338732289926957547911626886256897827001365547985759505616272926349568891355
8562831223559141089440466515036443528176199890910759368501343282682974461750393173101202764688799219852845919020120080
7070185233159750780123185160819091646400936985950392407039244338175991662534595625395488583292423822619883881864503470
4954809459846998334733454302981147329668664653528406390265197049742205651078526824023577363871250984665883310179485561

85

86

APPENDIX A. LONG NUMBERS

7499894538883789608125217274047342725866002980594018747466851220571169231727976437786057542886253388211175777184532974
0097807725744660482321882173588171354627553777957698845085607342183893688251692953969941520950882052522343896268489645
1843528101446517675707267415042455599019883283339558541688065427183997115570834690403560066553996743620423989810568650
93478267549010607624458865335641803809983523615425661564050572655797463969035632765319003568724422092246113150247680847
60245352529184009447393680250033091706757429330465364557780338293921481286094759371532055008900514743998856986885030941
91475036691348461406451701323929971265805322391678938406092585700594422238173587726371003619257477202060605163720647257
76321389183300261634559215968402380498297428289627199648829558956696920465788418722155161944161599165876481822800747683
04020711819357463997330754842644323228422117483437665868762375270 + 336250136183625364202932160036862298944250263111188
95354530161837124127112469529996680342505275948543518811289395593209080591347765543281735079959677891726256451843017645
44786974635366237556039880986259234296216540298959003914463639472675931963841765539919351072818271291720579899402868720
07433971272862866865900471687318469643201220648321523457304086088001872019027740855289211437855748375833807999314921617
17193374593913064478308254572417837581007087268308950138415961877016302751381909072882344823695322999477252363206261850
26247949135607176493794676658613872571174090306361814691030399638992375793380108144503305416893643627431950240834063620
68358807581122398650274649312831942350751639085398081039387709655922856255684350864345977353244903063755118911122913299
93006519172805971020006171852575134596426734391094280961922529068355495108552724847030102611838698112721849097818998934
49999874502906926513589141996822031461363635368353885826431702138645244705915921870748815699420531951961614615186632437
11249856028291720168364919788551870919506160681506116765339286901921118998758934774152278639468018988352289177955027606
28671673866488792263640321463650331414478557384782644255774128699054673355535700945915666816481930947948208850471650231
01355757873533048945333213963234952446045866962999944553952343159900675939115882624753622737884652084804723067314073844
67276229998054568549950770518453645310850127809054945639149223381700113829153303352673375793838508344445072860728264210
75076406161903002304686491289129138556094627141785903356455082815542637128033625018254825936090736721896773663169299213
99237242462906806407927113647134132673254750143923471187924868354593280681537541951832304171107774810606839838236872448√
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1 = {(100(225208401581761492694498913218691156073311728330384347171411473194627149232103455205124317623659911132
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87

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2 = {−(25(803684790594336842600138578401913118694951524383141666061859691754580235847067331844147519594894293153
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50254592727910459468279563180496685541155488983181081214600976154868605238983414701670948046074470971894699508359139326

88

APPENDIX A. LONG NUMBERS

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4 = {−(510897194771571248996315453159377346622421912466581513388130046309267707911908821588275973732833185671954
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90

APPENDIX A. LONG NUMBERS

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c1
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92

APPENDIX A. LONG NUMBERS

87169210913858487594050495244392254463208946366066752089983104396204097538273420942299705187130063522641821868436199995
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3289 5)}c1
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c1
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8548714541533805254933173247760611693109001275123987079136397504400704444314578127860577451315102970345624278193214739
1450208276777565520865092612992810686025004722876211443712035188982202598432876551798630602453172113746425887913365543
9331790211937663219884779179641263660789328243953775337900430795571772387994502380538147022211801266431028270866659650
1713806028523041154216877312820133922828562688279645030348345487440246703466125013753223544383290440874612176009139780
026063745757556852232772173874297719582184824048787660 + 68235467807881397185690518715660657031677603780500738668731168
1936465298862454849693950244248084644497936199789305177073297812045500998045769894620640661627348089933567215414670590
78567427273638481092709884273212175599667637207693775359135940234858438843579140251039308694641506986392674887735941315
08067505870931581228131732452222498285672075852715481627753058538650562480006696878877970345774400279779490901417897848
65848304537759629043753495840956250277202489641180358302779068383102712458454995495748583602706581467624581174626520482
73930186154940242340343522791542010726317092196166644877413670950631713870104724239414441029863014822580135108308041338
12706577690260426145839350550909460590766396284208993929812361533965006045893382474939843450082701817702901025272759622
99768002380238503920831498759113921878216251280270278924010702779780672775055235273795000322932685657318467013216714325

94

APPENDIX A. LONG NUMBERS

88134424977758930231314947060928861258445829738427153194911211329400400373181294952726242484029389117606046675554190390
61655446109076300726353563715675860220799499106493897333975807609975760732652213553633542680586292481648227165891278065√
28732384710343943939191607508423005607215985382043462118004381903746228706283889055316771 5)/(6116939332376492591759064
42352645692834444239759510254364165936336953792043114754808641577242840108183063803666802529045437992204918863523579011
23794706031912405497322674862222407606605134410795620563530871607518962466781355924136372689563507241427576556716358292
60699033515350043056577067440822069711572095802733013540252738084001383225258154542024612996940029817256456221477328517
77967195996965341644090026768140366814877776076104181881726090965217318313599425195452790537245921936822863292278417800
55205390095529844878405244417904047430252935611151940201919561967326726788687777840424483191571138428883957319202637485
02646793726632403545472567590535073294379017791890522093119750574342684130210175159450587478910018952393071499815206450
95401191538665575463236045579891271759318971102998066191687066024358373723016075152432923895708474270123303388601719089
90275313231918908394249482856781074179175785373863445348342377275494058709312789902924023252528435137229108105122897074
29690482184517980339807783458081392343249600397409105042508744093888122690820055025598275086301295883914741199611468012
85026810130341976365418408671192545554625442846904458147848302491090799385476297288868260287007198022375594193572384137
72 + 273557843228720352877839175098882473227266165385248225714694351670921656526037755228508314374566902157870541651901
13336349984016060926664411822547053260847357182694180616873517960543144285794892873181599685296522156972637466431136732
60044859500286809690889649898971027139567128727611830381427471271973871398440229552297780458910507831652519792752615130
22323570484680575565434009989456005149214660602157420941318648914320794165398996880671676586207976560039608437268117190
15855387884077441786540952522653213504047600245052151432954543260340957614143571722353060893268210685713146273567235045
36671566626405354402040827983976869215319646092106177148471112384630373853378734364368336933809289248169474841780432253
37741210737524397199068841600805948519815779210689214709292843673823010809583510061263894811661189688501200370499825168
50875941674570924246090243151395889248357337983957040170601666137459321322502715118714981150104656659862107536265789522
51564209925032925958586741472072403534854298767760245269472112721349003333996563747622839931441466484216376109987489269
00018024581688924964526313635728714620065788381716594522717314349833982639314082080284258050352509399991662289695340444√
64870346819395181664493129788208 5)}c1
5 = {−(758964876433740262031460718319859215532806936026666243479301424618139779789053771319823857971460849033722
7303469247803183379533688098212646853730125551322679328395246062099625244708457568617129719141086784343588262217852289
7297869762035152779032553880613832119815815230431431218782712005722015761262575275212200940875742021510280033018127667
3923853537522494631331655073577197857792405969483804590299739868358277726893297291904519201337402057425703184280722763
2042754581975260952073722548795190712235290862725467187735902744707923360115685703167815327175306351737822078028988660
5093297074173437070632883064709413805982566015840695201221740850325200430469586353881976542555314869945854491581862852
4247315864784444071075891715766590453923895064773212000408573964674040264595461852587731671951709864964841467429827075
1697545991949790118605540823467372097879311163129115156059291557417351001388877589645013976741732200460042856317483234
41893046596823697908474527906538970416125925463722993946862 + 339419411248114277488436666280138353898217645764557330409
3521166181747521702683802540814082637670199525324829508360942576678735243080765424853565615113202643918754864888493640
3754872211849907202021726862518732022020118872643630622861933678488880582056385815154136363599420052251008588147704345
7912436316802351796598762547909781066389614831634312439891966269973801990459707460746822365655081654847004590244840847
46622740637967752747398966025161559547876415018596363418195179445725861516452053181968475114350725563551779669186184123
24458007441454552040801656552085571429710189967358176561025579558147228549535458777461407390355983732071619489985932631
65013640184091092948456102977712371713800228131172720845517973341162889361725874092723092082522596587777542106686390462
84425987882108097864171069522977879462834546906318213393442911048390486077419004007476025281111606801873045523974694163√
7881991669394864920343485299296611762251585586991009385279355878748208928367234642912430901494139027963 5)/(20866808530
18132339494135112080450623464259875759365604406691205350859739423504084403377537639693147527018470615672569385755505842
80571001731305306837206576856053955754465595542566262891397045903469073441959871573425679255265291280366362034470273488
23191871137782100145071699999545519343284517625815168031152845100939083963609804253972707653720268600410259190202458745
58968089301322294231010403754851667653457684304641540557207140119418955033659420684559651454319620146469939006962774969
60301297439021030389125572641823687651049895187016142359849478464166787450837104472087937897447159110704932028253784998
51844295969164053980132501431042328745194036213918679168625712852677601483464279246352532658687607499060850802048789135
01771858570363734741085016452877238642898841433637623637984292305504882208069936462915381057339427760425051452343821464
16557221016552281153290321672957124524938546357899116868457786077143987069769754518201698328015552769147920247596328619
95407048558621093338428180 + 933192046939158222944739161595736453385817056715983921339188037014303527662572565684395305
59367482418514144158870826529244674018849627850286866855858667835250950983031178581484705913729213764083618847252436229
45403267167348583445327651289032648780761000479001547594663921932358191696032779823407345763042815343633401967322108708
30620609746103760670556618405524275723397272406529226092577577562978303396017048840500029969284657100223111364875703663
97216681475098416233626282368364406427877925512010740917461317812296661150701683453721583413558146557293103259434025662
85153690011485004278407300406633531497308093058555950158262971444560317164092817708966989293329998580582213399414389573
42180680930716021226515583986816313375264502953585125756948815717691062823721534957435182995246471062611670071257659037
57490132136537682950462170469981348451483166709674202399307690041584198635816696002948967626623829325193890450610109445√
44955400312574402981124481461434675538535934767356743682779297203 5)}

d = {−(3764319823697395338782061174654769962354702143320794389501549491665992010269267493962197924404400906079826
5927219231767307032288562417212254002545543245931439906078865795054438368087130519338651804695358877773497228165749151

95

3568147399296078922306459972290915327791236855066380509744069209930350224827914950436833573512836340978806761463449978
332330509577172489 + 16834550029674799434464777079204745420680874908687501789605775660910144249522120161450582966787127
0457610141346106322263800705595386801943841618053503043031786610415437041613174325882611791082042601849576372897753754
1724118741790334499226344658546378106443648676306206500773460569396276749840467492050819410568828332262915716166293909√
04814440091816507713977468 5)/(48(469078659588690679450273363325955145969894222677870263195000640775808917464190847283
6648624540535843280317513093814840962529407992264502207230329898775014241625292150786170517104729142673096155614415916
1225699473304608704066693557260596972517329502033178112487553192152152962367401861175675099217670388338823056024926393
903172187361447650237485132655585536280 + 20977835392695918010899641348343881983056947251214592630239410317363959337916
7122758615386065354063196411065276044137816187940628815102260278680288312731938593387891807226359666101895109188378103
8559077154165967545146918613853265759521202098366538353773094059750871869405050344890406938969221703934959573279384549√
7089695636476673035074761003154644431915003041 5))}

Bibliography[BB11]

D. H. Bailey and J. M. Borwein. High-precision numerical integration:
progress and challenges. J. Symbolic Comput., 46(7):741754, 2011.

[Bec89] Sybilla Beckmann. Ramied primes in the eld of moduli of branched coverings of curves. J. Algebra, 125(1):236255, 1989.
[Bel79] G. V. Bely. Galois extensions of a maximal cyclotomic eld. Izv. Akad.

Nauk SSSR Ser. Mat., 43(2):267276, 479, 1979.
[Bil95]Patrick Billingsley. Probability and measure. Wiley Series in Probability andMathematical Statistics. John Wiley & Sons Inc., New York, third edition,
1995. A Wiley-Interscience Publication.

[BST10] Manjul Bhargava, Arul Shankar, and Jacob Tsimerman. On the DavenportHeilbronn theorem and second order terms. 2010.
[CF86]

J. W. S. Cassels and A. Fröhlich, editors. Algebraic number theory, London,
1986. Academic Press Inc. [Harcourt Brace Jovanovich Publishers]. Reprintof the 1967 original.

[Chi09] Nancy Childress. Class eld theory. Universitext. Springer, New York, 2009.
[CL84]

H. Cohen and H. W. Lenstra, Jr. Heuristics on class groups of number elds.
In Number theory, Noordwijkerhout 1983 (Noordwijkerhout, 1983), volume1068 of Lecture Notes in Math., pages 3362. Springer, Berlin, 1984.

[CM87] H. Cohen and J. Martinet. Class groups of number elds: numerical heuristics. Math. Comp., 48(177):123137, 1987.
96

BIBLIOGRAPHY

97[CM90] Henri Cohen and Jacques Martinet. Étude heuristique des groupes de classes
des corps de nombres. J. Reine Angew. Math., 404:3976, 1990.
[Dem09] Lassina Dembélé. A non-solvable Galois extension of Q ramied at 2 only.

C. R. Math. Acad. Sci. Paris, 347(3-4):111116, 2009.
[DGV11] Lassina Dembélé, Matthew Greenberg, and John Voight. Nonsolvable number elds ramied only at 3 and 5. Compos. Math., 147(3):716734, 2011.
[Die12] Luis V. Dieulefait. A nonsolvable extension of Q unramied outside 7. Com-pos. Math., 148(3):669674, 2012.
[Don11] Simon Donaldson. Riemann surfaces, volume 22 of Oxford Graduate Texts

in Mathematics. Oxford University Press, Oxford, 2011.
[EV10] Jordan S. Ellenberg and Akshay Venkatesh. Statistics of number elds andfunction elds. Proceedings of the ICM, 2010. To appear.
[EVW09] Jordan S. Ellenberg, Akshay Venkatesh, and Craig Westerland. Homological stability for Hurwitz spaces and the Cohen-Lenstra conjecture overfunction elds. 2009.
[EVW12] Jordan S. Ellenberg, Akshay Venkatesh, and Craig Westerland. Homological stability for Hurwitz spaces and the Cohen-Lenstra conjecture overfunction elds, II. 2012.
[Frö83] Albrecht Fröhlich. Central extensions, Galois groups, and ideal class groupsof number elds, volume 24 of Contemporary Mathematics. American Mathematical Society, Providence, RI, 1983.
[FT93]

A. Fröhlich and M. J. Taylor. Algebraic number theory, volume 27 of Cam-bridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1993.
[FW89] Eduardo Friedman and Lawrence C. Washington. On the distribution ofdivisor class groups of curves over a nite eld. In Théorie des nombres(Quebec, PQ, 1987), pages 227239. de Gruyter, Berlin, 1989.

98

BIBLIOGRAPHY[Ger89] Frank Gerth, III. The 4-class ranks of quadratic extensions of certain imaginary quadratic elds. Illinois J. Math., 33(1):132142, 1989.
[Gro97] Alexandre Grothendieck. Esquisse d'un programme. In Geometric Galois

actions, 1, volume 242 of London Math. Soc. Lecture Note Ser., pages 548.
Cambridge Univ. Press, Cambridge, 1997. With an English translation on
pp. 243283.
[Gro98] Benedict H. Gross. Modular forms (mod p) and Galois representations.

Internat. Math. Res. Notices, (16):865875, 1998.
[Har77] Robin Hartshorne. Algebraic geometry. Springer-Verlag, New York, 1977.
Graduate Texts in Mathematics, No. 52.
[Har94] David Harbater. Galois groups with prescribed ramication. In Arithmetic

geometry (Tempe, AZ, 1993), volume 174 of Contemp. Math., pages 3560.
Amer. Math. Soc., Providence, RI, 1994.
[J+ 10]Fredrik Johansson et al.

precisionoating-point

mpmath:arithmetica Python library for arbitrary(version0.14),

February2010.

http://code.google.com/p/mpmath/.
[Jon]John Jones. Number elds. http://hobbes.la.asu.edu/NFDB/.

[Kar87] Gregory Karpilovsky. The Schur multiplier, volume 2 of London Mathemati-

cal Society Monographs. New Series. The Clarendon Press Oxford University
Press, New York, 1987.
[KZ01]

Maxim Kontsevich and Don Zagier. Periods. In Mathematics unlimited

2001 and beyond, pages 771808. Springer, Berlin, 2001.
[LZ04]

Sergei K. Lando and Alexander K. Zvonkin. Graphs on surfaces and their ap-

plications, volume 141 of Encyclopaedia of Mathematical Sciences. SpringerVerlag, Berlin, 2004. With an appendix by Don B. Zagier, Low-Dimensional
Topology, II.

BIBLIOGRAPHY99[MZM86] B. Heinrich Matzat and Andreas Zeh-Marschke. Realisierung der Mathieugruppen M11 und M12 als Galoisgruppen über Q. J. Number Theory,23(2):195202, 1986.
[Rob04] David P. Roberts. An ABC construction of number elds. In Number theory,
volume 36 of CRM Proc. Lecture Notes, pages 237267. Amer. Math. Soc.,
Providence, RI, 2004.
[Rob11] David P. Roberts. Nonsolvable polynomials with eld discriminant 5A . Int.

J. Number Theory, 7(2):289322, 2011.
[S+ 10]W. A. Stein et al. Sage Mathematics Software (Version 4.5.1). The Sage
Development Team, 2010. http://www.sagemath.org.

[af54]I. R. afarevi£. On the problem of imbedding elds. Izv. Akad. Nauk SSSR.

Ser. Mat., 18:389418, 1954.
[Ser73]Jean-Pierre Serre.

Congruences et formes modulaires [d'après H. P. F.

Swinnerton-Dyer]. In Séminaire Bourbaki, 24e année (1971/1972), Exp.

No. 416, pages 319338. Lecture Notes in Math., Vol. 317. Springer, Berlin,
1973.
[Shi74]

Kuang-yen Shih. On the construction of Galois extensions of function elds
and number elds. Math. Ann., 207:99120, 1974.

[Sym12] SymPy Development Team. SymPy: Python library for symbolic mathemat-

ics, 2012.
[The08] The PARI Group, Bordeaux. PARI/GP, version 2.3.3, 2008. available
from http://pari.math.u-bordeaux.fr/.
[Tho84] John G. Thompson. Some nite groups which appear as Gal L/K , where

K ⊆ Q(µn ). J. Algebra, 89(2):437499, 1984.
[TT11] Takashi Taniguchi and Frank Thorne. Secondary terms in counting functions
for cubic elds. 2011.

100

BIBLIOGRAPHY[Völ96] Helmut Völklein. Groups as Galois groups, volume 53 of Cambridge Studies

in Advanced Mathematics. Cambridge University Press, Cambridge, 1996.
An introduction.
[Was97] Lawrence C. Washington. Introduction to cyclotomic elds, volume 83 of

Graduate Texts in Mathematics. Springer-Verlag, New York, second edition,
1997.
[Wol07] Wolfram Research, Inc., Champaign, Illinois. Mathematica, Version 6.0.1.0,
2007.

BIBLIOGRAPHY

101

I certify that I have read this dissertation and that, in my opinion, it
is fully adequate in scope and quality as a dissertation for the degreeof Doctor of Philosophy.

(Akshay Venkatesh)Principal Adviser

I certify that I have read this dissertation and that, in my opinion, it
is fully adequate in scope and quality as a dissertation for the degreeof Doctor of Philosophy.

(Brian Conrad)

I certify that I have read this dissertation and that, in my opinion, it
is fully adequate in scope and quality as a dissertation for the degreeof Doctor of Philosophy.

(Persi Diaconis)

Approved for the University Committee on Graduate Studies