Arboreal Representations, Sectional Monodromy Groups

Home , Monodromy, Splitting field

Arboreal representations, sectional monodromy groups, and abelian varieties over finite fields by Borys Kadets Submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2020 ○c Massachusetts Institute of Technology 2020. All rights reserved.

Author...... Department of Mathematics March 31, 2020

Certified by...... Bjorn Poonen Distinguished Professor in Science Thesis Supervisor

Accepted by ...... Davesh Maulik Graduate Co-Chair, Department of Mathematics 2 Arboreal representations, sectional monodromy groups, and abelian varieties over finite fields by Borys Kadets

Submitted to the Department of Mathematics on March 31, 2020, in partial fulfillment of the requirements for the degree of Doctor of Philosophy

Abstract

This thesis consists of three independent parts. The first part studies arboreal representations of Galois groups – an arithmetic dynamics analogue of Tate modules – and proves some large image results, in particular confirming a conjecture of Odoni. Given a field 퐾, a separable polynomial 푓 ∈ 퐾[푥], and an element 푡 ∈ 퐾, the full − ⋃︀ ∘−푘 backward orbit 풪 (푡) := 푘 푓 (푡) has a natural action of the Galois group Gal퐾 . For a fixed 푓 ∈ 퐾[푥] with deg 푓 = 푑 and for most choices of 푡, the orbit 풪−(푡) has − the structure of complete rooted 푑-ary tree 푇∞. The Galois action on 풪 (푡) thus defines a homomorphism 휑 = 휑푓,푡 : Gal퐾 → Aut 푇∞. The map 휑푓,푡 is the arboreal representation attached to 푓 and 푡. In analogy with Serre’s open image theorem, one expects im 휑 = Aut 푇∞ to hold for most 푓, 푡, but until very recently for most degrees 푑 not a single example of a degree 푑 polynomial 푓 ∈ Q[푥] with surjective 휑푓,푡 was known. Among other results, we construct such examples in all sufficiently large even degrees. The second part concerns monodromy of hyperplane section of curves. Given a 푛 geometrically integral proper curve 푋 ⊂ P퐾 , consider the generic hyperplane 퐻 ∈

푛⋀︀

P퐾(푡1,...,푡푛). The intersection 퐻 ∩푋 is the spectrum of a finite separable field extension 퐿/퐾(푡1, ..., 푡푛) of degree 푑 := deg 푋. The Galois group 퐺푋 := Gal(퐿/퐾(푡1, ..., 푡푛)) is known as the sectional monodromy group of 푋. When char 퐾 = 0, the group 퐺푋 equals 푆푑 for all curves 푋. This result has numerous applications in algebraic geometry, in particular to the degree-genus problem. However, when char 퐾 > 0, the sectional monodromy groups can be smaller. We classify all nonstrange nondegenerate 푛 curves 푋 ⊂ P , for 푛 > 3 such that 퐺푋 ̸= 퐴푑, 푆푑. Using similar methods we also completely classify Galois group of generic trinomials, a problem studied previously by Abhyankar, Cohen, Smith, and Uchida.

In part three of the thesis we derive bounds for the number of F푞-points on simple abelian varieties over finite fields; these improve upon the Weil bounds. For example, when 푞 = 3, 4 the Weil bound gives #퐴(F푞) > 1 for all abelian varieties 퐴. We prove

3 dim 퐴 dim 퐴 that 퐴(F3) > 1.359 , and 퐴(F4) > 2.275 hold for all but finitely many simple abelian varieties 퐴 (with an explicit list of exceptions).

Thesis Supervisor: Bjorn Poonen Title: Distinguished Professor in Science

4 Acknowledgments

First and foremost I want to thank my advisor Bjorn Poonen for teaching me how to be a mathematician. I thank my fellow graduate students, and especially Bjorn’s army: Padma Srinivasan, Renee Bell, Isabel Vogt, Nicholas Triantafillou, Vishal Arul and Atticus Christensen for all their help and support. I thank my family for encouraging me to do mathematics professionally. I thank Davesh Maulik and Andrew Sutherland for serving on my thesis committee.

This research was supported by Simons Foundation grants #402472 (to Bjorn Poonen) and #550033, and National Science Foundation grant DMS-1601946.

5 6 Contents

1 A primer on Galois and monodromy groups 13

1.1 Étale fundamental groups ...... 14

1.2 Permutation groups ...... 17

1.3 Discretely valued fields and Newton polygons ...... 21

2 Arboreal Galois representations 25

2.1 Introduction ...... 25

2.2 Large arboreal representations over an arbitrary field ...... 28

2.3 Iterated monodromy groups ...... 35

2.4 Surjective arboreal representations over Q ...... 36

3 Sectional monodromy groups 43

3.1 Introduction ...... 43

3.2 Transitivity of Sectional Monodromy Groups ...... 46

3.3 Tangents and inertia groups ...... 50

3.4 Sectional monodromy groups of nonstrange curves ...... 55

3.5 Galois groups of trinomials ...... 63

4 Counting points on simple abelian varieties 79

7 4.1 Introduction ...... 79

4.2 Abelian varieties over large fields ...... 82

4.3 Abelian varieties over small fields ...... 86

A Tables 91

A.1 Cycle types of Mathieu groups ...... 91

A.2 Factors of trinomials over finite fields ...... 92

8 List of Figures

3-1 ...... 74

3-2 ...... 74

9 10 List of Tables

4.1 Lower and upper bounds on 푎(푞) and 퐴(푞), respectively...... 82

1/푔 4.2 Lower and upper bounds on #퐴(F푞) ...... 87

4.3 Auxiliary polynomials for Theorem 4.3.2...... 88

4.4 Auxiliary parameters for Theorem 4.3.2 ...... 89

A.1 Cycle types of Mathieu groups ...... 91

A.2 Factorization of some trinomials ...... 92

11 12 Chapter 1

A primer on Galois and monodromy groups

A finite degree 푑 topological covering of manifolds 푓 : 푌 → 푋 gives rise to a homo-

−1 1 morphism 휑: 휋1(푋, 푥) → Aut(푓 (푥)) = 푆푑 for any choice of basepoint 푥 ∈ 푋. In this way, the finite covering 푓 produces the group Mon(푓) := im 휑, the monodromy group of the covering. This construction is superficially similar to the formation of the Galois group of a finite separable field extension. The notion of the étale fundamental group unifies geometric monodromy groups and arithmetic Galois groups and allows us to define monodromy groups attached to finite separable coverings of algebraic varieties.

This chapter introduces background and tools for determining algebraic monodromy groups. Section 1.1 contains a brief introduction to the theory of étale fundamental groups. Section 1.2 collects various group-theoretic statements useful for computing monodromy groups. Section 1.3 collects basic facts about Galois groups of extensions of nonarchimedean local fields that are useful for studying local monodromy groups.

1Topological spaces often have basepoints, while schemes are more commonly equipped with base points.

13 1.1 Étale fundamental groups

Étale fundamental groups are an algebraic geometry analogue of fundamental groups in topology. Like fundamental groups, étale fundamental groups naturally act on fibers of finite “unramified” coverings of schemes. The correct algebraic version ofthe topological notion of unramified map is the following.

Definition 1.1.1. A morphism of schemes 휙: 푌 → 푋 is called étale if it is smooth of relative dimension 0.

In other words, locally on 푌 the morphism 휙 corresponds to an extension of rings

퐴 → 퐵 of the form 퐵 = 퐴[푥1, . . . , 푥푛]/(푓1, . . . , 푓푛) such that the Jacobian determinant

det(휕푓푖/휕푥푗) is invertible in 퐵. A morphism of C-varieties 푋 → 푌 is étale if and only if it is biholomorphic.

Example 1.1.2. The following are examples of étale morphisms.

1. An open immersion 푌 → 푋.

2. A finite separable extension of fields Spec 퐿 → Spec 퐾.

1 1 푝 3. The Artin-Schreier cover 휙: A퐾 → A퐾 given by 휙(푥) = 푥 − 푥, where 푝 = char 퐾 > 0.

Suppose 푋 is a connected scheme. Fix a geometric point 푥 of 푋, that is a mor-

phism 푥: Spec 퐾 → 푋 for some algebraically closed field 퐾. Let FÉt푋 denote the

category of finite étale coverings of 푋. Let 퐹푥 : FÉt푋 → Sets denote the functor sending a covering 휙: 푌 → 푋 to the fiber 휙−1(푥).

Fact 1.1.3 ([1] Section V.7). The category FÉt푋 is a Galois category and 퐹푥 is its fiber functor, so 퐹푥 defines an equivalence of categories between FÉt푋 and the category of Aut(퐹푥)-sets.

In other words, there is a Galois correspondence between finite étale coverings of

푋 and open subgroups of the profinite group Aut(퐹푥).

14 et The group 휋1 (푋, 푥) := Aut(퐹푥) is called the étale fundamental group of 푋 with base point 푥. The following theorem supports the claim that étale fundmental groups are similar to fundamental groups in topology. Recall that for a C-variety 푋 the set 푋(C) inherits the natural analytic topology from C.

Theorem 1.1.4 (“Riemann’s existence theorem”, [1] Theorem XII.5.1). Suppose 푋 is a connected separated scheme of finite type over C and 푥 ∈ 푋(C) is a point. There et ∼ top is a canonical isomorphism 휋1 (푋, 푥) → 휋̂︀1(푋(C) , 푥) between the étale fundamental group of 푋, and the profinite completion of the topological fundamental group of 푋(C).

Remark 1.1.5. The hard part of Theorem 1.1.4 is to construct a canonical algebraic structure on a finite topological covering of 푋(C). When 푋 is proper, this can be done using GAGA, but for the general case more care is needed.

For varieties over fields that are not algebraically closed the structure of theétale fundamental group is more complicated. The simplest case is that of a point Spec 퐾. ∏︀푛 Finite étale coverings of Spec 퐾 are of the form Spec 퐿 for 퐿 = 푖=1 퐿푖 a finite product of finite separable field extensions 퐿푖/퐾. So the fundamental group of Spec 퐾 is just et sep the absolute Galois group of 퐾, 휋1 (Spec 퐾) = Gal(퐾 /퐾). For a general variety, the étale fundamental group is an extension of the Galois group of the ground field by the geometric fundamental group.

Theorem 1.1.6 ([1] Theorem V.6.1). Suppose 푋/퐾 is a connected separated geometrically irreducible scheme of finite type and 푥 ∈ 푋(퐾) a geometric point. Then there is a canonical exact sequence

et et 1 → 휋1 (푋퐾 , 푥) → 휋1 (푋, 푥) → Aut(퐾/퐾) → 1.

et Remark 1.1.7. If char 퐾 = 0 and 푋 is normal, the group 휋1 (푋퐾 , 푥) is isomorphic to et 휋1 (푋C, 푥).

Remark 1.1.8. If 퐾 is an algebraically closed field of positive characteristic, and 푋/퐾

et a variety, then the structure of 휋1 (푋, 푥) can be much wilder than what one might

15 et 1 expect from the topological intuition. For example, the group 휋1 (A퐾 , 푥) is trivial when char 퐾 = 0, but not even finitely generated when char 퐾 > 0.

et Given a finite étale covering 푓 : 푌 → 푋 of degree 푑, we get a map 휑: 휋1 (푋, 푥) →

Aut(퐹푥(푌 )) = 푆푑. The image Mon(푓) := im 휑 ⊂ 푆푑 is called the monodromy group of the covering 푓; it is a permutation group of degree 푑: a finite group with a specified action on a set of size 푑. Given two base points 푥, 푥′ ∈ 푋, there is a bijection

′ 휓 : 퐹푥(푌 ) → 퐹푥′ (푌 ) and an isomorphism 휋1(푋, 푥) → 휋1(푋, 푥 ) making the following diagram commute. 휋1(푋, 푥) −−−→ Aut(퐹푥(푌 )) ⎮ ⎮ ⎮ ⎮휓 ⌄ ⌄ ′ 휋1(푋, 푥 ) −−−→ Aut(퐹푥′ (푌 )) In this way the monodromy group of the covering is independent of the choice of the base point. We will use the following properties of monodromy groups.

Proposition 1.1.9. Suppose that 푋 is a normal, connected, separated scheme and 푓 : 푌 → 푋 is a finite étale covering. Suppose 푈 ⊂ 푋 is an open subset and 휓 : 푋′ → 푋 a morphism.

1. The monodromy group Mon(푓) is transitive if and only if 푌 is irreducible.

2. The monodromy group Mon(푓) is imprimitive if and only if 푓 factorizes as a

composition of étale covers 푌 → 푌1 → 푋 of degrees strictly greater than 1 (primitivity is defined in Section 1.2).

3. ([1] Proposition V.8.2). Suppose 푈 ⊂ 푋 is an open subscheme. The monodromy

group of the restricted covering 푓푈 : 푌푈 → 푈 is equal to Mon(푓).

4. Suppose 푋′ is a connected scheme and 휓 : 푋′ → 푋 is a morphism. The monodromy group of the pullback 푋′ × 푌 → 푋′ of 푓 under 휓 is a permutation 푋 subgroup of Mon(푓).

5. ([1] Proposition V.8.2). If 푋, 푌 are of finite type over a field 퐾, and 푓 is a 퐾- morphism, then Mon(푓) is equal to the Galois group of the extension of function fields 퐾(푌 )/퐾(푋).

16 Remark 1.1.10. Normality of 푋 is an important assumption. For example, a nodal cubic over C admits cyclic étale covers 푌푛 → 푋 with 푌푛 isomorphic to a cyclic chain of projective lines. Then property 1 does not hold, since the total space 푌푛 of the 푛 cover is not irreducible. Property 5 is false, since the Galois group of C(푌푛) = C(푡) over C(푡) is trivial, as is the monodromy of the covering restricted to the complement of the node, contradicting property 3. See [22] Exercise 10.6.

Monodromy groups are often calculated by combining the information on transitivity of 푌 → 푋 and related covers (for example, 푌 ×푌 → 푋) with a computation of 푋 monodromy of various pullbacks. Suppose 푋 is a variety over a field 퐾, 푓 : 푌 → 푋 is a generically étale cover, and 푥 ∈ 푋 is a point over which 푓 is not étale. The monodromy of the pullback of 푌 → 푋 to the fraction field of the completed local ˆ ring Frac 풪푋,푥 will be referred to as the local monodromy group of 푓 at 푥. These local monodromy groups can often be described using the theory of local fields discussed in Section 1.3.

1.2 Permutation groups

We record here a selection of theorems on permutation groups that are useful for computing Galois groups.

Definition 1.2.1. A finite permutation group is a triple (퐺, 휑, 푆), where 퐺 is a finite group, 푆 is a finite set, and 휑: 퐺 → Aut(푆) is an injective homomorphism. The size of 푆 is called the degree of 퐺.

We will often abuse notation by referring to “a permutation group 퐺” whenever the pair (휑, 푆) is clear from the context. A permutation group is 푘-transitive if 푘 6 #푆 and the group acts transitively on the 푘-tuples of distinct elements of 푆. A permutation group is primitive if it preserves no nontrivial partition of 푆. For example. the group of symmetries of the rooted tree below acting on the 6 vertices at the lower level is not primitive.

17 Example 1.2.2. Here is a list of some common permutation groups.

1. 퐴푛, the alternating group on 푛 letters, it is an (푛 − 2)-transitive group.

2. AGL푛(푞), the affine general linear group, where 푛 > 1 and 푞 is a prime power. ⃗ ⃗ 푛 It is a group of affine maps ⃗푥 → 퐴⃗푥 + 푏, where 퐴 ∈ GL푛(푞), and 푏 ∈ F푞 . It 푛 acts on A (F푞). It is a primitve and doubly transitive group. Since elements of

AGL푛(푞) preserve collinearity, the group is 3-transitive if and only if 푞 = 2.

푛 3. AΓL푛(푞), the affine semilinear group. It is a permutation group acting on F푞

generated by AGL푛(푞) and the coordinate-wise 푝-power Frobenius.

4. PGL푛(푞), the projective general linear group, the group of linear symmetries of the projective space 푛−1. It is a 2-transitive group. Since the group preserves PF푞 collinearity, it is 3-transitive if and only if 푛 = 2. The only 4-transitive group

in the family is PGL2(3) = 푆4.

5. 푀11, the Mathieu group on 11 letters. It is defined as a one-point stabilizer of

the larger Mathieu group 푀12, which itself is the group of symmetries of the

unique 푆(5, 6, 12) Steiner system. The group 푀11 is a 4-transitive group. See [15] for the construction of this sporadic simple group.

Many nontrivial results about permutation groups can be obtained as conse- quences of the classification of finite simple groups. We now outline someargu- ments that demonstrate the relationship between permutation groups and finite simple groups.

18 Primitive groups can be naturally categorized into several classes. Suppose 퐺 is primitive, and consider a minimal normal subgroup {1} ̸= 푁 ⊂ 퐺. The orbits of 푁 on 푆 form a 퐺-invariant partition of 푆. Since 퐺 is primitive, 푁 has only one orbit, and so 푁 is transitive. Consider the centralizer 퐶퐺(푁); it is a normal subgroup of 퐺. Any two minimal normal subgroups 푁1, 푁2 of 퐺 centralize each other, since

[푁1, 푁2] is a normal subgroup of 푁1 ∩ 푁2. If 퐶퐺(푁) = {1}, then 푁 is the unique minimal normal subgroup and the conjugation action homomorphism 퐺 → Aut 푁 is an injection. In this case 푁 is characteristically simple (see [50] Exercise 1.5.23),

푛 and so 푁 = 푇 for some simple group 푇 . Suppose 퐶퐺(푁) ̸= {1}; since 퐶퐺(푁) is a nontrivial normal subgroup it acts transitively on 푆. If an element 푛 ̸= 1 of 푁 has a fixed point 푠 ∈ 푆, then 푛 also fixes every point of 퐶퐺(푁)푠 = 푆, a contradiction. Thus Stab(푠) ∩ 푁 = {1} and the action of 푁 on 푆 is regular (i.e. isomorphic to the action of 푁 on itself). This reasoning did not use minimiality of 푁, so it applies also to the normal subgroup 퐶퐺(푁), Thus the action of 퐶퐺(푁) on 푆 is regular. Since

퐶퐺(푁) acts by automorphisms of the 푁-torsor 푆, the group 퐶퐺(푁) is isomorphic to 푛 푁. If 푁 = 퐶퐺(푁), then 푁 is elementary abelian 푁 = F푝 since it is characteristically simple, 푁 = 퐶퐺(푁), and the conjugation action of 퐺 on 푁 defines an injection

퐺/푁 → Aut(푁) = GL푁 (F푝). Finally, suppose 푁 ̸= 퐶퐺(푁) and 퐶퐺(푁) ̸= 1. If ′ ′ ′ 푁 ̸= 푁 is a minimal normal subgroup of 퐺, then 푁 ⊂ 퐶퐺(푁) and the action of 푁 ′ ′ on 푆 is regular, so #푁 = #푆 = #퐶퐺(푁) and 푁 = 퐶퐺(푁). Hence 퐺 has exactly two minimal normal subgroups, and the socle of 퐺 – the product of its minimal normal subgroups – is a power of a finite simple group.

The arguments outlined above can be developed much further, giving a complete classification of primitive permutation groups in terms of finite simple groups; there- sulting statement is the O’Nan–Scott Theorem; see [16] Theorem 4.1A. Classification of finite simple groups combined with the O’Nan–Scott theorem gives very powerful tools for computing Galois and monodromy groups.

Given a finite étale covering 푓 : 푌 → 푋, the transitivity degree of the monodromy group can be read off the description of irreducible components of the fiber products

19 푌 ×푋 푌 ×푋 · · · ×푋 푌 . There are very few groups of high transitivity degree:

Theorem 1.2.3. [[13] Sections 7.3, 7.4] If 퐺 is a triply transitive permutation group

of degree 푑, then either 퐺 ⊃ 퐴푑 or one of the following holds:

푛 1. 퐺 = AGL푛(2), 푑 = 2 ;

2. 퐺 = 퐺1, 푑 = 16 (a special permutation subgroup of AGL4(2));

3. PSL2(푟) ⊂ 퐺 ⊂ PΓL2(푟), 푑 = 푟 + 1, 푟 is a power of a prime;

4. 퐺 = 푀11, 푑 = 11 (a Mathieu group, see [15]);

5. 퐺 = 푀11, 푑 = 12;

6. 퐺 = 푀12, 푑 = 12

7. 퐺 = 푀22, 푑 = 22;

8. 퐺 = Aut(푀22), 푑 = 22;

9. 퐺 = 푀23, 푑 = 23;

10. 퐺 = 푀24, 푑 = 24.

Remark 1.2.4. The classification of 2-transitive groups is also known, see [16] Sec- tion 7.7.

Another common input to the computation of a monodromy group is a local cal- culation. Local monodromy groups often contain cycles. Primitive groups containing cycles can be classified:

Theorem 1.2.5 (Jones [25]). Let 퐺 be a primitive permutation group of finite degree

푑, not containing 퐴푑. Suppose that 퐺 contains a cycle fixing exactly 푘 points, where 0 6 푘 6 푑 − 2. Then one of the following holds:

1. 푘 = 0 and either

(a) 퐶푝 ⊂ 퐺 ⊂ AGL1(푝) with 푑 = 푝 prime, or

20 푛 (b) PGL푛(푞) ⊂ 퐺 ⊂ PΓL푛(푞) with 푑 = (푞 − 1)/(푞 − 1) and 푛 > 2 for some prime power 푞, or

2. 푘 = 1 and either

푛 (a) AGL푛(푞) ⊂ 퐺 ⊂ AΓL푛(푞) with 푑 = 푞 and 푛 > 1 for some prime power 푞, or

(b) 퐺 = PSL2(푝) or 퐺 = PGL2(푝) with 푑 = 푝 + 1 for some prime 푝 > 5, or

3. 푘 = 2 and PGL2(푞) ⊂ 퐺 ⊂ PΓL2(푞) with 푑 = 푞 + 1 for some prime power 푞.

Even when local monodromy does not have cycles, but has sufficiently many fixed points, one can still conclude that the monodromy group is large. The minimal degree 휇(퐺) of a permutation group 퐺 is the minimum over 푔 ∈ 퐺 ∖ {1} of the number of points not fixed by 푔.

Theorem 1.2.6 (Liebeck-Saxl [37] Corollary 3). Let 퐺 be a primitive permutation √ group of degree 푑, and assume that 퐺 ̸⊃ 퐴푑. Then 휇(퐺) > 2( 푑 − 1).

Remark 1.2.7. There is a classification of groups with 휇(퐺) < 푑/2, see [19].

1.3 Discretely valued fields and Newton polygons

We briefly summarize the basics of the theory of local fields and their extensions, a detailed treatment can be found in [48].

Definition 1.3.1. A complete discretely valued field is a pair (퐹, 푣퐹 ), where 퐹 is a field complete with respect to the discrete valuation 푣 = 푣퐹 : 퐹 → Q (the image of

푣 is an infinite cyclic group). The ring 풪퐹 = {푥 ∈ 퐹 |푣(푥) > 0} ⊂ 퐹 is called the

valuation ring of 퐹 . An element 휋 ∈ 풪퐹 is called a uniformizer if 푣(휋) is a generator

for 푣(퐹 ) . The field 푘퐹 := 풪퐹 /(휋) is called the residue field of 퐹 .

21 Example 1.3.2.

1. 퐹 = Q푝 with 푣 = 푣푝 the 푝-adic valuation. The element 휋 = 푝 is a uniformizer,

and F푝 = Z푝/(푝) is the residue field.

2. 퐹 = 푘((푡)), and 푣 is the 푡-adic valuation. The element 푡 is a uniformizer and 푘 is the residue field.

√ (︀ )︀ 3. 퐹 = Q푝( 푝), the valuation 푣 is defined by 푣(푥) = 푣푝 Norm퐹/Q푝 (푥) /2. The √ element 푝 is a uniformizer and F푝 is the residue field.

Given a finite extension 퐿 of a complete discretely valued field 퐹 , there is a unique extension of 푣 to a discrete valuation 푣퐿 on 퐿, defined by 푣퐿 : 퐿 → Q, 푣퐿(푥) :=

푣퐹 (Norm퐿/퐹 (푥))/[퐿 : 퐹 ]. The pair (퐿, 푣퐿) is a discretely valued field. The ramification

index 푒퐿/퐹 of 퐿/퐹 is the index 푒퐿/퐹 := [푣퐿(퐿): 푣퐹 (퐹 )] of the extension of value

groups. An extension 퐿/퐹 is totally ramified if 푒퐿/퐹 = [퐿 : 퐹 ], and is tamely ramified

if char 푘퐹 - 푒퐿/퐹 . The field 푘퐹 = 풪퐹 /휋퐹 = 풪퐹 / (휋퐿풪퐿 ∩ 풪퐹 ) is a subfield of 푘퐿; thus an extension of discretely valued fields 퐿/퐹 gives rise to an extension of residue fields

푘퐿/푘퐹 . The degree 푓퐿/퐾 := [푘퐿 : 푘퐹 ] is called the inertia degree of 퐿/퐾. An extension

퐿/퐹 is unramified if 푒퐿/퐹 = 1 and the residue field extension 푘퐿/푘퐹 is separable.

Proposition 1.3.3. Suppose 퐹 is a discretely valued field with perfect residue field

푘퐹 .

1. Every separable extension 퐿/퐹 can be extended to a tower 퐿/퐿′/퐹 , where 퐿/퐿′ is totally ramified and 퐿′/퐹 is unramified.

2. The functor sending a finite field extension 퐿/퐹 to the residue field extension

푘퐿/푘퐹 defines an equivalence of categories

∼ {unramified field extensions of 퐹 } → {finite extensions of 푘퐹 }.

3. For any extension 퐿/퐹 the identity [퐿 : 퐹 ] = 푒퐿/퐹 푓퐿/퐹 holds.

22 4. If 퐹 = 푘((푡)), then any finite extension 퐿/퐹 is isomorphic to 푘퐿((푢)).

5. If 퐹 = 푘((푡)) and 푘 is algebraically closed, then any tamely ramified extension

퐿/퐹 is of the form 퐿 = 푘((푡1/푒)) with char 퐾 - 푒.

Newton polygons, defined below, are a versatile tool for determining valuations of roots of polynomials.

푛 Definition 1.3.4. Suppose 퐹 is a discretely valued field, and 푓 = 푎0 + ··· + 푎푛푥 ∈

퐹 [푥] is a polynomial with 푎0 ̸= 0. Consider the set of points 푆 = {(푖, 푣(푎푖)) : 0 6 2 푖 6 푛} in R . The Newton Polygon NP(푓) of 푓 is the lower convex hull of 푆. The Newton polygon consists of segments of distinct slopes connecting vertices in 푆.A

width of a segment from (푥1, 푦1) to (푥2, 푦2) is defined to be 푥2 − 푥1, and the slope of this segment is 푦1−푦2 . 푥1−푥2

2 2 3 2 4 Example 1.3.5. Consider the polynomial 푓 = 푝+푝푥+푥 +푝 푥 +푝 푥 ∈ Q푝[푥]. The Newton polygon of 푓 consists of two segments connecting consecutive vertices (0, 1), (2, 0), and (4, 2). One segment has slope −1/2 and width 2, and the other segment has slope 1 and width 2.

The Newton polygon of 푓 determines valuations of the roots of 푓.

Proposition 1.3.6 (see [34] Section IV.3). Suppose 퐹 is a discretely valued field and

푓 ∈ 퐹 [푥] is a polynomial with nonzero constant term. Extend the valuation 푣퐹 on 퐹 to a valuation 푣 : 퐹 → Q. Consider the Newton polygon 푁푃 (푓) of 푓. Then the number of zeros of 푓 with valuation 훼 is the width of the segment of slope −훼 (the width is 0 if such segment does not exist).

23 푛−1 Example 1.3.7 (Eisenstein polynomials). Consider a polynomial 푎0+···+푎푛−1푥 + 푛 푥 ∈ 퐹 [푥] such that 푣(푎0) = 1, 푣(푎푖) > 1 for 1 6 푖 6 푛 − 1. The Newton polygon of 푓 consists of one segment of width 푛 and slope −1/푛. Let 훼 ∈ 퐹 be a root of 푓. From the Newton polygon we conclude 푣(훼) = 1/푛. Therefore the extension 퐹 (훼)/퐹 has ramification index at least 푛. Since the degree deg 퐹 (훼)/퐹 is at most 푛 we conclude that 푓 is an irreducible polynomial and 퐹 (훼)/퐹 is a totally ramified extension of degree 푛.

Remark 1.3.8. In a similar fashion, Newton Polygons can be used to determine valuations of roots of convergent power series; see [34] Section IV.4.

24 Chapter 2

Arboreal Galois representations

The results of this chapter have appeared in [31].

2.1 Introduction

Let 퐾 be a field, let 푓 ∈ 퐾[푥] be a polynomial of degree 푑, and let 푡 ∈ 퐾 be an arbitrary element. Write 푓 ∘푛 = 푓 ∘ 푓 ∘ · · · ∘ 푓 for the 푛th iterate of 푓 (with the convention 푓 ∘0(푥) = 푥). Assume that the polynomials 푓 ∘푛(푥) − 푡 are separable for all 푛 ≥ 0. Fix a separable closure 퐾sep of 퐾. Then the roots of 푓 ∘푛(푥) − 푡 in 퐾sep for varying 푛 have a natural tree structure. Namely, define a graph with the set of vertices equal to the disjoint union of roots of 푓 ∘푛(푥) − 푡 for all 푛 ≥ 0. Draw an edge between two roots 훼, 훽 when 푓(훼) = 훽. Then when 푡 is not a periodic point of 푓, the resulting graph is a complete rooted 푑-ary tree 푇∞. Vertices at level 푛 correspond ∘푛 to the roots of 푓 (푥) − 푡 (the root of the tree 푡 is at level zero). Let 푇푛 be the tree formed by levels 0 through 푛 of 푇∞. For example, the next figure is the tree 푇2 when 푓 = 푥2 − 2 and 푡 = 0.

25 0

√ √ − 2 2

√︀ √ √︀ √ √︀ √ √︀ √ 2 − 2 − 2 − 2 2 + 2 − 2 + 2 Let 퐺 := Gal(퐾sep/퐾) be the absolute Galois group of 퐾. The Galois action on

sep 퐾 defines a homomorphism 휑: 퐺 → Aut(푇∞) known as the arboreal Galois representation attached to 푓 and 푡. This construction is parallel to that of the Tate module of an abelian variety, where one replaces 푓 by the multiplication by 푝 endomorphism [푝]. With this analogy in mind, it is natural to seek a counterpart of Serre’s open image theorem in arithmetic dynamics. The crucial difference is that while the closed

subgroups of GL2(Z푝) are easy to describe using Lie theory, the subgroup structure of

the profinite group Aut(푇∞) is complicated; this makes analyzing possibilities for the image of an arboreal representation a hard problem. Our goal is to study conditions

for having the simplest kind of Galois image, namely the whole group Aut(푇∞).

Example 2.1.1. Let 퐾 = Q, 푡 = 0 and 푓 = 푥2 +1. Then the arboreal representation is surjective (see [54]).

It is conjectured that when 퐾 is a number field and 푑 = 2, the image of an arboreal

representation has finite index in Aut(푇∞) unless some degeneracy conditions are satisfied (see [26, Conjecture 3.11]). However, very little is known when the degreeof 푓 is greater than 2. We will show that when the degree of 푓 is even, there is a criterion for the surjectivity of the arboreal representation; this generalizes the method of Stoll in [54]. In Section 2.2 we prove the following theorem.

Theorem 2.1.2. Assume that the degree of 푓 is even and char 퐾 ̸= 2. Assume that for any 푛 and any 훼 ∈ 푓 −푛(푡) the splitting field of 푓(푥) − 훼 over 퐾(훼) has

∘푛 Galois group 푆푑. If the discriminants disc(푓 (푥) − 푡) are linearly independent in × ×2 the F2-vector space 퐾 /퐾 , then the arboreal representation attached to 푓 and 푡 is surjective.

26 This theorem allows us to prove surjectivity of some arboreal representations attached to even degree polynomials. The following conjecture was made by Odoni in [45, Conjecture 7.5.]

Conjecture 2.1.3. Let 퐹 be a Hilbertian field. Then for any integer 푑 there exists a degree 푑 monic polynomial 푓 such that the associated arboreal representation (for 푡 = 0) is surjective.

Until recently, the conjecture was open even for 퐹 = Q (see [26, Conjecture 2.2]). Looper [38] proved Odoni’s conjecture for 퐹 = Q and prime degrees 푑, and showed that for many other values of 푑 Odoni’s conjecture can be deduced from Vojta’s conjecture. In Section 2.4 we prove the following theorem unconditionally.

Theorem 2.1.4. Let 푑 ≥ 20 be an even number. Then there exist infinitely many polynomials 푓 ∈ Q[푥] of degree 푑 such that the arboreal representation associated to 푓 and 푡 = 0 is surjective.

Independently of our work, Joel Specter [53] and Robert Benedetto and Jamie Juul [10] proved related results on Odoni’s conjecture for number fields. Specter shows that Odoni’s conjecture holds for any number field. Benedetto and Juul prove Odoni’s conjecture for even degree polynomials over an arbitrary number field, and for √ √ odd degree 푑 polynomials over number fields 퐹 that do not contain Q( 푑, 푑 + 1).

Consider the case when 퐾 = 퐹 (푢) for some field 퐹 and indeterminate 푢, 푓 is a

∘푛 polynomial defined over 퐹 , and 푡 = 푢. Let 퐾푛 be the splitting field of 푓 (푥) − 푡

over 퐾. Then the groups Gal(퐾푛/퐾) are the monodromy groups of the ramified 1 1 ∘푛 coverings P퐹 → P퐹 induced by 푓 . In this case the image of the resulting arboreal representation is known as the iterated monodromy group of 푓. These monodromy groups have been studied in the case when 푓 is post-critically finite (see [44]), when 푓 is quadratic (see [46]), and for various rational functions 푓 in [28]. Recall that a polynomial 푓 is called post-critically finite if for every root 훾 of 푓 ′ the orbit of 훾 under 푓 is finite. In Section 2.3 we prove the following theorem.

27 Theorem 2.1.5. Let 푓 ∈ 퐹 [푥] be a degree 2푚 polynomial such that the Galois group of the splitting field of 푓(푥) − 푡 over 퐾 = 퐹 (푡) is the symmetric group 푆2푚. Assume that 푓 ′(푥) is irreducible over 퐹 and 푓 is not post-critically finite. Then the arboreal representation attached to 푓 and 푡 over 퐾 is surjective.

In fact we prove a stronger statement that provides a criterion for the surjectivity of the iterated monodromy group attached to an even degree polynomial 푓 in terms of the critical orbit of 푓. Jamie Juul [28, Proposition 3.2.] proves a version of Theorem 2.1.5 for finite level arboreal representations and polynomials of arbitrary degree. However, we do not know if there are many polynomials for which Juul’s result implies surjectivity of the infinite level arboreal representation is large. Our result shows that when 퐹 is a number field most polynomials have surjective infinite level iterated monodromy groups, see Remark 2.3.3.

2.2 Large arboreal representations over an arbitrary field

We begin by introducing some arboreal notation. Fix a field 퐾 of characteristic not 2, a polynomial 푓 ∈ 퐾[푥] of degree 푑, and an element 푡 ∈ 퐾. Assume 푡 is not periodic under 푓, and assume that the polynomials 푓 ∘푛(푥) − 푡 are separable for all 푛. Write

푇∞ for the preimage tree of 푡 under 푓 as in the introduction. Let 퐾푛/퐾 denote the ∘푛 splitting field of 푓 (푥) − 푡 over 퐾. Denote by 푇푛 the tree formed by the first 푛 levels of 푇∞. The Galois group Gal(퐾푛/퐾) injects into the automorphism group of the tree

Aut(푇푛). We call the extension 퐾푛/퐾 maximal if Gal(퐾푛/퐾) = Aut(푇푛).

The aim of this section is to prove a criterion for surjectivity of arboreal representations 휑: 퐺 → Aut(푇∞). We establish surjectivity in three steps. Step one is to show that for all 푛 and 훼 ∈ 푓 −푛(푡) the Galois group of the splitting field of 푓(푥) − 훼 over 퐾(훼) is 푆푛. Step two is to show that the splitting field of 푓(푥)−훼 is disjoint from 푛 퐾푛/퐾(훼). Step three is to show that the 푑 extensions of 퐾푛 given by the splitting

28 −푛 fields of 푓(푥) − 훼푖 for different 훼푖 ∈ 푓 (푡) are linearly disjoint. We show that steps two and three can be reduced to the arithmetic of forward orbits of 푓.

The main idea is simple: any proper normal subgroup of 푆푑 is contained in 퐴푑; therefore to show that some 푆푑 extensions are linearly disjoint it is enough to show that their (unique) quadratic subfields are linearly disjoint, as we will now prove.

Lemma 2.2.1. Let 퐿/퐾 be a Galois extension of fields with Galois group 푆푑. Let 퐹 denote the unique quadratic extension of 퐾 contained in 퐿. Let 푀/퐾 be an arbitrary Galois extension. Then the fields 퐿 and 푀 are linearly disjoint over 퐾 if and only if the fields 퐹 and 푀 are linearly disjoint over 퐾.

Proof. If 퐿 and 푀 are disjoint, then so are 퐹 and 푀. Assume that 퐹 and 푀 are disjoint. The field extension 퐾 ⊂ (퐿 ∩ 푀) ⊂ 퐿 corresponds to a normal subgroup of

푆푑. Any nontrivial normal subgroup of 푆푑 is contained in 퐴푑 and therefore if 퐿 ∩ 푀 were not 퐾 it would contain 퐹 .

Lemma 2.2.2. Let 퐿푖/퐾 for 푖 = 1, . . . , 푛 be 푆푑-Galois extensions of 퐾. Let 퐹푖 ⊂ 퐿푖 denote the unique quadratic extension of 퐾 inside 퐿푖. Then the fields 퐿푖 are linearly disjoint if and only if the fields 퐹푖 are linearly disjoint.

Proof. If 퐿푖 are linearly disjoint, then 퐹푖 are also linearly disjoint. Assume that 퐹푖 are linearly disjoint. We use induction on 푛. The case 푛 = 2 follows from Lemma 2.2.1.

Suppose that the result holds for 푛 − 1. Then the fields 퐿1, . . . , 퐿푛−1 are linearly disjoint. Let 푀 denote the compositum 퐿1 . . . 퐿푛−1. The Galois group Gal(푀/퐾) 푛−1 is isomorphic to (푆푑) . We need to show that 푀 and 퐿푛 are disjoint. By Lemma

2.2.1 it is enough to show that 퐹푛 and 푀 are disjoint.

Let 퐹/퐾 be a quadratic extension of 퐾 contained in 푀. Then 퐹 defines a 푛−1 character 휒:(푆푑) = Gal(푀/퐾) → {±1}. The character 휒 can be written as

휒(푎1, . . . , 푎푛−1) = 휒1(푎1) ····· 휒푛−1(푎푛−1), (푎1, . . . , 푎푛−1) ∈ 푆푑 × · · · × 푆푑,

where 휒푖 : 푆푑 → {±1} are quadratic characters. The group 푆푑 has only two quadratic

29 characters: the trivial character and the sign character. Let 푏푖 ∈ 퐾 be elements √ such that 퐹푖 = 퐾( 푏푖). Let 퐼 ⊂ {1, . . . , 푛 − 1} be the set of indices 푖 for which 휒푖 is nontrivial. Then the extension 퐹 is obtained by adjoining to 퐾 a square root of ∏︀ 푖∈퐼 푏푖. In particular, since 퐹푛 is disjoint from the compositum 퐹1 . . . 퐹푛−1, it is not

a subfield of 푀 and therefore 푀 and 퐿푛 are disjoint.

Lemma 2.2.2 allows us to reduce proving surjectivity of an arboreal representation to Kummer theory. We also need a standard fact about discriminants.

Lemma 2.2.3. Let 푆 be a commutative ring, let 푃, 푄 ∈ 푆[푥] be arbitrary polynomials. Then disc(푃 푄) = (−1)deg 푃 deg 푄 disc(푃 ) disc(푄)푅(푃, 푄)2 where 푅(푃, 푄) is the resultant of 푃 and 푄.

Proof. We recall the normalization of discriminant and resultant for non-monic polynomials from [35, Chapter IV §8]. Assume 푃, 푄 have degrees 푛, 푚, leading coef-

ficients 푎, 푏, and roots 훼1, . . . , 훼푛 and 훽1, . . . , 훽푚. Then the resultant is given by 푚 푛 ∏︀ 푅(푃, 푄) = 푎 푏 푖,푗(훼푖−훽푗), and the relevant discriminants are given by the formulas 푛(푛−1)/2 2푛−2 ∏︀ 푚(푚−1)/2 2푛−2 ∏︀ disc(푃 ) = (−1) 푎 푖̸=푗(훼푖−훼푗), disc(푄) = (−1) 푏 푖̸=푗(훽푖−훽푗), and

(푛+푚)(푛+푚−1)/2 2(푛+푚)−2 ∏︁ ∏︁ ∏︁ 2 disc(푃 푄) = (−1) (푎푏) (훼푖 − 훼푗) (훽푖 − 훽푗) (훼푖 − 훽푗) . 푖̸=푗 푖̸=푗 푖,푗

Fix a field 퐾 of characteristic not 2 and fix an element 푡 ∈ 퐾. Let 푓 ∈ 퐾[푥] be a

degree 푑 polynomial where 푑 is even. Consider the corresponding tree 푇∞, arboreal

representation 휑: 퐺 → Aut(푇∞), and the extensions 퐾푛/퐾.

Theorem 2.2.4. Assume that for some 푛 ∈ Z>0 the field extension 퐾푛−1/퐾 is maximal. Assume also that for every 훼 ∈ 푓 −(푛−1)(푡) the polynomial 푓(푥) − 훼 is

irreducible over 퐾(훼) and has Galois group 푆푑 or 퐴푑. Then 퐾푛/퐾 is maximal if and ∘푛 ×2 only if disc(푓 (푥) − 푡) ̸∈ 퐾푛−1.

30 Proof. Assume that 퐾푛/퐾 is maximal. Then the Galois group of 퐾푛/퐾푛−1 acts on ∘푛 푛−1 the roots of 푓 (푥) − 푡 as a product of 푑 copies of 푆푑. In particular there is an ∘푛 element of Gal(퐾푛/퐾푛−1) that induces an odd permutation of the roots of 푓 (푥) − 푡. ∘푛 Therefore disc(푓 (푥) − 푡) is not a square in 퐾푛−1.

∘푛 ×2 ∘푛−1 Now assume disc(푓 (푥) − 푡) ̸∈ 퐾푛−1. For any choice of the root 훼 of 푓 (푥) − 푡 consider the discriminant disc(푓(푥) − 훼). Assume that disc(푓(푥) − 훼) is a square

in 퐾푛−1. Since the extension 퐾푛−1/퐾 is maximal, the Galois action on the roots −(푛−1) 훼푖 ∈ 푓 (푡) is transitive. Therefore for every 훼푖 the discriminant disc(푓(푥) − 훼푖) ×2 is in 퐾푛−1 as well. By Lemma 2.2.3 the following identity holds

⎛ ⎞ ∏︁ ∏︁ disc (푓(푥) − 훼) ≡ disc ⎝ (푓(푥) − 훼)⎠ 훼∈푓 −(푛−1)(푡) 훼∈푓 −(푛−1)(푡) ∘푛 ×2 ≡ disc(푓 (푥) − 푡) ̸≡ 1 (mod 퐾푛−1).

×2 Therefore disc(푓(푥) − 훼) ̸∈ 퐾푛−1 and the Galois group of 푓(푥) − 훼 over 퐾(훼) is

equal to 푆푑. By Lemma 2.2.1, applied to the splitting field of 푓(푥) − 훼 over 퐾(훼)

and 퐾푛−1/퐾(훼), the polynomial 푓(푥) − 훼 is irreducible over 퐾푛−1 and has the Galois

group equal to 푆푑. In order to apply Lemma 2.2.2 we need to show that disc(푓(푥)−훼푖) × ×2 are linearly independent in the F2 vector space 퐾푛−1/퐾푛−1.

−푚 푛−푚 We claim that for 푚 ≤ 푛 − 1 and 훽푖 ∈ 푓 (푡) the elements disc(푓 (푥) − 훽푖) × ×2 are linearly independent in the F2 vector space 퐾푛−1/퐾푛−1. We prove the statement by induction. When 푚 = 0 the statement holds by the assumption of the theorem. Assume the statement is true for 푚 = 푙 − 1. Assume that there is a nonempty subset −푙 ∏︀ ∘푛−푙 퐽 ⊂ 푓 (푡) such that 훽∈퐽 disc(푓 (푥) − 훽) is a square. Say that 훽푖 and 훽푗 belong

to the same cluster if 푓(훽푖) = 푓(훽푗) (in other words 훽푖 and 훽푗 have the same parent in the tree). If every cluster of roots is either contained in 퐽 or has an empty intersection

31 with 퐽, then 푓 −1(푓(퐽)) = 퐽, and so by Lemma 2.2.3

∏︁ ∏︁ ∏︁ disc(푓 ∘푛−푙(푥) − 훽) ≡ disc(푓 ∘푛−푙(푥) − 훽) 훽∈퐽 훾∈푓(퐽) 훽∈푓 −1(훾) ∏︁ 푛−푙+1 ×2 ≡ disc(푓 (푥) − 훾) (mod 퐾푛−1). 훾∈푓(퐽)

The right hand side is not a square by the induction hypothesis. Therefore, we can assume that there is a cluster 퐼 such that 퐼 has a nontrivial intersection with 퐽. Choose two elements 훽′, 훽′′ ∈ 퐼 such that 훽′ is in 퐽 and 훽′′ is not in 퐽. Since the extension 퐾푛−1/퐾 is assumed to be maximal, there exists an element 휎 ∈ Gal(퐾푙/퐾) that acts on the roots of 푓 ∘푙(푥) − 푡 as a transposition of 훽′ and 훽′′. Then

(︃ )︃ (︃ )︃ ∏︁ ∏︁ 1 ≡ disc(푓 ∘푛−푙(푥) − 훽) · 휎 disc(푓 ∘푛−푙(푥) − 훽) 훽∈퐽 훽∈퐽 ∘푛−푙 ′ ∘푛−푙 ′′ ×2 ≡ disc(푓 (푥) − 훽 ) disc(푓 (푥) − 훽 ) (mod 퐾푛−1).

Since Gal(퐾푙/퐾) acts doubly-transitively on the cluster, for every pair of elements ∘푛−푙 ∘푛−푙 훽푖, 훽푗 ∈ 퐼 the product disc(푓 (푥)−훽푖) disc(푓 (푥)−훽푗) is a square in 퐾푛−1. Since ∏︀ ∘푛−푙 푑 is even, the product 훽∈퐼 disc(푓 − 훽) is a square in 퐾푛−1. By Lemma 2.2.3 we have the identity

∏︁ ∘푛−푙 푛−푙+1 ×2 disc(푓 (푥) − 훽) ≡ disc(푓 (푥) − 푓(훽)) (mod 퐾푛−1), 훽∈퐼 where the right hand side is not a square by the induction hypothesis. Therefore the

∘푛−푙 ×2 elements disc(푓 − 훽푖) are linearly independent mod퐾푛−1.

Now we can apply Lemma 2.2.2 to the extensions given by the splitting fields of

푓(푥) − 훼푖 over 퐾푛−1. We have proved that each extension is an 푆푑 extension. The (︁√︀ )︁ unique quadratic subextension of this splitting field is 퐾푛−1 disc (푓(푥) − 훼푖) . ×2 Since disc(푓(푥) − 훼푖) are linearly independent in 퐾푛−1/퐾푛−1, the corresponding quadratic extensions are linearly disjoint.

32 Proposition 2.2.5. Assume that the extension 퐾푛/퐾 is maximal. Then the quadratic subextensions 퐾 ⊂ 퐹 ⊂ 퐾푛 are contained in the compositum of quadratic extensions 퐾(√︀disc(푓 ∘푚(푥) − 푡)) for 푚 = 1, . . . , 푛.

Proof. The group Aut(푇푛) fits into an exact sequence

푑푛−1 1 → (푆푑) → Aut(푇푛) → Aut(푇푛−1) → 1. (⋆)

Let 푠푚 : Aut(푇푛) → 푆푑푚 be the homomorphism given by the action of the automorphism group on 퐼푚 (the 푚’th level of the tree). For 휎 ∈ 푆푑 let 휎푖 be the element 푑푛−1 (1,..., 1, 휎, 1,..., 1) ∈ 푆푑 with 휎 inserted in position 푖. Let 푔 ∈ Aut(푇푛) be −1 푑푛−1 an element such that 푠푛−1(푔)푖 = 푗. Then 푔휎푖푔 is conjugate to 휎푗 in 푆푑 . Let

휒: Aut(푇푛) → {±1} be an arbitrary homomorphism. Consider the restriction of 휒 푑푛−1 to 푆푑 . The map 휒푖 : 푆푑 → {±1} given by 휒푖(휎) = 휒(휎푖) is a quadratic character of 푆푑 which is either the sign character or the trivial character. Since 휎푖 and 휎푗 are conjugate in Aut(푇푛) for all 푖, 푗, the characters 휒푖 and 휒푗 are equal for all 푖, 푗. 푑푛−1 Therefore the restriction of 휒 to 푆푑 is either the trivial character or the product of sign characters. Applying this inductively we can describe all quadratic characters of Aut(푇푛).

We claim that any quadratic character 휒: Aut(푇푛) → {±1} is a product of characters 휒ˆ푚 = sign ∘푠푚, 푚 = 1, . . . , 푛. We prove the claim by induction. The case

푛 = 0 is trivial. Assume the statement is proved for 푛 − 1. Let 휒: Aut(푇푛) → {±1} 푑푛−1 be a character. Consider the exact sequence (⋆). The restriction of 휒 to 푆푑 is either trivial or is equal to the restriction of 휒ˆ푛. Therefore 휒 or 휒 · 휒ˆ푛 descends to a character of Aut(푇푛−1) which is a product of 휒ˆ푚’s by the induction hypothesis.

Quadratic subextensions of 퐾푛/퐾 correspond to quadratic characters of Aut(푇푛).

√︀ ∘푚 The quadratic character corresponding to the extension 퐾( disc(푓 (푥) − 푡)) is 휒ˆ푚.

Since any quadratic character is a product of 휒ˆ푚’s, any quadratic extension is contained in the compositum of 퐾(√︀disc(푓 ∘푚(푥) − 푡)).

The following corollary implies Theorem 2.1.2.

33 Corollary 2.2.6. Let 푓 ∈ 퐾[푥] be a polynomial of even degree 푑. Assume that for every 푘 and every 훼 ∈ 푓 −푘(푡) the Galois group of the splitting field of 푓(푥) − 훼 over 퐾(훼) is either 푆푑 or 퐴푑. Then the arboreal representation 휑푛 : 퐺 → Aut (푇푛) associated to 푓 is surjective if and only if the elements disc(푓 ∘푖(푥) − 푡), 푖 = 1, . . . , 푛 are multiplicatively independent in 퐾×/퐾×2.

Proof. We prove the statement using induction on 푛. The case 푛 = 0 is trivial. As- sume Gal(퐾푛−1/퐾) = Aut(푇푛−1). Then by Theorem 2.2.4 the equality Gal(퐾푛/퐾) = ∘푛 ×2 ∘푛 Aut(푇푛) holds if and only if disc(푓 (푥) − 푡) ̸∈ 퐾푛−1. Since disc(푓 (푥) − 푡) is an ∘푛 ×2 element of 퐾 the condition disc(푓 (푥) − 푡) ̸∈ 퐾푛−1 holds if and only if the extension √︀ ∘푛 퐾( disc(푓 (푥) − 푡)) is not a subfield of 퐾푛−1. By Proposition 2.2.5 any quadratic

subextension of 퐾푛−1/퐾 is inside the field obtained by adjoining to 퐾 square roots of disc(푓 푖(푥) − 푡) for 푖 = 1, . . . , 푛 − 1. Therefore the field 퐾(√︀disc(푓 ∘푛(푥) − 푡)) is

푛 푖 contained in 퐾푛−1 if and only if disc(푓 (푥) − 푡) is in the span of disc(푓 (푥) − 푡), 푖 = 1, . . . , 푛 − 1 in 퐾×/퐾×2.

Remark 2.2.7. Theorem 2.2.4, Proposition 2.2.5 and Corollary 2.2.6 can be stated as results about maximal subgroups of the group Aut(푇푛) without any reference to field theory.

We record here the formula for the discriminant disc(푓 ∘푛(푥) − 푡). The proof can be found in [45, Lemma 3.1].

Lemma 2.2.8. Let 푓 ∈ 퐾[푥] be a polynomial of even degree. Choose an algebraic

′ closure 퐾 of 퐾. Let 휆푖 for 푖 ∈ 퐼 be the roots of 푓 (푥) and 푚푖 be the corresponding multiplicities. Then for all 푡 ∈ 퐾,

∘푛 ∏︁ ∘푛 푚푖 ×2 disc(푓 (푥) − 푡) ≡ (푓 (휆푖) − 푡) (mod 퐾 ) 푖∈퐼

34 2.3 Iterated monodromy groups

Let 푓 ∈ 퐾[푥] be a polynomial and 푡 ∈ 퐾(푡) be an indeterminate. Then the arboreal representation associated to the preimages of 푡 under 푓 is known as the iterated monodromy group of 푓; see [44]. Using Corollary 2.2.6 we can show that, under mild hypotheses, the iterated monodromy group is maximal for even degree polynomials.

Theorem 2.3.1. Let 푓 ∈ 퐾[푥] be an even degree polynomial and let 푡 ∈ 퐾(푡) be an indeterminate. Assume that the Galois group of the splitting field of 푓(푥) − 푡 over

sep sep 퐾(푡) is either 퐴푑 or 푆푑. Choose a separable closure 퐾 . Let Γ ⊂ 퐾 be the multiset ′ ∘푛 ′ of roots of 푓 (푥) and denote by Γ푛 the multiset 푓 (Γ). Let Γ푛 be the sub(multi)set of

Γ푛 which consists of the elements that have odd multiplicity in Γ푛 and assigns to them multiplicity one. Then the arboreal representation attached to 푓 and 푡 is surjective if ′ ⋃︀ ′ and only if Γ푛 ̸⊂ 푖<푛 Γ푖 for all 푛.

−푛 Proof. For every 푛 ∈ Z≥0 any 훼 ∈ 푓 (푡) is transcendental over 퐾. Therefore there is an isomorphism of fields 퐾(훼) ≃ 퐾(푢) for an indeterminate 푢. Since 푓(푥) − 푡 is

irreducible over 퐾(푡) and has Galois group 푆푑 or 퐴푑, the same is true for 푓(푥) − 훼 over 퐾(훼). By Lemma 2.2.8 applied to the field 퐾(푡) the odd multiplicity roots of

∘푛 ′ the polynomial disc(푓 (푥) − 푡) ∈ 퐾[푡] are precisely the elements of Γ푛.

′ Assume that for every 푛 the set Γ푛 contains an element that is not in the union of ′ ∘푛 Γ푖 for 푖 < 푛. Then the polynomials disc(푓 (푥) − 푡) are linearly independent modulo 퐾(푡)×2. Therefore, by Corollary 2.2.6 the arboreal representation associated to 푓 and 푡 is surjective.

′ ⋃︀ ′ ′ ′ Assume that Γ푛 ⊂ 푖<푛 Γ푖 for some 푛. Since 푓(Γ푖) ⊂ Γ푖+1, for every 푚 > 푛 the set ′ ⋃︀ ′ ′ ′ Γ푚 is contained in 푖<푛 Γ푖. Hence for some 푖 ̸= 푗 the sets Γ푖 and Γ푗 are equal. In this case disc(푓 ∘푖(푥) − 푡) ≡ disc(푓 ∘푗(푥) − 푡) (mod 퐾(푡)×2) and the arboreal representation is not surjective by Corollary 2.2.6.

Remark 2.3.2. Assume that 푓(푥) has coefficients in a subfield 푘 ⊂ 퐾 and 푓 ′(푥) is

sep irreducible over 푘. Then Gal(푘 /푘) acts transitively on the multiset Γ푛. Since 푑 − 1

35 is odd, every element of Γ푛 has odd multiplicity, so the support of the multiset Γ푛 is ′ ′ ′ ′ ′ Γ푛. If for some 푛 ̸= 푚 the sets Γ푛 and Γ푚 intersect, then Γ푛 = Γ푚. Therefore the conditions of Theorem 2.3.1 are satisfied if 푓(푥) − 푡 has Galois group 퐴푑 or 푆푑 and 푓 is not post-critically finite (the orbit of the set of critical points is infinite). Thisis the form in which Theorem 2.1.5 was stated.

Remark 2.3.3. If 푓 is an indecomposable polynomial of degree > 31 over a field of characteristic 0, then the Galois group of the splitting field of 푓(푥)−푡 is either 푆푑, 퐴푑, a cyclic group or a dihedral group [41]; see [20] for a positive characteristic version. Moreover, the Galois group can be cyclic or dihedral only when 푓 is conjugate to a Dickson polynomial; see [55, Theorem 3.11]. Together with the previous remark this shows that when 퐾 is a number field most polynomials of even degree satisfy the conditions of Theorem 2.3.1.

2.4 Surjective arboreal representations over Q

In this section we assume 푡 = 0. We can use the results from Section 2.2 to construct surjective even degree arboreal representations over Q. In order to do so we first need a way of showing that the “tiny” extensions given by the splitting fields of 푓(푥) − 훼 over 퐾(훼) for 훼 ∈ 푓 −푛(0) are maximal. In Proposition 2.4.3 we will show that this can be achieved by combining local arguments at two primes.

We use the following convention: if 퐿 is a field with a discrete valuation 푣, and 푓 ∈ 퐿[푥] is a polynomial, then the Newton polygon of 푓 is considered with respect to the equivalent valuation 푣′ that satisfies 푣′(퐿) = Z.

Lemma 2.4.1. Let 퐾 be a field with a discrete valuation 푣, and let 푓 ∈ 퐾[푥] be an

−푛 Eisenstein polynomial of degree at least 2. Then for any 푛 ∈ Z≥0, any 훼푛 ∈ 푓 (0), and any extension of 푣 to 퐾(훼푛) the polynomial 푓(푥) − 훼푛 is Eisenstein over 퐾(훼푛).

Proof. Using induction, if 푔(푥) := 푓(푥) − 훼푛 is Eisenstein and 훼푛+1 is a root of 푔, then 훼푛+1 has the minimal valuation in 퐾(훼푛+1), and thus 푓(푥) − 훼푛+1 is Eisenstein

36 over 퐾(훼푛+1).

Lemma 2.4.2. Let 푝 be a prime number and let 푣 denote the p-푎푑푖푐 valuation on the field Q. Let 푓 ∈ Q[푥] be an irreducible polynomial of degree 푑 ≥ 2. Assume that for some odd integer 푘 > 0 the Newton polygon of 푓 consists of two line segments: one from (0, 2) to (푘, 0) and one from (푘, 0) to (푑, 0). Assume that for every 푛 ∈ Z≥0 and every 훼 ∈ 푓 −푛(0) the polynomial 푓(푥) − 훼 is irreducible over Q(훼). Then for −푛 any 푛 ∈ Z≥0 and any 훼푛 ∈ 푓 (0) there is an extension of 푣 to Q(훼푛) such that the polynomial 푓(푥) − 훼푛 has the same Newton polygon over Q(훼푛)푣 as 푓 has over Q푝.

Proof. We prove the statement by induction. Assume that 푣푛−1 is a discrete valuation on Q(훼푛−1) such that 푔(푥) = 푓(푥) − 훼푛−1 has the Newton polygon of the desired shape. Let 훼푛 be any root of 푔. Consider the extension 푣푛 of 푣푛−1 to Q(훼푛) such that 푣푛(훼푛) = 2/푘. Since the local extension Q(훼푛)푣푛 /Q(훼푛−1)푣푛−1 has degree 푘 and 푣푛(훼푛) = 2/푘, the ramification index of Q(훼푛)/Q(훼푛−1) at 푣푛 is 푘. Therefore ′ ′ the valuation 푣푛 = 푘푣푛 satisfies 푣푛(Q(훼푛)) = Z and hence the Newton polygon of ′ 푓(푥) − 훼푛 at 푣푛 has the two-segment shape as claimed.

Proposition 2.4.3. Let 푓 ∈ Q[푥] be a polynomial of even degree 푑. Let 푝 be a prime satisfying 푑/2 < 푝 < 푑 − 2, and let 푞 ̸= 푝 be an arbitrary prime. Assume that 푓 is Eisenstein at 푞 and that the Newton polygon of 푓 at 푝 is the union of two segments: one from (0, 2) to (푝, 0) and one from (푝, 0) to (푑, 0). Then for every 푛 ∈ Z≥0 and every 훼 ∈ 푓 −푛(0) the Galois group of the splitting field of 푓(푥)−훼 over Q(훼) is either

퐴푑 or 푆푑.

Proof. Fix 푛 and 훼 ∈ 푓 −푛(0). Since 푓 is Eisenstein at 푞, by Lemma 2.4.1 푓(푥) − 훼 is irreducible (and, in particular, separable). Let 퐺 be the Galois group of the splitting field of 푓(푥) − 훼 over Q(훼) and 푟 : 퐺 → 푆푑 be the transitive permutation representation associated to the roots of 푓(푥) − 훼. By Lemma 2.4.2 there is an extension 푣 of the 푝-adic absolute value to Q(훼) such that the Newton polygon of 푓(푥)−훼 contains a segment (0, 2) to (푝, 0). In particular, the splitting field of 푓(푥)−훼 is wildly ramified at 푣 and therefore there exists 푔 ∈ 퐺 of order 푝. Since 푝 > 푑/2 the

37 permutation 푟(푔) is a cycle of length 푝. Since the length of the cycle 푟(푔) is greater

than 푑/2, the permutation group 푟 : 퐺 → 푆푛 is primitive. A theorem of Jordan (see [16, Theorem 3.3E]) states that any primitive permutation group that contains a cycle

fixing at least three points is either 퐴푑 or 푆푑.

We want to construct examples of polynomials over Q with surjective arboreal representation. Corollary 2.2.6 breaks this problem into two: showing that the extensions given by the splitting fields of 푓(푥) − 훼푖 over Q(훼푖) are 푆푑 or 퐴푑 and showing that the discriminants disc(푓 ∘푛(푥)) are independent modulo squares. We use Propo- sition 2.4.3 for the former problem. To deal with the latter we consider the case when 푓 ′(푥) = (푥−퐶)푔(푥)2. In this case Lemma 2.2.8 shows that disc(푓 ∘푛(푥)) equals 푓 ∘푛(퐶) modulo squares. To show that 푓 ∘푛(퐶) are distinct modulo squares we will need to know that they have enough distinct prime divisors. For this the following lemma is useful.

Lemma 2.4.4. Assume 푓 ∈ Q[푥] is integral at a prime 푝, and 푓(0) = 푓(푓(0)).

Assume that for some positive integers 푛 > 푘 and for some 푝-integral 푐 ∈ Z(푝) both 푓 ∘푘(푐) and 푓 ∘푛(푐) are divisible by 푝. Then 푝 divides 푓(0).

Proof. Reducing modulo 푝 gives 0 ≡ 푓 ∘푛(푐) ≡ 푓 ∘푛−푘(푓 ∘푘(푐)) ≡ 푓 ∘푛−푘(0) ≡ 푓(0) (mod 푝).

Now we give explicit examples of polynomials with surjective arboreal representation. Note that when 푑 ≥ 20 is an even number there exists a prime 푝 that satisfies 푑 − 3 ≥ 푝 ≥ 푑/2 + 5 (see for example [43]). This is why we have the condition 푑 ≥ 20 in the following theorem.

Theorem 2.4.5. Fix an even integer 푑 ≥ 20 and a prime number 푝 satisfying 푑−3 ≥

푝 ≥ 푑/2 + 5. Let 푘 := 푑/2 − 1 and 푢 := 푝 − 푘 − 2. Given 퐴, 퐵, 퐶, 퐷 ∈ Q consider the polynomial 푓(푥) such that 푓 ′(푥) = (2푘 + 2)(푥 − 퐶)(푥푘 + 퐴푥푢 + 퐵)2 and 푓(0) = 퐷. Then for infinitely many choices of (퐴, 퐵, 퐶, 퐷) the arboreal representation associated to 푓 and 0 is maximal.

38 Proof. The polynomial 푓 is given by the formula

2푘 + 2 2푘 + 2 2푘 + 2 푓(푥) = 푥2푘+2 − 퐶푥2푘+1 + 2퐴푥푘+푢+2 − 2퐴퐶푥푘+푢+1 2푘 + 1 푘 + 푢 + 2 푘 + 푢 + 1 2푘 + 2 2푘 + 2 2푘 + 2 2푘 + 2 2푘 + 2 + 퐴2푥2푢+2− 퐶퐴2푥2푢+1+ 2퐵푥푘+2− 2퐵퐶푥푘+1+ 2퐴퐵푥푢+2 2푢 + 2 2푢 + 1 푘 + 2 푘 + 1 푢 + 2 2푘 + 2 − 2퐴퐵퐶푥푢+1 + (푘 + 1)퐵2푥2 − (2푘 + 2)퐵2퐶푥 + 퐷. (⋆) 푢 + 1

The monomials are not necessarily ordered by decreasing degree, but the four largest degree and four smallest degree monomials are the first four and the last four respectively (this is one place where we use the condition 푑 − 3 ≥ 푝 ≥ 푑/2 + 5).

We will use 푠-adic properties of the coefficients of 푓 for various primes 푠 to prove surjectivity of the associated arboreal representation. For this we will first choose some appropriate primes, and then use weak approximation to choose 퐴, 퐵, 퐶, 퐷, forcing 푓 to have a specific local behavior.

Choose a prime number ℓ > 푑5, ℓ ≡ 3 (mod 4). Choose another prime 푞 such that 푞 ≡ 1 (mod ℓ) (in particular 푞 > ℓ > 푑5). In order to apply Lemma 2.4.4 we want 퐷 to be a fixed point of 푓. The equation 푓(퐷) = 퐷 is equivalent to the equation 푈(퐴, 퐵, 퐷) − 푉 (퐴, 퐵, 퐷)퐶 = 0 where

2푘 + 2 2푘 + 2 2푘 + 2 푈(퐴, 퐵, 퐷) = 퐷2푘+2 + 2퐴퐷푘+푢+2 + 퐴2퐷2푢+2 + 2퐵퐷푘+2 푘 + 푢 + 2 2푢 + 2 푘 + 2 (2.1) 2푘 + 2 + 2퐴퐵퐷푢+2 + (푘 + 1)퐵2퐷2 푢 + 2

2푘 + 2 2푘 + 2 2푘 + 2 2푘 + 2 푉 (퐴, 퐵, 퐷) = 퐷2푘+1 + 2퐴퐷푘+푢+1 + 퐴2퐷2푢+1 + 2퐵퐷푘+1 2푘 + 1 푘 + 푢 + 1 2푢 + 1 푘 + 1 (2.2) 2푘 + 2 + 2퐴퐵퐷푢+1 + (2푘 + 2)퐵2퐷 푢 + 1

We choose 퐴, 퐵, 푁1 ∈ Z satisfying the following local conditions.

39 prime ℓ Consider the quadratic equation 푈(퐴, 퐵, −푝2) + 푝2푉 (퐴, 퐵, −푝2) = 0. The discriminant of this quadric 푄 in 퐴, 퐵 is

(︂ 푢2(푢 + 1)(12푢2 + 37푢 + 29) )︂ −(2푘 + 2)2푝4푢+4푡+8 2(2푢 + 1)(2푢 + 2)(푢 + 1)2(푢 + 2)2

which is a nonzero element of Q. Since ℓ is large (ℓ > 푑5), the discriminant of 푄 is nonzero modulo ℓ. Also (since ℓ is large) the quadric 푉 (퐴, 퐵, −푝2)

is not proportional to 푄. Let (퐴ℓ, 퐵ℓ) be an Fℓ point on the (affine) conic 2 푄 = 0 that does not lie on 푉 (퐴, 퐵, −푝 ) = 0. Choose 푁1 ≡ 푝 (mod ℓ) and

(퐴, 퐵) ≡ (퐴ℓ, 퐵ℓ) (mod ℓ).

prime 푝 Let 푣푝(푁1) = 푣푝(퐴) = 푣푝(퐵) = 1.

prime 푞 Let 푣푞(퐴) = 푣푞(퐵) = 1 and 푣푞(푁1) = 0.

primes < 푑 For every prime 푠 less than 푑 and not equal to 2 let 푣푠(퐴) = 푣푠(퐵) =

푣푠(푁1) = 푣푠(푑!). For 푠 = 2 assume that 푣2(퐴) = 푣2(퐵) = 0. There exists a

constant 푀 such that when 푣2(푧) > 푀 and 푣2(푥) = 푣2(푦) = 0 the valuation of 푈(푥, 푦, 푥)/푉 (푥, 푦, 푧) is

(︂ (푘 + 1)푦2푧2 )︂ 푣 = 푣 (푧) − 1. 2 (2푘 + 2)푦2푧 2

Let 푣2(푁1) = 푀.

Let 퐾 be the product of primes that are greater than 푑, not equal to 푝, 푞, ℓ, and

divide 푁1. After raising 퐾 to a sufficiently large power we can assume that 퐾 is 푚 1 modulo every prime less than 푑 and also modulo 푝푞ℓ. The triple (퐴, 퐵, 푁1퐾 ) also satisfies the local conditions above for every sufficiently large 푚. Let 퐶(푚) :=

2 2푚 2 2푚 푈(퐴, 퐵, −푞푁1 퐾 )/푉 (퐴, 퐵, −푞푁1 퐾 ). Consider any prime 푠|퐾. From the tri-

angle inequality and formulas (2.1), (2.2) there exists 푀0 such that for 푚 > 푀0 푚 2 2푚 we have 푣푠(퐶(푚)) = 2푣푠(푁1퐾 ). Let 퐷(푚) = −푞푁1 퐾 . Consider the polynomial 푓푚 associated to 퐴, 퐵, 퐶(푚), 퐷(푚) (given by the formula (⋆)). When 푚 2푘+1 1 2푘+2 ∘2 tends to infinity we have 퐶(푚) ∼ 2푘+2 퐷, 푓푚(퐶(푚)) ∼ − 2푘+1 퐶(푚) , 푓푚 (퐶(푚)) ∼

40 1 (2푘+2)2 ′ 푘 푢 2 (2푘+1)2푘+2 퐶(푚) . Since 푓푚(푥) = (푥 − 퐶(푚))(푋 + 퐴푋 + 퐵) the function 푓푚 is decreasing when 푥 ≤ 퐶(푚) and increasing when 푥 ≥ 퐶(푚). There exists 푀1 such ∘2 that for 푚 > 푀1 we have 퐶(푚), 푓푚(퐶(푚)) < 0 and 푓푚 (퐶(푚)) > 0 > 퐶(푚) which ∘푛 implies that 푓푚 (퐶(푚)) > 0 for all 푛 > 1. Finally fix an arbitrary 푚 > max(푀0, 푀1) 푚 and set 퐶 = 퐶(푚), 퐷 = 퐷(푚), 푁 = 푁1퐾 .

Local conditions on 퐴, 퐵, 푁 imply local properties of 퐷 = −푞푁 2. We can also find valuations of 퐶 = 푈(퐴, 퐵, 퐷)/푉 (퐴, 퐵, 퐷) using the non-archimidean triangle inequality.

To summarize, we have chosen 퐴, 퐵, 푁, 퐷 ∈ Z and 퐶 ∈ Q such that the following conditions hold.

1. 푓(퐷) = 퐷, since 퐶 = 푈(퐴, 퐵, 퐷)/푉 (퐴, 퐵, 퐷);

2. 퐷 = −푞푁 2;

3. 퐶 is ℓ-integral and 퐶 ≡ 퐷 (mod ℓ). Since (퐴, 퐵, 푁, 푞) ≡ (퐴푙, 퐵푙, 푝, 1) (mod ℓ) both 퐶 and 퐷 are 푝2 modulo ℓ;

4. Valuations at the prime 푝 satisfy 푣푝(퐷) = 2, 푣푝(퐴) = 푣푝(퐵) = 1, 푣푝(퐶) = 2;

5. Valuations at the prime 푞 satisfy 푣푞(퐴), 푣푞(퐵), 푣푞(퐶) ≥ 1, 푣푞(퐷) = 1;

6. There is a finite set of primes 푆 ⊂ N ∖ {1, . . . , 푑} such that 푓 has coefficients in Z[푆−1], 퐶 is in Z[푆−1], and every prime in 푆 divides the denominator of 퐶. Indeed our conditions for primes less than 푑 imply that the valuation of 퐶 at these primes is nonnegative;

7. 푓(퐶) is negative, and 푓 ∘푛(퐶) is positive for 푛 > 1;

8. For every prime 푠 ̸= 푞 dividing 퐷 we have 푣푠(퐶) ≥ 푣푠(퐷)/2. Indeed, primes dividing 퐷 are primes dividing 푝푞퐾 and primes less than 푑. For each of them the property holds.

We now show that conditions 1 – 8 imply surjectivity of the arboreal representation. Condition 5 implies that 푓 is Eisenstein at 푞, therefore by Lemma 2.4.1 all

41 iterates of 푓 are irreducible. Note that since 퐷 ̸= 0, by condition 1 the point 0 is not periodic under 푓 and the arboreal representation is well-defined. Condition 4 implies that the Newton polygon of 푓 at 푝 is a union two segments: one from (0, 2) to (푝, 0) and one from (푝, 0) to (푑, 0). Hence by proposition 2.4.3 for any 푛 and 훼 ∈ 푓 −푛(0) the

Galois group of the splitting field of 푓(푥) − 훼 over 퐾(훼) is 퐴푑 or 푆푑. Now consider the discriminants disc(푓 ∘푛(푥)) as elements of Q/Q×2. If they are linearly independent, Corollary 2.2.6 would imply that the arboreal representation is surjective. By

Lemma 2.2.8 we have the equality disc(푓 ∘푛(푥)) ≡ 푓 ∘푛(퐶) (mod Q×2). We claim that the numbers 푓 ∘푛(퐶) are independent modulo squares. Indeed 푓(퐶) is negative by condition 7, and therefore is not a square. Consider the number 푓 ∘푛(퐶) for some 푛 > 1. By condition 3 we have the formula 푓 ∘푛(퐶) ≡ 푓 ∘푛(퐷) ≡ 퐷 ≡ −푁 2 (mod ℓ). Therefore 푓 ∘푛(퐶) is not a square modulo ℓ since ℓ ≡ 3 (mod 4). By condition 6 the number 푓 ∘푛(퐶) is 푆-integral. For any prime 푠 ∈ 푆 when (⋆) is evaluated at 푥 = 퐶 the valuation of the sum of the first two terms is smaller than the valuation ofthe

rest of the terms. The non-Archimedean triangle inequality implies that 푣푠(푓(퐶)) 2푘+2 2푘+2 2푘+1 1 2푘+2 is equal to the 푠-adic valuation of (푥 − 2푘+1 퐶푥 )|푥=퐶 = − 2푘+1 퐶 which is

(2푘+2)푣푠(퐶) < 푣푠(퐶). If 푣푠(푥) < 푣푠(퐶), then 푣푠(푓(푥)) = (2푘+2)푣푠(퐶). Therefore the 푠-adic valuation of 푓 ∘푛(퐶) is even and negative. Since 푓 ∘푛(퐶) is positive, has square denominator, and is not a square modulo ℓ, the numerator of 푓 ∘푛(퐶) has an odd

∘푚 multiplicity prime divisor 훾 ̸∈ 푆 that is not a square modulo ℓ. If 푣훾(푓 (퐶)) is even for all 푚 < 푛, then 푓 ∘푛(퐶) is linearly independent from 푓 ∘푚(퐶) when 푚 < 푛. Assume

∘푛 that 푣훾(푓 (퐶)) is odd. Then by Lemma 2.4.4, 훾 divides 퐷. The prime 훾 does not equal 푞, since 푞 ≡ 1 (mod ℓ). Hence condition 2 implies 훾 divides 푁 and condition 8

implies 푣훾(퐶) > 0. By the formula (⋆) and the triangle inequality if 푣훾(푥) ≥ 푣훾(퐷)/2,

then 푣훾(푓(푥)) ≥ 푣훾(퐷), and if 푣훾(푥) ≥ 푣훾(퐷), then 푣훾(푓(푥)) = 푣훾(퐷). Therefore by ∘푛 condition 8 the valuation 푣훾(푓 (퐶)) is equal to 푣훾(퐷) which is even, contradiction. Thus the numbers 푓 ∘푘(퐶) are independent modulo squares.

42 Chapter 3

Sectional monodromy groups

The results of this section have appeared in [29].

3.1 Introduction

Let 퐾 be an algebraically closed field. Consider an integral nondegenerate proper

푟 푟 curve 푋 in the projective space P over 퐾. Choose coordinates 푥0, ..., 푥푟 on P . 푟 Extend scalars from 퐾 to 퐿 = 퐾(푡1, ..., 푡푟) and consider the subscheme 푍 ⊂ P퐿 given by the intersection of 푋퐿 with the hyperplane 푥0 + 푡1푥1 + ··· + 푡푟푥푟 = 0. The scheme 푍 is the spectrum of a finite separable extension 푀 of 퐿 (see Lemma 3.2.1). The Galois group 퐺 of the Galois closure of 푀 over 퐿 is our main object of interest and will be referred to as the sectional monodromy group. A folklore result is the following.

Proposition 3.1.1 (see Remark 3.3.3). Assume that the characteristic of 퐾 is zero. ∼ Let 푑 be the degree of 푋. Then 퐺 = 푆푑.

The result is important in the study of curves in projective space. It can be used to restrict the possibilities for Hilbert polynomials of space curves. For more details see [21], [17].

The char 퐾 = 0 assumption of Proposition 3.1.1 cannot be removed. For example,

푝 푝−1 the sectional monodromy group of the rational plane curve 푥 = 푦푧 is AGL1(푝).

43 Motivated by geometric applications, Rathmann [47] proved a version of Proposi- tion 3.1.1 valid in all characteristics.

Theorem 3.1.2 (Rathmann). Assume that 푋 ⊂ P푟 is a smooth nondegenerate curve of degree 푑. Assume 푟 > 4. Then the sectional monodromy group of 푋 contains the alternating group 퐴푑.

Remark 3.1.3. Gareth Jones noticed that Rathmann’s paper [47] contains a group- theoretic error: the list of triply transitive groups in [47] Theorem 2.4 is incomplete. It is unclear if this gap is easy to fix. For instance, one of the omitted Mathieu groups

Aut(푀22) is not contained in the alternating group, so the proof of [47] Proposi- tion 2.14 requires a separate analysis of this case.

Remark 3.1.4. The conclusion of Theorem 3.1.2 does not hold for 푟 = 3: Rathmann

3 gave an example of a smooth rational curve in P with 퐺 = PGL2(푞).

Rathmann’s theorem is important for understanding statistics over function fields, for example the large characteristic version of the Bateman-Horn conjecture (see [18] Section 6 for examples of such statistical problems). It is thus desirable to have a version of Theorem 3.1.2 that applies to a wider class of curves, especially to curves in the projective 3-space. In Section 3.3 we prove an extension of Rathmann’s theorem that applies to nonsmooth curves in P3. Recall that a projective curve is called strange if all of its tangent lines pass through a common point. Strange curves are rare: for example, the only smooth strange curves are lines in any characteristic and conics in characteristic 2. The tangent variety of 푋 (also called the tangent developable surface) is defined to be the Zariski closure of the union of lines tangent to 푋.

Theorem 3.1.5. Assume that 푋 ⊂ P푟 is an integral nondegenerate curve of degree 푑. Assume that 푟 > 3 and 푋 is not strange. Then exactly one of the following is true.

1. The sectional monodromy group of 푋 contains 퐴푑.

2. 푟 = 3 and 푋 is projectively equivalent to the projective closure of the rational

44 푞 푞+1 3 curve {(푡, 푡 , 푡 ), 푡 ∈ 퐾} ⊂ A퐾 for some power 푞 of the characteristic. The

sectional monodromy group of 푋 is PGL2(푞).

3. 푟 = 3, char 퐾 = 2, the tangent variety of 푋 is a smooth quadric, and 퐺 ⊂

AGL푛(2).

Note that the curves that satisfy the assumptions of Theorem 3.1.5 and have small sectional monodromy groups are either rational or contained in a smooth quadric. Rational curves and curves lying on quadrics are not important for the applications to the degree-genus problem.

For plane curves, sectional monodromy groups are not known even in the following very simple case.

푛 푚 푛−푚 Example 3.1.6. Let 푋푛,푚 be the rational curve 푥 = 푦 푧 for some relatively prime integers 푛 > 푚 > 0. Then the sectional monodromy group of 푋 is the Galois group of the trinomial 푥푛 + 푎푥푚 + 푏 over 퐾(푎, 푏). Classifying possible Galois groups of generic trinomials is an old problem; see [14], [51], [56], [52]. A closely related problem of classifying Galois groups of trinomials over the function field 퐾(푡) when 푎 and 푏 are specialized to powers of 푡 has also been widely studied; see [2], [4], [3], [5], [7].

The curves 푋푚,푛 do not satisfy both of the assumptions of Theorem 3.1.5: they are evidently plane curves (푟 = 2) and many of them are strange. Nevertheless, we can use methods similar to those in the proof of Theorem 3.1.5 to study the curves

푋푚,푛. In section 3.5 we prove the following theorem.

Theorem 3.1.7. Let 퐾 be an algebraically closed field of characteristic 푝. Let 푛, 푚 be a pair of relatively prime integers. Let 퐺 be the Galois group of the polynomial 푥푛 + 푎푥푚 + 푏 over 퐾(푎, 푏).

푑 푑 푑 1. If 푛 = 푝 and 푚 = 1 or 푚 = 푝 − 1, then 퐺 = AGL1(푝 ).

2. If 푛 = 6, 푚 = 1 or 푚 = 5, and 푝 = 2, then 퐺 = PSL2(5).

45 3. If 푛 = 12, 푚 = 1 or 푚 = 11, and 푝 = 3, then 퐺 = 푀11 (a Mathieu group, see [15]).

4. If 푛 = 24, 푚 = 1 or 푚 = 23, and 푝 = 2, then 퐺 = 푀24.

5. If 푛 = 11, 푚 = 2 or 푚 = 9, and 푝 = 3, then 퐺 = 푀11.

6. If 푛 = 23, 푚 = 3 or 푚 = 20, and 푝 = 2, then 퐺 = 푀23.

7. Let 푞 := 푝푘 for some integer 푘. If 푛 = (푞푑 − 1)/(푞 − 1), 푚 = (푞푠 − 1)/(푞 − 1) or

푠 푚 = 푛 − (푞 − 1)/(푞 − 1) for some integers 푠, 푑, then 퐺 = PGL푑(푞).

8. If none of the above holds, then 퐴푛 ⊂ 퐺.

3.2 Transitivity of Sectional Monodromy Groups

From now on, let 퐾 denote a fixed algebraically closed field. We first give amore geometric definition of the sectional monodromy groups from the introduction. Let

푟 푟 > 2 be an integer and let 푋 ⊂ P be a degree 푑 projective nondegenerate integral 푟 * curve. Consider the incidence scheme 퐼 = 퐼푋 ⊂ 푋 ×(P ) parameterizing pairs (푃, 퐻) of points 푃 ∈ 푋 and hyperplanes 퐻 ⊂ P푟 subject to the condition 푃 ∈ 퐻. The scheme 푟 * 푍 from the introduction is the generic fiber of the projection 휋 = 휋푋 : 퐼푋 → (P ) .

We call the covering 휋푋 the sectional covering associated to 푋.

Lemma 3.2.1. The incidence scheme 퐼 is irreducible. The morphism 휋 is finite of degree 푑 and generically étale.

Proof. The fibers of the first coordinate projection 퐼 → 푋 are projective spaces of dimension 푟 −1; hence 퐼 is irreducible. The morphism is generically étale by Bertini’s theorem [27] Proposition 6.4, and finite since it is quasi-finite and proper.

Lemma 3.2.1 implies that the sectional monodromy group 퐺 is well-defined and

acts transitively on a set of 푑 points. If 푈 ⊂ (P푟)* is a dense open subset over which 휋 is étale and Ω is a fiber of 휋 over a geometric point of 푈, then 퐺 can be identified with

46 et the image of 휋1 (푈) in Sym(Ω). If 푍 ⊂ 푈 is a closed subset, then the monodromy group of 휋−1(푍) → 푍 is a permutation subgroup of 퐺. We now show that the action of 퐺 on Ω is always 2-transitive, and has even higher transitivity degree if the curve is not “ strange”.

Lemma 3.2.2. The sectional monodromy group acts doubly transitively.

Proof. Choose a hyperplane 퐻 that intersects 푋 at 푑 distinct smooth points. Let 푃 be one of the points in 퐻 ∩푋. Choose an (푟 −2)-dimensional subspace 푉 in P푟 such that 푉 ∩ 푋 = 푃 . Consider the line 푙 ⊂ (P푟)* corresponding to the family of hyperplanes containing 푉 . Let 푍 = 휋−1(푙). Projection of 푍 to 푋 is an isomorphism over 푋 ∖ 푃 . The fiber above 푃 is isomorphic to 푙 (via 휋). Therefore 푍 decomposes into a union of two irreducible components 푍 = 푍1 ∪ 푍2, where 휋 : 푍1 → 푙 is an isomorphism ′ and 푍2 is isomorphic to 푋. Since the fiber of 휋 = 휋|푍2 : 푍2 → 푙 above the point of 푙 corresponding to 퐻 is smooth, the covering 휋′ is generically étale. Therefore, the monodromy group 퐺 contains a subgroup that acts via fixing one point and acting transitively on the others. Thus 퐺 is doubly transitive.

For every 푟 there are examples of integral nondegenerate curves 푋 ⊂ P푟 for which

퐺푋 is not triply transitive (see [47] Example 1.2). However we will prove that 퐺푋 is 푟-transitive if 푋 is not “strange”. We recall the definition of a strange curve.

Definition 3.2.3. An integral curve 푋 in P푟 is called strange if the tangent lines to 푋 at the smooth points pass through a common point.

Remark 3.2.4. Equivalently, an integral curve is strange if the tangent lines to 푋 at all but finitely many smooth points pass through a common point.

A secant of a curve 푋 is a line intersecting it in at least 2 points.

Proposition 3.2.5. If every secant intersects 푋 in at least three points, then 푋 is strange.

Proof. See [23] Proposition 3.3.8.

47 Most of the results that follow concern curves that are not strange. It is natural to expect higher transitivity for nonstrange curves. This is the case, as we show in Proposition 3.2.8. We precede the proof of Proposition 3.2.8 with two lemmas.

푟 Lemma 3.2.6. Let 푋 ⊂ P , 푟 > 3 be a proper nondegenerate nonstrange integral curve. Let ℒ denote the set of lines in P푟 that have nonempty intersection with every tangent to 푋. Then either ℒ is finite or 푟 = 3, 푋 is contained in a smooth quadric, and ℒ is one of the quadric’s rulings.

Proof. Assume that ℒ is infinite. Consider two distinct lines ℓ1, ℓ2 ∈ ℒ. If ℓ1 and ℓ2

intersect, then either every tangent to 푋 passes through ℓ1 ∩ ℓ2 or every tangent to 푋

is in the plane containing ℓ1 and ℓ2. Since 푋 is nondegenerate and nonstrange, neither of these is possible. Therefore any two distinct lines from ℒ are skew. Consider two

skew lines ℓ1, ℓ2 ∈ ℒ. Since every tangent of 푋 crosses both of them, the curve 푟 푋 lies in the dimension three projective subspace of P that contains ℓ1, ℓ2. Hence 푟 = 3. A classical fact, which we prove below, is that the union of lines that intersect three pairwise skew lines is a smooth quadric. Thus there exists a smooth quadric 푄 containing every tangent line of 푋. Elements of ℒ are pairwise skew and lie on 푄, therefore ℒ forms one of the rulings of 푄.

We now show that the union of lines intersecting three fixed lines in P3 is a smooth quadric. Since 10 = dim 퐻0(P3, 풪(2)) > 3 dim 퐻0(P1, 풪(2)) = 9, there exists a quadric containing any triple of lines. If the lines are pairwise skew, the quadric cannot be a cone or a pair of planes; therefore it must be smooth. Every line intersecting a smooth quadric in at least three points has to lie on it.

푟 Lemma 3.2.7. Let 푋 ⊂ P , 푟 > 3 be a nonstrange nondegenerate integral curve. If 푟 = 3 assume additionaly that 푋 is not contained in a quadric. Then the projection from a general point 푃 of 푋 is birational onto its image. Moreover, the image of a general projection is nonstrange.

Proof. Consider the set ℒ of Lemma 3.2.6. Since 푋 is not contained in a quadric, ℒ is finite. Proposition 3.2.5 implies that a general point of 푋 lies on finitely many

48 multisecants. Therefore projection from a general point is generically one-to-one on 퐾 points and is thus birational. Let 푃 ∈ 푋(퐾) be a point such that the projection

휋푃 : 푋 → 푌 from 푃 is birational onto its image 푌 , and such that 푃 does not lie on a line from ℒ. Suppose that 푌 is strange and all the tangents pass through 푄 ∈ P푟−1. −1 푟 Then the line 휋푃 (푄) ⊂ P is an element of ℒ that contains 푃 . Thus 푌 is nonstrange.

푟 Proposition 3.2.8. If 푋 ⊂ P , 푟 > 2 is an integral nondegenerate nonstrange curve, then 퐺 is 푟-transitive.

Proof. We prove the statement by induction on 푟. The case 푟 = 2 is Lemma 3.2.2. Assume 푟 > 2. Assume first that it is not the case that 푟 = 3 and 푋 lies on a quadric. Choose a point 푃 ∈ 푋(퐾) such that the conclusions of Lemma 3.2.7 hold.

Let 푌 ⊂ P푟−1 be the projection of 푋 from 푃 . Consider the hyperplane 퐻 ∈ (P푟)* corresponding to the family of hyperplanes in P푟 passing through 푃 . Let 푍 denote 휋−1(퐻). The scheme 푍 has an irreducible component 푍′ = 푃 × 퐻 ⊂ 푋 × (P푟)*. The covering 푍 ∖푍′ → 퐻 is birational to the sectional covering of 푌 , since the preimages of hyperplanes in P푟−1 are hyperplanes in P푟 containing 푃 . By the induction hypothesis the monodromy group of the covering 푍 ∖ 푍′ → 퐻 is 푟 − 1-transitive. The covering 푍′ → 퐻 is an isomorphism. Therefore the monodromy group of 푍 → 퐻 acts by fixing one point and acting (푟 − 1)-transitively on the rest. Since 퐺 acts transitively and contains a subgroup acting (푟 − 1)-transitively on all but one (fixed) point, 퐺 is 푟-transitive.

We have to show that when 푟 = 3 and 푋 lies on a quadric 푄 the group is 3- transitive. Consider a generic plane 퐻. We can assume that both 퐻 ∩ 푋 and 퐻 ∩ 푄 are smooth. Take two points of 퐻 ∩ 푋 and a line ℓ through them. The line ℓ contains

* 3 * exactly two points 푃1, 푃2 of the curve. Consider the line ℓ ∈ (P ) corresponding to the hyperplanes containing ℓ. The restriction of the sectional covering of 푋 to ℓ* is generically étale since 퐻 ∩ 푋 is smooth. Let 푌 ⊂ 푋 × ℓ* denote the preimage of

ℓ under 휋푋 . The projection of 푌 to 푋 is an isomorphism over 푋 ∖ {푃1, 푃2}. The * preimages of 푃1, 푃2 in 푌 are each isomorphic to ℓ . Therefore 푌 decomposes into a

49 ⋃︀ ⋃︀ union of irreducible components 푌 = 푌1 푌2 푌3, where 푌1 is isomorphic to 푋 and * 푌2, 푌3 are isomorphic to ℓ . Hence the monodromy group contains a subgroup that acts by fixing two points and acting transitively on the rest. Therefore 퐺 is triply transitive.

Multiple transitivity is a very strong condition on the group. For example, the

only 5-transitive groups are 퐴푛 and 푆푛 in their natural actions and the Mathieu

groups 푀12 and 푀24 acting on 12 and 24 points respectively. The classification of 2-transitive groups is known; see, for example, [16] §7.7 for the statement.

Remark 3.2.9. It is common to state classification results on permutation groups without specifying the action (or by referring to the “natural” action). This is relatively harmless except for the cases of 푀11, which has two multiply-transitive actions on

11 and 12 points, and of 푀12, which has two multiply transitive actions on 12 points exchanged by an outer automorphism.

The following proposition due to Rathmann describes the behavior of the sectional monodromy group with respect to projections from points outside of the curve.

푟 Proposition 3.2.10 ([47] Proposition 1.10.). Let 푋 ⊂ P , 푟 > 3 be a projective nondegenerate integral curve. Let 푌 be the projection of 푋 from a general point of

P푟. Then the sectional monodromy groups of 푋 and 푌 are isomorphic.

3.3 Tangents and inertia groups

We will compute sectional monodromy groups by restricting the sectional monodromy

푟 * covering 휋푋 to a curve in (P ) and computing inertia groups. A natural choice of a point in (P푟)* above which the restricted covering ramifies is a point corresponding to a hyperplane containing a tangent line of 푋. Tangent hyperplanes can behave pathologically in characteristic 푝 (even for nonstrange curves).

For the statements and precise definitions in this paragraph see [24] and references therein. Let 푋 ⊂ P푟 be an integral projective curve that is not a line. The dual curve

50 푋* ⊂ Gr(2, 푟 + 1) is defined to be the closure of the set of tangent linesto 푋. The

* Gauss map is the rational map γ : 푋 99K 푋 that sends a point to the tangent line at that point. In characteristic 0 the Gauss map is birational. In characteristic 푝 both the separable degree 푠 and the inseparable degree 푞 of γ can be greater than one. A general tangent line of 푋 has 푠 points of tangency with multiplicity max(푞, 2) at each one. It is a theorem of Kaji that Gauss maps can be arbitrary bad in the sense that any finite inseparable extension of function fields can be realized as the generic fiber of a Gauss map; see [32] Corollary 3.4.

In Lemma 3.3.2 we use inertia groups to construct special subgroups of the sectional monodromy group; together Proposition 3.2.8 and Lemma 3.3.2 will reduce most of the computation of sectional monodromy groups to pure group theory. First we need to prove the following lemma.

Lemma 3.3.1. Let (퐹, 푣) denote the valued field 퐾((푡)). Let 풪 denote its ring of integers 풪 = 퐾[[푡]]. Let 푃 ∈ 풪[[푥]] be an arbitrary power series and let 푞 be a power

푞+2 푞+1 푞 of 푝. Consider a power series 푄 = 푥 푃 (푥) + 푎푞+1푥 + 푎푞푥 + 푎1푥 + 푎0 ∈ 풪[[푥]], where 푣(푎0) = 푣(푎1) = 1, 푣(푎푞) = 0. Assume that 푣(푎1푎푞 − 푎0푎푞+1) = 1. Assume that all roots of 푄 in the open unit disk 퐷 := {푥 ∈ 퐹 : 푣(푥) > 0} lie in 퐹 sep, so 푄 has exactly 푞 roots in 퐷. The Galois action on these roots defines a homomorphism

휙: Gal(퐹 /퐹 ) → 푆푞. Let 퐻 denote the image of 휙. Then 퐻 equals AGL1(푞) embedded via its action on the affine line..

Proof. The Newton polygon of 푄 has a unique segment of positive slope, it connects (0, 1) and (푞, 0). Therefore 푄 has 푞 roots in the open unit disk each with valuation 1/푞. Let 훼 be a root of 푄. The field extension 퐿 = 퐹 (훼)/퐹 has ramification index at least 푞; therefore [퐿 : 퐹 ] = 푞 and 퐻 acts transitively. Consider the roots of 푄(푥 + 훼) over 퐿. We have

푞 2 푞 2 푞 푄(푥 + 훼) = 푥 (푥 + 훼) 푃 (푥 + 훼) + 훼 (푥 + 훼) 푃 (푥 + 훼) + 푎푞+1푥 (푥 + 훼)

푞 푞 푞 + 푎푞+1훼 (푥 + 훼) + 푎푞푥 + 푎1푥 + 푎1훼 + 푎푞훼 + 푎0.

51 The constant term must be zero since 훼 is a root of 푄. The constant term of 푄(푥 + 훼)/푥 is

푞+1 푞+2 ′ 푞 2훼 푃 (훼) + 훼 푃 (훼) + 푎푞+1훼 + 푎1.

푞 푞 푞+1 Assume 푣(푎푞+1훼 + 푎1) > 1. Then 푎푞+1훼 + 푎1 ≡ 0 (mod 훼 ). Reducing the 푞+1 푞 푞+1 equation 푄(훼) = 0 modulo 훼 , we conclude that 푎0 + 푎푞훼 ≡ 0 (mod 훼 ). Hence 푞 푞+1 푞 푞+1 훼 ≡ −푎0/푎푞 (mod 훼 ) and 푎1 − 푎0푎푞+1/푎푞 ≡ 푎푞+1훼 + 푎1 ≡ 0 (mod 훼 ). But 푞 푣(푎푞+1푎0/푎푞 −푎1) = 1, contradiction. Thus 푣(푎푞+1훼 +푎1) = 1, and the constant term of 푄(푥 + 훼)/푥 has valuation 1.

∑︀ 푘 The power series 푄(푥 + 훼)/푥 =: 푘 푏푘푥 has constant term 푏0 of valuation 1. 푞 푞−1 2 푞−1 푞−1 Modulo 훼 we have 푄(푥 + 훼)/푥 ≡ 푥 (푥 + 훼) 푃 (푥) + 푎푞+1푥 (푥 + 훼) + 푎푞푥 푞 (mod 훼 ). So for 1 6 푘 6 푞 − 2 the valuation of 푏푘 is at least one. The valuation of

푏푞−1 is zero. Therefore the Newton polygon of 푄(푥 + 훼)/푥 has only one segment of positive slope, it connects (0, 1) to (푞 − 1, 0). The field 퐾((푡)) has a unique extension of degree 푞 − 1, namely 퐾((푡1/푞−1)). We conclude that the stabilizer of a point in 퐻 is a cyclic group of order (푞 − 1). A cycle in an imprimitive group has to either belong to a block or contain the same number of elements in every block. Since 푞 − 1 and 푞 are relatively prime, a (푞 − 1)-cycle belongs to a single block. So the group cannot be imprimitive. Since 퐹 is a nonarchimedean local field, 퐻 is solvable. Theorem 1.2.5 implies that in this case AGL1(푞) ⊂ 퐻 ⊂ AΓL1(푞). The group AΓL1(푞) is the group of semilinear transformation of the affine line, namely the group generated by AGL1(푞) and the Frobenius automorphism. Since the stabilizer of a point in 퐻 is a cyclic group, 퐻 = AGL1(푞).

Lemma 3.3.2. Let 푋 be a nondegenerate nonstrange integral plane curve of degree 푑. Assume that the Gauss map of 푋 has separable degree 푠 and inseparable degree 푞.

Choose a geometric point 푥 above which the covering 휋푋 is étale. Denote by Ω the 2 * fiber of 휋푋 above 푥 ∈ (P ) , so the group 퐺푋 acts on Ω by permutations and #Ω = 푑. Then the following hold:

52 1. If 푞 = 1, then 퐺 contains a transposition.

2. If 푞 > 1, then there is a decomposition Ω = 퐴1 ⊔ ... ⊔ 퐴푠 ⊔ 퐵, #퐴푖 = 푞, #퐵 =

푑 − 푠푞, a subgroup 퐻 ⊂ 퐺 and a collection of subgroups 퐻푖 ⊂ 퐺, 푖 = 1, ..., 푠 such that the following properties hold.

∙ 퐻 fixes every point of 퐵 and preserves the decomposition Ω = 퐴1 ⊔ ... ⊔

퐴푠 ⊔ 퐵.

∙ The image of 퐻 in the symmetric group Sym(퐴푖) equals AGL1(푞) embedded via its action on the affine line.

∙ There exists an element ℎ ∈ 퐻 that acts on each 퐴푖 as a (푞 − 1)-cycle (fixing one point).

∙ 퐻푖 preserves the decomposition Ω = 퐴1 ⊔...⊔퐴푠 ⊔퐵, fixes 퐵∪퐴푖 pointwise,

and acts transitively on each 퐴푗 for 푗 ̸= 푖.

2 * Proof. Let 푈 ⊂ (P ) be a dense open subset over which the covering 휋푋 is étale. The complement of 푈 contains finitely many lines. A line in (P2)* defines a point (P2)** = P2. Let 푆 ⊂ P2 be the set of points corresponding to lines in the complement of 푈. Then for every point 푃 ̸∈ 푆 the restriction of 휋 to the family of lines passing through 푃 is generically étale. A general tangent to 푋 is tangent at exactly 푠 points with multiplicity max(2, 푞) at each one. Since 푋 is not strange, a general tangent line does not intersect the set of singular points Sing푋 of 푋 and does not intersect

푆. Choose a tangent line ℓ to 푋 that does not intersect 푆 ∪ Sing푋 and such that ℓ ∩ 푋 has 푠 points with multiplicity max(2, 푞) and 푑 − 푞푠 points with multiplicity 1. In particular for every point 푃 on ℓ the restriction of 휋 to the family of lines passing through 푃 is generically étale.

Let 푃1, ..., 푃푠 be the points of tangency and 푄1, ..., 푄푑−푞푠 be the rest of the inter- 2 ⋃︀ ⋃︀ section. Choose a line in P that is disjoint from both 푖 푃푖 and 푗 푄푗. Consider the affine plane A2 obtained by removing the chosen line. Choose coordinates 푥, 푦 in A2 so that ℓ is the line 푦 = 0, and the coordinates of 푃1, ..., 푃푠 are (훼1, 0), (훼2, 0), ..., (훼푠, 0). Choose an algebraic closure of the fraction field of the local ring at [ℓ] ∈ (P2)*; identify

53 Ω with the fiber at the geometric point corresponding to this choice. Let 퐴푖 ⊂ Ω be ⋃︀ the set that reduces to 푃푖, and let 퐵 = Ω ∖ 푖 퐴푖 be the rest of Ω. For an element

훼 ∈ 퐾 consider the family of lines ℓ푡 = ℓ푡(훼) given by the equation 푦 = 푡(푥 − 훼).

The family ℓ푡(훼) is the family of lines passing through (훼, 0), so the covering 휋 is 2 * generically étale above the line corresponding to ℓ푡(훼) in (P ) . We want to analyze inertia groups at 푡 = 0 of the restriction of 휋 to these families.

1. Assume 푞 = 1. In this case char 퐾 ̸= 2, see [33] Proposition 3.3. We need to describe the extension of 퐾((푡)) obtained by adjoining the coordinates of the

intersection of 푋 with 푦 = 푡(푥 − 훼). Choose a uniformizer 푥푖 = 푥 − 훼푖. For

each point 푃푖 consider the local uniformization 푦 = 푝푖(푥푖) of 푋 with respect

to 푥푖, where 푝푖(푥푖) is a power series with coefficients in 퐾. Since 푞 = 1 the

power series 푝푖(푥푖) vanishes at 0 with multiplicity max(푞, 2) = 2. When 훼 ̸= 훼푖

the Newton polygon of 푄푖 = 푝푖(푥푖) − 푡푥푖 − 푡훼 + 푡훼푖 has a unique positive slope

segment, it connects (0, 1) and (2, 0). Hence the positively valued roots of 푄푖

have valuation 1/2. Therefore when 훼 ̸= 훼푖 the extension obtained by adding

the positively valued roots of 푄푖 to 퐾((푡)) is the (unique) ramified degree 2

extension. Similarly, if 훼 = 훼푖 the equation 푄푖 = 0 has two positively valued

roots, both lying in 퐾((푡)). Taking 훼 to be distinct from each 훼푖 gives an element

푔 ∈ 퐺 that fixes 퐵 and acts on each 퐴푖 as a transposition. Taking 훼 = 훼1 gives

an element 푔1 ∈ 퐺 that acts by identity on 퐵 ∪ 퐴1 and as a transposition on

퐴푗 for all 푗 ̸= 1. The product 푔푔1 is a transposition.

2. Now assume 푞 > 1 is a power of the characteristic. This statement is similar to part (1), except the degrees of the extensions are no longer 2 and some wild

ramification appears. Take 푥푖 = 푥 − 훼푖 as a uniformizer for 푋 at 푃푖. Let 푞 푞+1 (푥푖, 푝푖(푥푖)) be the uniformization and define 푐푖, 푑푖 by 푝푖(푥푖) = 푑푖푥 + 푐푖푥 + 푞+1 푂(푥 ). Choose 훼 ∈ 퐾 so that both 푑푖 + (훼 − 훼푖)푐푖 ̸= 0 and 훼 ̸= 훼푖 hold for every 푖. Consider the intersection of 푋 with the line 푦 = 푡(푥 − 훼) over the field 퐾((푡)). The 푥-coordinates of the intersection points that reduce to

푃푖 are the positively valued roots of 푄푖(푥) = 푝푖(푥푖) − 푡푥푖 + 푡(훼 − 훼푖). Since

54 훼 ̸∈ 푆, the intersection of 푋 with 푦 = 푡(푥 − 훼) over 퐾(푡) defines a separable

extension of 퐾(푡). Therefore the positively valued roots of 푄푖 lie in a separable

extension of 퐾((푡)). Lemma 3.3.1 applies to the power series 푄푖 (condition

푣(푎1푎푞 −푎0푎푞+1) = 1 becomes 푑푖 +(훼−훼푖)푐푖 ̸= 0). Let 퐿푖/퐾((푡)) be the extension

obtained by adjoining all positive valuation roots of 푄푖 to 퐾((푡)). Then the

Galois group of 퐿푖/퐾((푡)) is AGL1(푞). The ramification index of 퐿푖/퐾((푡)) is 1/(푞−1) divisible by 푞(푞 − 1). In particular every field 퐿푖 is an overfield of 퐾((푡 )).

Let 퐻 be the Galois group of the compositum of 퐿푖, 푖 = 1, ..., 푠. Take an element ℎ ∈ 퐻 that surjects onto a generator of Gal(퐾((푡1/(푞−1)))/퐾((푡))). Every such

element ℎ will surject onto an element of order 푞 − 1 on each Gal(퐿푖/퐾((푡))). We now examine the possibilities for the action of ℎ on the positive valued

roots of 푄푖. Assume that for some fixed 푖 the element ℎ has an orbit of length

less than 푞 − 1 on the roots of 푄푖. The extension 퐿푖/퐾((푡)) is generated by

adjoining any two roots of 푄푖. Assume that ℎ has an orbit of length 푘 with 1 < 푘 < 푞 − 1. Then ℎ푘 fixes at least two roots and therefore is the identity

element in Gal(퐿푖/퐾((푡))). Contradiction. Therefore ℎ acts on each 퐴푖 by fixing one point and moving the rest cyclically.

Remark 3.3.3. When char 퐾 = 0 Lemmas 3.3.2 and 3.2.2 imply that 퐺 is 2-transitive

and contains a transposition, so 퐺 = 푆푑.

3.4 Sectional monodromy groups of nonstrange curves

The main result of this section is Theorem 3.4.3 that describes sectional monodromy groups of nonstrange space curves.

The sectional monodromy group of a nonstrange space curve is triply transitive by Proposition 3.2.8. Triply transitive permutation groups are completely classified in Theorem 1.2.3. We will need two geometric statements before proving Theorem 3.4.3.

55 푟 Proposition 3.4.1 (Kaji). Let 푋 ⊂ P , 푟 > 3 be a projective integral nonstrange * 푟−1 curve, let γ푋 : 푋 → 푋 be its Gauss map. Let 푌 ⊂ P be the projection of 푋 from a general point 푃 ∈ P푟. Identify 퐾(푋) and 퐾(푌 ) via the projection from 푃 . Let * * γ푌 : 푌 → 푌 denote the Gauss map of 푌 . Then the subfield 퐾(푋 ) of 퐾(푋) defined * by γ푋 and the subfield 퐾(푌 ) of 퐾(푋) = 퐾(푌 ) defined by γ푌 : 푌 coincide.

Proof. See [32] Theorem 2.1 and the remark following the theorem.

Proposition 3.4.2 ([9] Theorem 3.1). Let 푋 ⊂ P3 be a smooth curve of degree 푑. Assume that for a general point 푃 of 푋 there is no tangent line to 푋 at a point 푄 ̸= 푃 that intersects the tangent line at 푃 . Then either 푑 = 3 and 푋 is a twisted cubic or 푑 = 푞 + 1 for some power 푞 of the characteristic. In the latter case 푋 is rational and isomorphic to the projective closure of the parametrized curve (푡, 푡푞, 푡푞+1).

The following theorem is the main result of this section.

푚 Theorem 3.4.3. Let 푋 ⊂ P , 푚 > 3 be a nondegenerate integral nonstrange curve of degree 푑. Then one of the following holds.

1. The sectional monodromy group 퐺 contains the alternating group 퐴푑.

2. 푚 = 3 and 푋 is projectively equivalent to the rational curve given as the pro-

jective closure of the parametrized curve (푡, 푡푞, 푡푞+1) ⊂ A3 for some power 푞 of

the characteristic. In this case 퐺 = PGL2(푞).

3. 푚 = 3, char 퐾 = 2, the tangent variety of 푋 is a smooth quadric and 퐺 is

contained in AGL푛(2).

Proof. Let 푠, 푞 be the separable and inseparable degrees of the Gauss map γ푋 of 푋. Recall that for a general tangent line ℓ of 푋 the degree of the scheme-theoretic intersection of 푋 and ℓ satisfies deg 푋 ∩ ℓ = deg γ = 푠푞. Choose one such general tangent line ℓ and choose a point 푃 of 푋 that does not lie on ℓ. Let 퐻 be a hyperplane containing ℓ and 푃 . The degree of the intersection of 퐻 with 푋 satisfies 푑 = deg 푋 =

deg 퐻 ∩ 푋 > deg ℓ ∩ 푋 + 1 = 푠푞 + 1. Therefore 푠푞 < 푑.

56 By Proposition 3.4.1 푠 and 푞 are also the separable and inseparable degrees of a

general projection of 푋. Let 푋′ be a general projection of 푋 to P2. Proposition 3.2.10 ′ implies that 퐺푋 = 퐺푋′ as permutation groups. Since 푋 is a plane curve, Lemma 3.3.2 applies. We conclude that if 푞 = 1 the group 퐺 contains a transposition, and therefore

퐺 = 푆푛. Assume 푞 > 1 and fix a decomposition Ω = 퐴1 ⊔ ... ⊔ 퐴푠 ⊔ 퐵, subgroups

퐻, 퐻푖 ⊂ 퐺, and an element ℎ ∈ 퐻 as in Lemma 3.3.2. We will use the properties of

(퐴푖, 퐵, 퐻푖, 퐻, ℎ) stated in Lemma 3.3.2 throughout this proof. Since 푠푞 < 푑 the set 퐵 is nonempty.

The group 퐺 is triply transitive by Proposition 3.2.8. Therefore it is on the list of Proposition 1.2.3. We analyze each possibility.

푛 Case (a): 퐺 ⊂ AGL푛(2) and Ω = F2 . This case also covers the 퐺1 group of Proposi-

tion 1.2.3, since 퐺1 is a permutation subgroup of AGL4(2). Assume first that 푞 ̸= 2. The element ℎ ∈ 퐺 acts as a product of 푠 (푞 − 1)-cycles. The element

푛 ℎ has a fixed point, which we may assume is the origin in F2 , so ℎ ∈ GL푛(F2). 푛 Consider Ω = F2 as an F2[푡] module, where 푡 acts as ℎ. Since ℎ has fixed points and is not equal to the identity, one of the following cases holds.

푛 (i) Ω has a submodule 푊 isomorphic to F2[푡]/(푡 − 1) for some 푛 > 1. Let 푀 be the restriction of ℎ to 푊 . Then 푀 푞−1 − 1 = 0. So 푡푞−1 − 1 is divisible

푛 by (푡 − 1) in F2[푡], therefore 푛 6 푣2(푞 − 1). This implies that the number 푛 of points in 푊 is 2 6 푞 − 1. But 푞 − 1 is the minimal size of a nontrivial orbit of ℎ, contradiction.

(ii) Ω = 푉1 ⊕ 푉2, where 푉1 ̸= 0 is indecomposable and 푉2 contains all nonzero

fixed points of ℎ. Then for every fixed point 푣 of ℎ, 푣 + 푉1 is invariant under the action of ℎ. Therefore, ℎ has at least as many orbits of length

greater than 1 as it has fixed points. On the other hand, since ℎ acts on 퐴푖 via fixing one point and permuting the rest cyclically and also fixes 퐵 ̸= ∅ pointwise, 퐴 has more fixed points than it has orbits of length greater than one. Contradiction.

57 Assume 푞 = 2. The set 퐵 is the set of common fixed points of the elements of 퐻.

Therefore 퐵 forms an affine space, and #퐵 is a power of 2. Similarly 퐵 ∪ 퐴1 is

the set of common fixed points of elements of 퐻1. Therefore #퐵+#퐴1 = #퐵+2 is a power of 2. Hence #퐵 = 2, and 2푛 = #Ω = 2푠 + #퐵, so 푠 = 2푛−1 − 1. We can assume that the tangent variety of 푋 is not a quadric, since otherwise we are in case 3. If 푛 = 2, then a nonidentity element of 퐻 acts on Ω as a

푛 transposition, hence 퐺 = 푆푑. Suppose 푛 > 2. Then 푋 has degree 2 > 4 and therefore 푋 lies on at most one quadric surface. For a general tangent line ℓ of 푋 there does not exist a quadric 푄 containing 푋 and ℓ. Consider a triple

ℓ1, ℓ2, ℓ3 of general tangents to 푋. There exists a smooth quadric 푄 containing

ℓ1, ℓ2, ℓ3 and not containing 푋. Then

푛+1 2 = deg 푄 ∩ 푋 > deg ℓ1 ∩ 푋 + deg ℓ2 ∩ 푋 + deg ℓ3 ∩ 푋 푛−1 = 3 deg γ푋 = 6푠 = 6 · (2 − 1).

This inequality is never satisfied for 푛 > 2, contradiction.

1 Case (b): PSL2(푟) ⊂ 퐺 ⊂ PΓL2(푟) and Ω = P (F푟). In this case deg 푋 = #Ω = 푟 + 1.

The group PΓL2(푟) is generated by PGL2(푟) and the Frobenius. In most of the

cases analyzed below, the subgroups 퐻, 퐻푖 contain elements that fix a triple

of points. The group PΓL2(푟) is triply-transitive; the pointwise stabilizer of

{0, 1, ∞} in PΓL2(푟) is the cyclic group generated by the Frobenius, and so the pointwise stabilizer of any triple of points is cyclic. Suppose 푞 < 푟. We consider several subcases.

(i) 푠 = 1. In this case the set 퐵 of Lemma 3.3.2 contains 푟 + 1 − 푞 > 2 elements. If 푞 = 2, then a nontrivial element of 퐻 is a transposition, so

퐺 = 푆푑. Suppose #퐵 = 푟 + 1 − 푞 = 2 and 푞 > 2. Then the element ℎ ∈ 퐻 acts as a (푞 − 1)-cycle and fixes 3 points. The stabilizer of a triple

of points in PΓL2(푟) is a cyclic group generated by the Frobenius, and thus cannot have exactly 3 fixed points when 푞 > 2. Therefore we can assume

58 that #퐵 = 푟 + 1 − 푞 > 3 and 푞 > 2. The group 퐻 stabilizes 퐵 pointwise; therefore 퐻 is abelian. On the other hand, Lemma 3.3.2 implies that 퐻

maps surjectively onto AGL1(푞), contradiction.

(ii) 푠 > 2, 푞 > 2. The element ℎ ∈ PΓL2(푟) fixes 푠 + #퐵 > 3 points, and therefore ℎ is a power of the Frobenius. Since ℎ only has orbits of sizes 1

and 푞−1, the number 푞−1 must be prime. Thus F푟 = Fℓ푞−1 for some prime

power ℓ. The action of ℎ on Fℓ푞−1 is such that the number of fixed points is 푠+#퐵−1 which is at least as large as the number of nontrivial orbits 푠. So

푞−1 (ℓ − ℓ)/(푞 − 1) = #{nontrivial orbits of ℎ} 6 #{ﬁxed points of ℎ} = ℓ + 1. The only pairs of prime powers (ℓ, 푞) with 푞 > 2 that satisfy this inequality are (2, 4), (2, 3), and (3, 3). These correspond to (푟, 푞) being (8, 4), (4, 3), and (9, 3). The case (푟, 푞) = (4, 3) is impossible since 푠 > 1. Suppose (푟, 푞) = (9, 3). The element ℎ must have exactly 3 + 1 fixed

points, and therefore 푠 = 3. Any element 푔 ∈ 퐻1, 푔 ̸= id fixes 퐵 ∪ 퐴1.

Since #퐵+#퐴1 > 3, the element 푔 must equal to the Frobenius in a degree 푞 −1 = 2 extension, and therefore be a product of transpositions. However

elements of 퐻1 preserve the decomposition 퐴2 ∪ 퐴3 and act transitively on

퐴2, contradiction. Similarly, in the case (푟, 푞) = (8, 4) the element ℎ must have 3 fixed points, and hence 푠 = 2. In this case any nontrivial element

of 퐻1 is a power of the Frobenius, and hence 퐻1 cannot act transitively on

퐴2, contradiction.

(iii) 푞 = 2, 푠 ̸= 1. If 푠 = 2, then a nontrivial element of 퐻1 is a transposition,

proving 퐺 = 푆푑. In particular we can assume 푠 > 3, 푟 > 7. Choose three

elements of 퐵 ∪ 퐴푖 and identify them with 0, 1, ∞. A nontrivial element of

푔 ∈ 퐻푖 fixes 0, 1, ∞ and has order 2, therefore 푔 is equal to the Frobenius 2 of the quadratic extension F푟/Fℓ, 푟 = ℓ . Hence 퐻푖 = Z/2Z. Since 퐻푖

acts transitively on 퐴푗 for 푗 ̸= 푖, the nontrivial element 푔 ∈ 퐻푖 acts as 1 a transposition on 퐴푗, 푗 ̸= 푖. In particular #퐵 + 2 = #P (Fℓ) = ℓ + 1. Suppose 퐵 has at least three points. Label them 0, 1, ∞. Every element

of 퐻 fixes 0, 1, ∞ and has order at most 2, so 퐻 = Z/2Z. Since 퐻

59 acts transitively on each 퐴푖, the set of fixed points of 퐻 is 퐵. Therefore

#퐵 = ℓ + 1 = #퐵 + 2. Contradiction. We proved that #퐵 6 2 and 퐻푖

is a group of order 2 fixing 퐵 ∪ 퐴푖 and acting as a transposition on 퐴푗 for 2 푗 ̸= 푖. Since ℓ + 1 = #퐵 + 2 6 4, either ℓ = 2 or ℓ = 3. Since ℓ = 푟 > 7,

we have #퐵 = 2, ℓ = 3, 푠 = 4. Let ℎ푖 denote the nontrivial element of 퐻푖.

Then ℎ1ℎ2 fixes 퐴3 ∪ 퐴4 ∪ 퐵 and acts on 퐴1 and 퐴2 as a transposition. No

element of PΓL2(9) fixes exactly 6 points. Contradiction.

We have shown that 푞 cannot be less than 푟. Suppose 푞 = 푟. The group PΓL2(푞)

is not quadruply transitive unless it contains 퐴푞+1 (i.e. unless 푞 = 2, 3, 4). Therefore Proposition 3.2.8 implies that the curve 푋 lies in P3. Take a smooth point 푃 of 푋 and consider the family of hyperplanes passing through the tangent line to 푋 at 푃 . Since the multiplicity of the intersection of such a hyperplane with 푋 at 푃 is at least 푞 and deg 푋 = 푞 +1, the curve 푋 is smooth. If a general tangent to 푋 is concurrent with another tangent line, then the plane containing both of them intersects 푋 with multiplicity greater than 푞 + 2 > 푞 + 1 which is impossible. Therefore we can apply Proposition 3.4.2. We conclude that 푋 is projectively equivalent to the projective closure of the rational curve (푡, 푡푞, 푡푞+1).

The monodromy group in this case is PGL2(푞), see [47] Example 2.15.

Case (c): 퐺 is one of the Mathieu groups and the action on Ω is in the list of Theorem 1.2.5. From Lemma 3.3.2 it follows that there is an element in 퐺 that acts via a product of 푠 (푞 − 1)-cycles. Take an element 푔 ∈ 퐺 that maps onto an element of order

푘 푝 in AGL1(푝). Replacing 푔 by 푔 is necessary we can assume that 푔 acts as a nontrivial product of cycles of length 푝. Cycle types of the Mathieu groups (in different permutation representations) are listed in Appendix A.1 (borrowing from [59]). Together the existence of cycle type (푞 − 1)푠, existence of a product of 푝-cycles, and the condition 푠푞 < 푑 leaves the following cases.

(i) 퐺 = 푀11, 푑 = 11, 푞 = 2, 푠 = 4. Any nontrivial element of 퐻1 has cycle 푡 type 2 , 1 6 푡 6 3. However there are no element with such cycle type in this representation.

60 푡 (ii) 퐺 = 푀11, 푑 = 12, 푠 = 5, 푞 = 2. All elements having cycle type 2 , 푡 > 0 have cycle type 24. These cannot form the subgroup 퐻, since a product of two distinct such elements is an element of type 22.

(iii) 퐺 = 푀11, 푑 = 11, 푞 = 5, 푠 = 2. Every element of 퐻1 fixes at least 6 points;

푀11 has no such elements.

푡 (iv) 퐺 = 푀11, 푑 = 12, 푞 = 2, 푠 = 5. Any element of type 2 with 푡 > 0 has 푡 = 4. Therefore every element of 퐻 is a product of 4 transpositions. A product of two distinct elements acting as products of four 2-cycles is not a product of four 2-cycles.

(v) 퐺 = 푀12, 푞 = 2, 푠 6 5. Any element of 퐻1 acts as a product of at 푡 most 푠 − 1 transpositions. Every element of 푀12 with cycle type 2 has 푡 = 0, 4, 6. Therefore 푠 = 5. Every element of 퐻 acts as a product of at most 5 transposition, so every nonidentity element of 퐻 acts as a product of 4 transpositions. A product of two 24 elements of 퐻 cannot be a 24 element. Contradiction.

(vi) 퐺 = 푀12, 푞 = 5, 푠 = 2. The group 퐻1 contains a nonidentity element that

fixes at least 7 points; 푀12 has no such elements.

(vii) 퐺 = 푀22, 푞 = 2, 푠 6 10. Any element of 퐻1 acts as a product of transpo- 푡 sitions. An element of 푀22 that has cycle type 2 has 푡 = 0, 8. Therefore 8 8 푠 > 9. Every element of 퐻 is a 2 element. A product of 2 distinct 2 elements of 퐻 cannot be a 28 element, contradiction.

(viii) 퐺 = Aut(푀22), 푞 = 2, 푠 6 10. Any element of 퐻1 acts as a product

of at most 푠 − 1 6 9 transpositions. An element of Aut(푀22) that has 푡 cycle type 2 has 푡 = 0, 7, 8. Therefore 푠 > 8. Suppose 푠 = 8. Then 퐻1

acts transitively on 퐴푖, for 푖 > 1. Since the only possible cycle type of a 7 nonidentity element of 퐻1 is 2 , 퐻1 = Z/2Z acting as a transposition on

every 퐴푖, 푖 ̸= 1. Consider now the possible cycle types of elements of 퐻. If 퐻 contains a 28 element 푔, then the product of 푔 with the nonidentity

element of 퐻1 is a transposition, contradiction. On the other hand, a

61 product of two different 27 elements of 퐻 cannot be a 27 element. Therefore 푠 ̸= 8. Suppose 푠 > 8. Then the group 퐻 contains two disctinct elements

푡1 푡2 푔1, 푔2 with cycle types 2 , 2 for 푡1, 푡2 ∈ {7, 8}. Then 푔1푔2 is not the

identity, and therefore fixes at most 2(푠−7) elements of ∪푖퐴푖. On the other

hand, 푔1 and 푔2 both act as a transposition on 퐴푖 for at least 푡1 + 푡2 − 푠 >

14 − 푠 different 푖. Therefore 푡1 + 푡2 − 푠 6 푠 − 7, which implies 푠 > 10. Contradiction.

(ix) 퐺 = Aut(푀22), 푞 = 3, 푠 = 7. Any element of the group generated by 푘 ℓ 퐻1, ..., 퐻7 has cycle type 2 3 , with 푘, ℓ 6 7. The only cycle types of this 6 7 form in Aut(푀22) are 3 and 2 . Since 퐻푖 fixes 퐴푖, an element of 퐻푖 cannot 7 6 have cycle type 2 , so every nonidentity element of 퐻푖 has cycle type 3 .

Consider nonidentity elements 푔1 ∈ 퐻1, 푔2 ∈ 퐻2. The product 푔1푔2 acts 6 as a 3-cycle on 퐴1, therefore 푔1푔2 has cycle type 3 . This means that 푔1

and 푔2 restrict to the same 3-cycle on 퐴푖 for 4 different values of 푖. Then −1 푔1푔2 fixes at least 3 · 4 = 12 points, however Aut(푀22) has no nonidentity elements fixing 12 points.

푡 (x) 퐺 = 푀23, 푑 = 23, 푞 = 2. Every element with cycle type 2 , 푡 > 0 has 푡 = 8.

Since every element of 퐻1 is a product of at most 푠−1 transpositions, 푠 > 9.

Also 푠푞 < 푑 implies 푠 6 11. Since 퐻 acts transitively on each 퐴푖, there

exist two distinct nontrivial elements ℎ1, ℎ2 ∈ 퐻 each having the cycle type 8 8 2 . But ℎ1ℎ2 cannot have cycle type 2 , contradiction.

(xi) 퐺 = 푀24, 푑 = 24, 푞 = 2, 푠 > 8. Since 푠푞 < 푑, we have 푠 6 11. In this representation every 2푡 element of 퐻 has 푡 = 0, 8. A product of two 28 elements of 퐻 cannot be a 28 element. Contradiction.

Remark 3.4.4. Suppose char 퐾 ̸= 2. Then the group 퐺 is contained in the alternating group if and only if the separability degree of the Gauss map is even [47] Theorem 2.10.

When char 퐾 = 2 we do not know under what conditions 퐺 is equal to 푆푛.

62 3.5 Galois groups of trinomials

In this section we study the Galois group 퐺 of the polynomial 푥푛 + 푎푥푚 + 푏 over the field 퐾(푎, 푏). The following theorem of Cohen covers most of the previously known results about 퐺.

Theorem 3.5.1 ([14] Corollary 3). Assume that 푛 and 푚 are relatively prime, 푝 -

푚(푛 − 푚) and if 푚 = 1 or 푚 = 푛 − 1 assume additionally that 푝 - 푛. Then 퐴푛 ⊂ 퐺.

Moreover, if 푝 is odd, then 퐺 = 푆푛.

Some cases not covered by Theorem 3.5.1, such as a trinomial with 퐺 = 푀11, are described in [56]. Our goal is to compute all the possibilities for 퐺.

We begin by fixing notation. Throughout this section, (푛, 푚) is a pair of relatively prime positive integers, 푋푛,푚 denotes the projective rational curve plane 푛 푚 푛−푚 푥 = 푦 푧 , and 퐺 = 퐺푛,푚 is the sectional monodromy group of 푋푛,푚. An affine 푚 푛 parametrization of 푋푛,푚 is (푡 , 푡 ). Therefore 퐺 is also the Galois group of the trinomial 푥푛 + 푎푥푚 + 푏 over 퐾(푎, 푏). Without loss of generality we assume that 푚 < 푛/2 since 푋푛,푚 ≃ 푋푛,푛−푚.

We start by describing properties of tangent lines and Gauss maps of the curves

푋푛,푚.

Proposition 3.5.2.

1. If 푛, 푚 and (푛 − 푚) are prime to 푝, then 푋푛,푚 is nonstrange. In this case the Gauss map has separability degree 1, and inseparability degree 1 when 푝 > 2 and 2 when 푝 = 2.

2. Assume that 푘 ∈ {푛, 푚, 푛 − 푚} is divisible by 푝. Write 푘 = 푞푑 where 푞 is the

largest power of 푝 that divides 푘. Then 푋푛,푚 is strange and the Gauss map has inseparability degree 푞 and separability degree 푑.

Proof. If 푝|푚, then 푝 - 푛, so we can replace 푋푛,푚 by the projectively isomorphic curve

푋푛,푛−푚, and 푝 - 푛 − 푚. Thus we may assume 푝 - 푚. An affine equation of 푋푛,푚 is

63 푛 푚 푚 푛 푥 = 푦 . The tangent line to 푋푛,푚 at the point (훼 , 훼 ) is given by the equation

푛훼푚(푛−1)푋 − 푚훼푛(푚−1)푌 = (푛 − 푚)훼푛푚.

푛 푚 푛 푛−푚 푛 푛 Restricting the equation 푥 = 푦 to the tangent line 푌 = 푚 훼 푋 −( 푚 −1)훼 gives the polynomial (︁ 푛 (︁ 푛 )︁ )︁푚 푝(푋) = 푋푛 − 훼푛−푚푋 − − 1 훼푛 푚 푚 Recall that the inseparability degree 푟 of the Gauss map is greater than 1 if and only if 훼푚 is a root of 푝(푋) with multiplicity greater than 2; in this case 푟 is equal to the multiplicity of the root 훼푚. Set 푋 = 훼푚 + 푍. Then

(︁ 푛 )︁푚 푝(푋) = (푍 + 훼푚)푛 − 훼푛−푚푍 + 훼푛 . 푚

The coefficient of 푍2 is

(︂푛(푛 − 1) 푛(푚 − 1))︂ − 훼푚(푛−2). 2 2

This coefficient is identically zero if and only if either 푝|푛 or 푝|(푛 − 푚).

1. 푝 - 푛, 푝 - 푛 − 푚. In this case the inseparability degree of the Gauss map is 1. We can recover the point (훼푛, 훼푚) from its tangent line

푛훼푚(푛−1)푋 − 푚훼푛(푚−1)푌 = (푛 − 푚)훼푛푚

using ratios of the coefficients of the equation. Therefore the separability degree of the Gauss map is also 1. Finally the dual curve (푛푎푚(푛−1) : 푚훼푛(푚−1) : (푛 − 푚)훼푛푚) is not a line; therefore the tangent lines do not pass through one common points and the curve is nonstrange.

2. 푝|푛. Let 푞 be the largest power of 푝 dividing 푛, write 푛 = 푞푑. In this case the equation of the tangent line becomes 푌 = 훼푛, and therefore the curve is strange. The Gauss map in affine coordinates is 훼 → 훼푛; therefore it has separable degree

64 푑 and inseparable degree 푞.

3. 푝|(푛 − 푚). Let 푞 be the largest power of 푝 dividing 푛 − 푚, and let 푛 = 푞푑. The equation of the tangent line becomes 푌 = 훼푛−푚푋. Therefore the curve is strange. The Gauss map in affine coordinates is just 훼 → 훼푛−푚 therefore the separable degree of the Gauss map is 푑 and the inseparable degree is 푞.

We now prove a stronger version of Lemma 3.3.2 for curves 푋푛,푚.

Lemma 3.5.3. Let 푋 = 푋푛,푚 be one of the trinomial curves. Choose a geometric 2 * point 훿 of (P ) above which the covering 휋푋 is étale. Denote by Ω the fiber of 휋푋 above 훿, so the group 퐺푋 acts on Ω by pertmutations. Then the following hold.

1. Assume 푝|푛. Write 푛 = 푞푑 with 푞 a power of 푝 and 푑 prime to 푝. Then there

exists a decomposition Ω = 퐴1 ⊔ ... ⊔ 퐴푑, a subgroup 퐻 ⊂ 퐺, a collection

of subgrops 퐻푖 ⊂ 퐺, 푖 = 1, ..., 푑, and an element ℎ ∈ 퐻 with the following properties.

∙ For every 푖 the set 퐴푖 is of cardinality 푞.

∙ The image of 퐻 in Sym(퐴푖) equals AGL1(푞) embedded via its action on 1 A (F푞).

∙ The element ℎ ∈ 퐻 acts on each 퐴푖 with one fixed point and one orbit of size 푞 − 1.

∙ The group 퐻푖 acts on 퐴푖 with one fixed point and one orbit of size 푞 − 1.

The image of 퐻푖 in Sym(퐴푗) for 푖 ̸= 푗 contains AGL1(푞) embedded via its 1 action on A (F푞).

2. Assume 푝|푛 − 푚. Write 푛 − 푚 = 푞푑 with 푞 a power of 푝 and 푑 prime to 푝.

Then there exists a decomposition Ω = 퐴1 ⊔ ... ⊔ 퐴푑 ⊔ 퐵, a subgroup 퐻 ⊂ 퐺,

a collection of subgrops 퐻푖 ⊂ 퐺, 푖 = 1, ..., 푑, and an element ℎ ∈ 퐻 with the following properties.

65 ∙ For every 푖 the set 퐴푖 is of cardinality 푞.

∙ The image of 퐻 in Sym(퐴푖) equals AGL1(푞) embedded via its action on 1 A (F푞).

∙ The element ℎ ∈ 퐻 acts on each 퐴푖 with one fixed point and one orbit of size 푞 − 1.

∙ The image of 퐻 in Sym(퐵) is Z/푚Z in its natural embedding; ℎ maps to a generator of Z/푚Z.

∙ The group 퐻푖 fixes 퐴푖 ∪ 퐵 and acts transitively on each 퐴푗 for all 푗 ̸= 푖.

Proof. Both parts of the statement are proved similarly to Lemma 3.3.2. We will pick a tangent line ℓ to 푋푛,푚. We will then restrict the covering to a family of lines that includes ℓ and compute the inertia group at [ℓ] ∈ (P2)*. To calculate the inertia groups we will use Lemma 3.3.1 and a computation of local uniformization.

1. Replacing 퐾 with a larger algebraically closed field if necessary, we can choose an element 푢 ∈ 퐾 that is not a root of unity. Consider the affine model 푥푛 = 푦푚 of 푋. The line 푦 = 푢푛 is tangent to 푋. Choose a root of unity

푛 푖 푚 휁푑 ∈ 퐾. The points of tangency are 푃푖 = (훼푖, 푢 ), where 훼푖 = 휁푑푢 . Let

푥푖 = 푥 − 훼푖. A local uniformization of 푋 at 푃푖 is given by (훼푖 + 푥푖, 푝푖(푥푖)), 푛 푛/푚 푞+1 where 푝푖(푥푖) = 푢 (1 + 푥/훼푖) . The 푥 coefficient of 푝푖 is zero. Choose an

element 훼 ∈ 퐾 and identify Ω with the intersection of 푋푛,푚 and the line ℓ given by the equation 푦 − 푢푛 = 푡(푥 − 훼) over the field 퐾((푡)). Let 퐻(훼) be the Galois

group of the field extension 퐿훼 obtained by adjoining to 퐾((푡)) the coordinates of the intersection. Then 퐻(훼) acts on Ω and is a subgroup of 퐺. The field

푛 푚 푛 퐿훼 is the field obtained by adding the roots of 푥 − 푡푥 + 푡훼 − 푢 to 퐾((푡)).

In particular it is separable. Let 퐴푖 ⊂ Ω be the set of intersection points that

reduce to 푃푖. The 푥-coordinates of the points in ℓ ∩ 푋 that reduce to 푃푖 are the 푛 positively valued roots of the equation 푝푖(푥푖)−푢 = 푡푥푖 +푡훼푖 −푡훼. Lemma 3.3.1 푛 −푞 applies with 푎0 = 푡(훼 − 훼푖), 푎1 = 푡, 푎푞 = 푢 푑훼푖 /푚, 푎푞+1 = 0. Hence if 훼 ̸= 훼푖,

the Galois group of the field 퐿푖,훼 obtained by adding these roots to 퐾((푡)) equals

66 AGL1(푞). When 훼 = 훼푖 the equation has a rational root 푥푖 = 0 and the rest

of the roots have valuation 1/(푞 − 1). Therefore when 훼 = 훼푖 the field 퐿푖,훼 is

the unique extension of 퐾((푡)) of degree 푞 − 1. Let 퐻푖 be the group 퐻(훼푖). Fix

훼 ∈ 퐾 distinct from each 훼푖 and let 퐻 be the group 퐻(훼). We have shown that

the actions of 퐻 and 퐻푖’s on 퐴푗’s are as claimed. Now we need to show the

existence of ℎ ∈ 퐻 that acts on each 퐴푖 as a (푞 − 1)-cycle. The extension 퐿푖,훼 has ramification index 푞(푞 −1) and therefore contains the field 퐾((푡1/(푞−1))). Let ℎ ∈ 퐻 be an element that surjects onto a generator of Gal(퐾((푡1/(푞−1)))/퐾((푡))).

Then the order of ℎ acting on 퐴푖 is 푞 − 1. Elements of AGL1(푞) of order (푞 − 1) act with two orbits of lengths 1 and 푞 − 1.

2. Consider the affine model 푥푛 = 푦푚 of 푋. Choose an element 푢 ∈ 퐾 that is not a root of unity. Let ℓ be the tangent line 푦 = 푢푛−푚푥 to 푋. Choose a root of

푖 푚 푖 푛 unity 휁푑. The points of tangency are 푃푖 := (휁푑푢 , 휁푑푢 ). The intersection ℓ ∩ 푋

consists of points 푃푖 and the point (0, 0). For an element 훼 ∈ 퐾 let ℓ훼 denote the line 푦 = (푡 + 푢푛−푚)푥 − 훼푡 over the field 퐾((푡)). Identify Ω with the intersection

ℓ훼 ∩ 푋 over 퐾((푡)). The absolute Galois group of 퐾((푡)) acts on Ω as a subgroup

of 퐺. Let 퐴푖 ⊂ Ω be the subset corresponding to points that reduce to 푃푖, and

let 퐵 be the subset that corresponds to points reducing to (0, 0). Let 퐿훼 be the

extension obtained by adding the coordinates of the intersection ℓ훼∩푋 to 퐾((푡)). 푛 푛−푚 푚 Then 퐿훼 is the splitting field of the trinomial 푥 −(푡+푢 )푥 +훼푡. In particular 푖 푚 퐿훼 is separable. Consider the uniformizer 푥푖 = 푥 − 휁푑푢 at 푃푖. Let the local 푖 푚 uniformization of 푋 at 푃푖 be given by (휁푑푢 +푥푖, 푝푖(푥푖)). The power series 푝푖(푥푖) 푖 푛 푖 푚 푛/푚 푖 푛 푖 푚 푖 푚 푑푞/푚 equals 휁푑푢 (1 + 푥푖/(휁푑푢 )) = 휁푑푢 (1 + 푥푖/(휁푑푢 ))(1 + 푥푖/(휁푑푢 )) . The

푥 coordinates of the points in the intersection ℓ훼 ∩ 푋 that reduce to 푃푖 are the 푛−푚 푖 푚 postiviely valued roots of the power series 푄푖 = 푝푖(푥푖)−(푡+푢 )(푥푖+휁푑푢 )+훼푡. We have

푞 푞+1 2푞 푄푖 = 푎0 + 푎1푥푖 + 푎푞푥푖 + 푎푞+1푥푖 + 푂(푥 ),

푖 푚 푑 −푖(푞−1) −푚푞+푛 푑 푛−(푞+1)푚 −푖푞 where 푎0 = (훼 −휁푑푢 ), 푎1 = −푡, 푎푞 = 푚 휁푑 푢 , 푎푞+1 = 푚 푢 휁푑 .

For a general 훼 the expression 푎0푎푞+1 − 푎1푎푞 has valuation 1. For each 푖 let

67 퐿훼,푖 be the field generated by the positively valued roots of 푄푖, and let 퐿훼 be the field obtained by adjoining to 퐾((푡)) the coordinates of the intersection of

푋 with ℓ훼. The field 퐿훼 is the compositum of 퐿훼,푖 for 푖 = 1, ..., 푑. Let 퐻(훼)

be the Galois group of the extension 퐿훼/퐾((푡)). Lemma 3.3.1 implies that for a

general 훽 ∈ 퐾 the image of 퐻(훽) in Sym(퐴푖) is AGL1(푞) for all 푖. Fix one such 훽 ̸= 0 and let 퐻 := 퐻(훽).

We now compute the action of 퐻(훼) on 퐵 for 훼 ̸= 0. The elements of 퐵 are the roots of 푄(푥) = 푥푛 − (푡 + 푢푛−푚)푥푚 + 훼푡 that reduce to 0 modulo 푡. The Newton polygon of 푄(푥) contains only one segment of positive slope, it connects (0, 1) and (푚, 0). Since the slope is 1/푚 and 푚 is prime to 푝, the elements of 퐵 are permuted cyclically by the group 퐻(훼) for every 훼 ̸= 0. Let ℎ ∈ 퐻 be an element that surjects onto a generator of Gal(퐾((푡1/퐿퐶푀(푞−1,푚)))/퐾((푡))). Since the field extension obtained by adjoining 퐵 to 퐾((푡)) is 퐾((푡1/푚)), the element ℎ

acts on 퐵 through a cyclic permutation. The field 퐿훼,푖 has ramification index 1/푞−1 푞(푞 − 1) and therefore contains 퐾((푡 )). Hence the action of ℎ on 퐴푖 has

order 푞 − 1, so ℎ acts on each 퐴푖 by fixing one point and permuting the rest cyclically.

Let 퐿푖 denote the field 퐿 푖 푚 . Let 퐻푖 be the Galois group of 퐿푖 over 휁푑푢 1/LCM(푚,푞−1) 푖 푚 퐾((푡 )). Then 퐻푖 fixes 퐵. When 훼 = 휁푑푢 the power series 푄푖

has root 0. The positive slope part of the Newton polygon of 푄푖/푥푖 is a single

segment from (0, 1) to (푞 − 1, 0). Therefore 퐻푖 fixes 퐴푖. For 푗 ̸= 푖 the posi-

tive slope part of the Newton polygon of 푄푗 is a segment from (0, 1) to (푞, 0);

therefore 퐻푖 acts transitively on 퐴푗 for 푗 ̸= 푖.

Theorem 3.5.4. Suppose 푚 and 푛 are relatively prime integers satisfying 푚 6 푛/2. Let 퐺 be the Galois group of the polynomial 푃 (푥) := 푥푛 + 푎푥푚 + 푏 over 퐾(푎, 푏).

푑 푑 1. If 푚 = 1 and 푛 = 푝 , then 퐺 = AGL1(푝 ).

2. If 푚 = 1, 푛 = 6, and 푝 = 2, then 퐺 = PSL2(5).

68 3. If 푚 = 1, 푛 = 12, and 푝 = 3, then 퐺 = 푀11.

4. If 푚 = 1, 푛 = 24, and 푝 = 2, then 퐺 = 푀24.

5. If 푚 = 2, 푛 = 11, and 푝 = 3, then 퐺 = 푀11.

6. If 푚 = 3, 푛 = 23, and 푝 = 2, then 퐺 = 푀23.

7. Let 푞 := 푝푘 for some 푘. If 푛 = (푞푑 − 1)/(푞 − 1), 푚 = (푞푠 − 1)/(푞 − 1) for some

푠, 푑, then 퐺 = PGL푑(푞).

8. If none of the above holds, then 퐴푛 ⊂ 퐺.

Proof. We break the proof into cases corresponding to the cases of Proposition 3.5.2. The case 푛 = 2 is trivial. In what follows we assume 푛 > 2 so that 푚 < 푛/2 since 푚, 푛 are relatively prime.

1. Assume 푝 - 푛푚(푛 − 푚). By Proposition 3.5.2 the curve 푋푚,푛 is nonstrange and the Gauss map is either birational or purely inseparable of degree two. In any

case Lemma 3.3.2 implies that 퐺 contains a transposition; hence 퐺 = 푆푛.

2. Assume 푝|푛. Consider the specialization of 푥푛 + 푎푥푚 + 푏 to the polynomial 푃 (푥) := 푥푛 + 푡−1푥푚 + 1 over the field 퐾((푡)). The Newton polygon of 푃 consists of two segments: one from (0, 0) to (푚, −1) and one from (푚, −1) to (푛, 0). Since 푛 − 푚 and 푚 are relatively prime and are both prime to 푝, the group

퐺 contains an 푚-cycle and an (푛 − 푚)-cycle. If 푚 ̸= 1, then 퐴푛 ⊂ 퐺 by

Theorem 1.2.5. If 푚 = 1 and 퐴푛 ̸⊂ 퐺, then Theorem 1.2.5 implies that one of the following holds

푑 푑 (a) 푛 = 푝 . In this case the Galois group is AGL1(푝 ), see [47] Example 2.17.

(b) 푛 = ℓ + 1 for some prime ℓ and 퐺 = PSL2(ℓ) or 퐺 = PGL2(ℓ). Write 푛 = 푞푑 for 푑 prime to 푝 and a power 푞 of 푝. Proposition 3.5.2 states that in this case the Gauss map has inseparable degree 푞 and separable degree 푑. If 푑 = 1, then 푛 is a power of 푝, which is precisely the case 2a above.

69 Assume 푑 > 1. Lemma 3.5.3 applies in this case with 퐵 = ∅. The group

퐻1 surjects onto AGL1(푞) and therefore has order divisible by 푝. Take an

element 푔1 ∈ 퐻1 of order 푝, so that it acts as a nontrivial product of cycles

of length 푝 and fixes 퐴1. Since no element of PGL2(ℓ) can fix three points,

#퐴1 = 푞 = 2. The element 푔1 fixes two points and acts as a products of transpositions. There is a unique such element, namely multiplication

by −1. In particular 푔1 acts as a transposition on 퐴푖 for 푖 > 1. Take an

element 푔2 ∈ 퐻 that acts on 퐴1 as a transposition. If 푑 > 4, then either

푔2 or 푔1푔2 fixes at least four points, contradiction. If 푑 = 2, then 푔1 acts a

transposition and so 퐺 = 푆푛. If 푑 = 3, then ℓ = 5 (and 푞 = 2). Suppose ℓ = 5, 푞 = 2. There is a factorization

(︁ 푎 )︁ (︁ 푎 )︁ 푥6+푎푥+푏 = 푥3 + 푐푥2 + 푐2훼푥 + + 훼푐3 푥3 + 푐푥2 + (푐2훼 + 푐2)푥 + + (훼 + 1)푐3 , 푐2 푐2

where 훼 ∈ 퐾 satisfies 훼2 + 훼 + 1 = 0 and 푐 is a root of 푦10 + 푎푦5 + 푎2 + 푏. So 푃 (푥) decomposes into product of two cubic polynomials after a degree

10 extension. If 퐴6 ⊂ 퐺, then the minimal degree of the extension over which 푃 (푥) splits into a product of two cubic polynomials is

#푆6 #퐴6 = ⋂︀ = 20. #(푆3 × 푆3) #((푆3 × 푆3) 퐴6)

Therefore 퐺 ̸⊃ 퐴6. The cubic polynomials appearing in the factorization are conjugate to each other over the field 퐾(푎, 푏)(푐) via an automorphism that fixes 푐 and sends 훼 to 1+훼. Therefore the total degree of the extension

is at most 10#푆3 = 60. Thus 퐺 = PSL2(5).

this case either 퐺 is one of the Mathieu groups 푀11, 푀12 or 퐺 contains 퐴푛.

Let 퐺̂︀ denote the Galois group of 푃 (푥) over F2(푎, 푏). Then 퐺 is a normal 12 subgroup of 퐺̂︀. Over F2 the polynomial 푥 +푥+1 factorizes into a product of irreducibles of degrees 3, 4 and 5. Therefore 퐺̂︀ is a 2-transitive group

70 containing a 4-cycle and thus 퐺̂︀ = 푆12. Since 퐺 is a normal subgroup of

퐺̂︀, the group 퐺 contains 퐴12.

(d) 푛 = 24. In this case either 퐺 = 푀24 or 퐺 ⊃ 퐴24. If 푝 = 2, then we can apply Serre’s linearization method [4]. We will find an additive polynomial that is divisible by 푥24+푎푥+푏. Consider the following sequence of equalities in 퐾(푎, 푏)[푥]/(푥24 + 푎푥 + 푏):

0 = 푥32 + 푎푥9 + 푏푥8 = 푥256 + 푎8푥72 + 푏8푥64

= 푥256 + 푎8(푎푥 + 푏)3 + 푏8푥64 = 푥256 + 푏8푥64 + 푎11푥3 + 푎10푏푥2 + 푎9푏2푥 + 푎8푏3

= 푥2048 + 푏64푥512 + 푎88푥24 + 푎80푏8푥16 + 푎72푏16푥8 + 푎64푏24

= 푥2048 + 푏64푥512 + 푎80푏8푥16 + 푎72푏16푥8 + 푎89푥 + 푎88푏 + 푎64푏24

The last polynomial in this chain of equalities is an additive polynomial

up to a constant. Therefore the group 퐺 is a subquotient of AGL11(2),

thus 퐺 cannot contain 퐴24 since 23 - # AGL11(2). Hence 퐺 = 푀24.

Suppose 푝 = 3. Let 퐺̂︀ denote the Galois group of 푃 (푥) over F9(푎, 푏). Then 퐺̂︀ contains 퐺 as a normal subgroups, and therefore 퐺̂︀ is also a primitive group containing an 푛-cycle. From Theorem 1.2.5 it follows that

퐺̂︀ is either one of the groups PSL2(23), PGL2(23), 푀24 or 퐺̂︀ ⊃ 퐴24. Let 푐 ∈ 2 24 F9 be a root of the polynomial 푦 −푦−1. The polynomial 푥 −푥+푐 factors

over F9 into a product of irreducible polynomials of degrees 1, 2, 3, 4, 6 and 8. Therefore 퐺̂︀ contains an element 푔 with cycle type 1, 2, 3, 4, 6, 8. Since 8 푔 fixes at least 15 points, 퐺̂︀ ̸⊂ PGL2(23). The group 푀24 has no elements

with the cycle type of 푔 (see Appendix A.1). Therefore 퐺̂︀ ⊃ 퐴푛. Since 퐺

is a normal subgroup of 퐺̂︀, the group 퐺 contains 퐴24.

3. Assume 푝|푚. Consider the specialization of 푥푛 + 푎푥푚 + 푏 to the polynomial 푃 (푥) = 푥푛 + 푡 over 퐾((푡)). The Galois group of this polynomial is cyclic and we deduce that 퐺 contains an 푛-cycle. Theorem 1.2.5 gives a finite list of possibilities for 퐺. Consider the specialization to the polynomial 푄(푥) = 푥푛 +

71 푡푥푚 + 푡2 over 퐾((푡)). The Newton polygon of this polynomials has two slopes 1/(푛 − 푚) and 1/푚. Since 푝|푚 the roots of 푄 span a wildly ramified extension. Any element of the wild inertia will fix at least 푛 − 푚 > 푛/2 points. No group from the case 1.2.5 of Theorem 1.2.5 can have nontrivial elements fixing at least half of the points.

4. Assume 푝|푛 − 푚. Consider the specialization 푥푛 + 푡 over 퐾((푡)). Since 푛, 푚 are relatively prime, the specialized polynomial is separable with cyclic Galois

group. Assume 퐴푛 ̸⊂ 퐺. Since 퐺 contains an 푛-cycle, Theorem 1.2.5 produces a list of possibilities for 퐺. We analyze each case separately.

(a) 퐶ℓ ⊂ 퐺 ⊂ AGL1(ℓ), 푛 = ℓ > 5 is prime. Consider the specialization 푥푛 + 푡−1푥푚 + 1 over 퐾((푡)). The Newton polygon of this trinomial consists of two segments with slopes 푚 and 푚 − 푛. Since 푝|(푛 − 푚) the extension is wildly ramified. Any element from the wild inertia subgroup fixesat least 푚 roots. Therefore 푚 = 1. In this case Lemma 3.5.3 applies with

#퐵 = 1. The stabilizer of 퐵 in 퐺 ⊂ AGL1(ℓ) is abelian and has AGL1(푞)

as a subquotient, thus 푞 = 2. Since ℓ > 5 the group 퐻1 has an element 푔 ̸= 푖푑. The element 푔 fixes at least two points, contradiction.

푟−1 푟 (b) PGL푟(ℓ) ⊂ 퐺 ⊂ PΓL푟(ℓ), 푛 = #P (Fℓ) = (ℓ − 1)/(ℓ − 1). We will prove 푟′ ′ that ℓ is necessarily a power of 푝 and that 푚 = #P (Fℓ) for some 푟 < 푟.

We can assume that PGL푟(ℓ) ̸⊃ 퐴푛. We call a point blue if it is in

the set 퐵 and azure if it is in one of the 퐴푖’s. Lemma 3.5.3 implies that

for every pair of points 푃, 푄 ∈ 퐴푖 there is an element of 퐺 that fixes all blue points and moves 푃 to 푄. Our goal is to show that 푝|ℓ and 퐵 forms a projective subspace of Ω. To this end we examine the Galois action on lines that contain blue points.

푟 Consider a line 푙 in P (Fℓ) that contains at least 3 blue points; identify 1 this line with P (Fℓ). Any element of 퐺 that fixes all blue points isa power of the Frobenius when restricted to 푙. Since any azure point can be moved by an element of 퐺 that fixes all blue points, the set of blue

72 points of 푙 is the set 푙(F푡) for some subfield F푡 ⊂ Fℓ. There is a subgroup of 퐻 that fixes all blue points and acts transitively on 퐴푖 for every 푖. Thus if 푙 contains a point from 퐴푖, then it contains the whole set 퐴푖. Assume

퐴푖, 퐴푗 ⊂ 푙 for some 푖 ̸= 푗. The group 퐻푖 fixes all blue points and 퐴푖, and acts transitively on the rest of the line. Therefore 퐴푖 equals 푙(F푡푘 ) ∖ 푙(F푡), where 푘 = log푡(푞 + 푡). Similarly 퐴푗 equals 푙(F푡푘 ) ∖ 푙(F푡), contradiction.

Thus 푙 contains points only from one group 퐴푖. Since the subgroup of

퐺 that fixes 퐵 acts transitively on 퐴푖, the extension Fℓ/F푡 is such that the Frobenius acts with only one orbit. There is only one such extension

F4/F2. So either ℓ = 4, 푞 = 2 or every line containing at least three blue points consists entirely of blue points. Suppose ℓ = 4, 푞 = 2. If 푟 = 2, then

PGL2(4) = 퐴5, contradiction. Therefore 푟 > 2. Suppose 푚 > 3. Take a blue point 푄 not on 푙 and let 푃 ∈ 푙 be an azure point 푃 ∈ 퐴푖. Then

퐴푖 ⊂ 푙. Consider the line connecting 푄 and 푃 . If this line contains a blue point 푅 ̸= 푄, then an element of 퐻 that induces a transposition on 퐴푖 does not preserve the collinearity of 푃, 푄 and 푅. If this line contains an azure point 푅 ∈ 퐴푗, then an element of 퐻푗 that induces a transposition on 퐴푖 does not preserve the collinearity of 푃, 푄 and 푅, contradiction. Suppose 푚 = 3, ℓ = 4, 푞 = 2, 푟 > 2. Consider blue points 푃, 푄 ∈ 푙 and an azure point 푅1 ∈ 퐴푖, 푅1 ̸∈ 푙 (as in Figure 3-1). Let 푅2 denote the point of ′ 퐴푖 ∖ {푅1}. The line 푙 connecting 푃 and 푅1 does not contain any blue ′ points except 푃 ; therefore there is a point 푆 ∈ 푙 , 푆 ∈ 퐴푗 for some 푗 ̸= 푖.

Let 푔 ∈ 퐻푗 be an element that maps 푅1 to 푅2. Since 푔 fixes 푆 and 푃 , ′ ′ 푔 preserves 푙 and therefore 푅2 ∈ 푙 and 푃, 푅1, 푅2 are collinear. Similarly

푄, 푅1 and 푅2 are collinear, contradiction. Therefore every line containing at least three blue points consists entirely of blue points.

Consider a line 푙 containing exactly two blue points 푃 and 푄 (if it exists). Consider a point 푅 ∈ 퐴푖 ∩ 푙. Since the pointwise stabilizer of 퐵 in 퐺 acts transitively on 퐴푖, the whole set 퐴푖 lies on 푙. We conclude that 푞|ℓ − 1. Suppose 푚 > 2, so there is a blue point 푆 ̸∈ 푙 (as in Figure 3-2).

73 Figure 3-1: Figure 3-2:

′ Let 푙 denote the line connecting 푆 and 푅. Since 퐴푖 ⊂ 푙, there is a point ′ 푇 ∈ (퐴푗 ∪ 퐵) ∩ 푙 for some 푗 ̸= 푖. Take an element 푔 ∈ 퐻푗 that moves 푅 to a different point on 푙. Then 푔 fixes 푆 and 푇 , and therefore fixes 푙′ but moves 푅 to a point not on 푙′, contradiction. If 푚 = 2 and 푟 > 2, there exists a line containing exactly one blue point. Therefore, assuming that 푙 exists, 푞|ℓ − 1 and either 푟 = 2 or there exists a line containing exactly one blue point.

Assume that there exists a line 푙 that passes through exactly one blue point. If all azure points on this line are from the same group 퐴푖, take two azure points 푃, 푄 on the line and consider an element of 퐺 that maps 푃 to 푄 and induces a translation on AGL1(F푞). This element preserves 푙 and decomposes all azure points into cycles of length 푝, hence 푝|ℓ. If there is a point 푃 ∈ 푙, 푃 ∈ 퐴푖 and a point 푄 ∈ 푙 ∩ 퐴푗, 푗 ̸= 푖, consider elements of 퐻푖 that fix 푃 and maps 푄 to a different point of 퐴푗. Such elements preserve

푙. We deduce that for every 푗 = 1, ..., 푑 if there is one point from 퐴푗 on the line, the whole set 퐴푗 is contained in the line. In this case we again get that 푞|ℓ.

We have proved the following statements.

∙ Every line that contains at least three blue point is entirely blue.

∙ If a line containing exactly two blue points exists, then 푞|ℓ − 1 and either 푟 = 푚 = 2 or a line containing exactly one blue point exists.

∙ If a line containing exactly one blue point exists, then 푝|ℓ.

74 Assume that it is not the case that 푟 = 푚 = 2. Then if a line containing two blue points exists, then 푞|ℓ − 1 and a different line passes through exactly one blue point, which in turn implies 푝|ℓ, contradiction. Therefore there are no lines that contain exactly two blue points. So every line that contains two blue points, contains at least three, thus is entirely blue. Take a line through a blue and an azure point. It must have exactly one blue point. Therefore 푝|ℓ and the blue points form a projective subspace.

Now we show that the case 푟 = 푚 = 2 is impossible. Suppose 푟 = 푚 = 2. Then 푛 = ℓ + 1 must be odd, therefore ℓ = 2푘. Since 푛 = 푞푑 + 2, both 푞 and 푑 are odd. Suppose 푑 = 1. If 푞 ̸= 3, then ℎ2 has exactly three fixed points, so ℎ2 is the degree 2 Frobenius. A nontrivial orbit of

1 2 the degree 2 Frobenius on A (F2푘 ) has size at most 푘. Since ℎ has two 푘 nontrivial orbits and two fixed points 2+2푘 > 2 . We conclude that 푘 6 3. Since the Frobenius must have two nontrivial orbits, 푘 is equal to 3. Then

푞 = 6 which is absurd. If 푑 = 1 and 푞 = 3, then 퐺 ⊂ PGL2(4) = 퐴5, contradiction. Thus we can assume that 푑 ̸= 1 and, since 푑 is odd, 푑 > 3. In this case the element ℎ has exactly 푑 fixed points, thus ℎ is a power of the Frobenius, so 푑−1 is a power of 2. Suppose 푞 ̸= 3. Since ℎ has a unique orbit of length 2, ℎ is the degree 2 Frobenius and the number of fixed points

1 of ℎ is 3 = #P (F2) = 푑. The group 퐻1 fixes 푞 +2 > 3 points, therefore 퐻1 is generated by a power of the Frobenius and 푞 + 1, which is the number

1 of fixed points of 퐻1 on A , is a power of 2. Therefore 푞 ≡ −1 (mod 4); hence 2푘 = ℓ = 푞푑 + 1 = 3푞 + 1 ≡ 2 (mod 4). Thus ℓ = 2, 푛 = 3. But if

ℓ = 2, then 퐺 ⊃ PGL2(2) = 푆3, contradiction. Finally we have to consider the case 푞 = 3. Write 푑 = 2푔 + 1, then 3(2푔 + 1) = 푞푑 = 푛 − 2 = 2푘 − 1. The last equation can be satisfied modulo 8 only if 푑 = 5, 푛 = 17. The group 퐻1 has 푞 + 푚 = 5 fixed points, and is therefore generated by the 1 degree 4 Frobenius. The degree 4 Frobenius on P (F16) acts with orbits of size 1 and 2, so it cannot act transitively on 퐴2. Contradiction.

Thus we proved that ℓ is a power of 푝, 푛 = 1 + ℓ + ··· + ℓ푟−1 and

75 푚 = 1 + ℓ + ··· + ℓ푠−1.

of the characteristic. We claim that for such 푛, 푚 the group 퐺 is PGL푟(ℓ). Consider the equation 푥푛 + 푎푥푚 + 푏. Let 푦 = 푥ℓ−1. The equation becomes 푦ℓ푟−1 + 푎푦ℓ푠−1 + 푏. Multiplying by 푦 gives 푦ℓ푟 + 푎푦ℓ푠 + 푏푦. The roots of the

latter form an Fℓ vector space, so its Galois group is a subgroup of GL푟(ℓ). Since the roots of 푥푛 + 푎푥푚 + 푏 correspond to lines in the space of roots

ℓ푟 ℓ푠 of 푦 + 푎푦 + 푏푦, 퐺 is contained in PGL푟(ℓ). Since 퐺 contains an 푛-cycle

and 푛 ̸= 11, 23, Theorem 1.2.5 implies that either PGL푠(푡) ⊂ 퐺 ⊂ PΓL푠(푡)

for some prime power 푡 or 푛 is prime and 퐶푛 ⊂ 퐺 ⊂ AGL1(푛). From the

case 4a of this proof we deduce that PGL푠(푡) ⊂ 퐺 ⊂ PΓL푠(푡) . From the case 4b we deduce that 푡 is the largest power of the characteristic that

divides 푛 − 1; hence 푡 = ℓ. Therefore 퐺 = PGL푑(ℓ).

(d) 푛 = 11 and 퐺 = 푀11 or 퐺 = PSL2(11). In this case 푚 can be any number from 1 to 5. Assume that 퐺̂︀ is the Galois group of 푃 over some 퐹 (푎, 푏) for some finite field 퐹 . The group 퐺 is a normal subgroup of 퐺̂︀. Since

퐺̂︀ contains an 푛-cycle, it is equal to one of 푀11, PSL2(11), 퐴11, 푆11. Hence

to show, that 퐴11 ⊂ 퐺 it suffices to show that 퐺̂︀ ̸= 푀11, PSL2(11). To

show that 퐺̂︀ ̸= 푀11, PSL2(11) it is enough to find s specialization of 푃 (푥) over a finite field, whose factorization pattern contradicts the possible cycle

types of 푀11 and PSL2(11). We do so in Table A.2 of Appendix A.2 (in every case the factorization shows that 퐺̂︀ has a cycle of length 2, 3 or 5 contradicting Theorem 1.2.5).

The only case remaining is that of the polynomial 푥11 + 푎푥2 + 푏 when

푝 = 3. But in this case Uchida [56, Example 4] proved that 퐺 = 푀11.

(e) 푛 = 23 and 퐺 = 푀23. Similarly to the previous case, for all but one possible pair (푝, 푚) we can find a trinomial over a finite field with factorization

pattern that is impossible for 퐺 = 푀23. The results are summarized in Table A.2 of Appendix A.2. The only case not covered in Table A.2 is that

76 of the polynomial 푥23 +푎푥3 +푏 for 푝 = 2. The polynomial 푥23 +푡푥3 +1 over

F2(푡) has Galois group 푀23, as proved by Abhyankar using the linearization method [4]. Consider the field 퐿 = 퐾(푎, 푏)(푏1/23). Let 훼 := 푏1/23 and 훽 := 푎훼−20. Then 퐿 is isomorphic to the field of rational functions 퐾(훼, 훽). Over 퐿 the equation 푥23 + 푎푥3 + 푏 = 0 can be simplified. Let 푥 = 훼푦, then the equation becomes 푦24 + 훽푦3 + 1 = 0. Since 퐾(훼, 훽)/퐾(훼) is a purely transcendental extension, the Galois group of 푦23 + 훽푦3 + 1 over 퐾(훼, 훽) is equal to the Galois group of the same equation over 퐾(훽), which is equal

to 푀23. Since 퐿/퐾(푎, 푏) is a cyclic extension, 퐺 ̸⊃ 퐴23, thus 퐺 = 푀23.

Remark 3.5.5. Theorem 3.5.4 does not distinguish between the cases 퐺 = 퐴푛 and

퐺 = 푆푛, but it is possible to do so. When 푝 is odd this is discussed in Remark 3.4.4.

If 푝 = 2 and 푛 is even, then 퐺 ⊂ 퐴푛 by [6, Proposition 2.24]. If 푝 = 2 and 푛 is odd, then 퐺 ⊂ 퐴푛 if and only if 푚 = 2 by [6, Proposition 2.23].

77 78 Chapter 4

Counting points on simple abelian varieties

The results of this section have appeared in [30].

4.1 Introduction

Let 퐴 be an abelian variety of dimension 푔 over a finite field F푞. The following classic

theorem of Weil gives an estimate for the size of the group 퐴(F푞).

Theorem 4.1.1 (Weil [58]). Suppose that 퐴/F푞 is an abelian variety of dimension 푔. Then

√ 2푔 √ 2푔 ( 푞 − 1) 6 #퐴(F푞) 6 ( 푞 + 1) .

Our goal is to improve upon the estimates of Theorem 4.1.1. For example, note that the lower bound is vacuous for 푞 = 2, 3, 4; our results will imply an exponential lower bound in the cases 푞 = 3, 4, while for 푞 = 2 there are infinitely many simple abelian varieties with one point, as proved in [39].

By Poincaré reducibility theorem (see [42, Theorem 1, Section 19]) any abelian variety is isogenous to a product of simple abelian varieties. Since isogenies preserve point counts, it is natural to consider only simple abelian varieties. Let 풜푞(푔) denote

79 the (finite) set of isogeny classes of simple abelian varieties of dimension 푔 over F푞. ᨀ Let 풜푞 denote the union 풜푞 := 푔 풜푞(푔). Define the quantities 푎(푞), 퐴(푞) by the formulas

1/푔 1/푔 푎(푞) := lim inf #퐴(F푞) , 퐴(푞) := lim sup #퐴(F푞) . 퐴∈풜푞 퐴∈풜푞

Serre (see [49, Section 4.6]) noticed that for general varieties the estimates coming from the Weil conjectures can be improved using some metric properties of totally positive algebraic integers. In the case of abelian varieties, Aubry, Haloui and Lachaud [8] observed that the asymptotic behavior of 퐴(푞) is related to the Schur–Siegel–Smyth trace problem. Let us briefly recall its statement.

Definition 4.1.2. Suppose that 훼 is an algebraic integer. Let {훼1, 훼2, ..., 훼푛} denote the Galois orbit of 훼. The normalized trace of 훼 is the average value of its Galois conjugates: tr(훼) := (훼1 + ··· + 훼푛)/푛.

Definition 4.1.3. Let TP ⊂ Q denote the set of all totally positive algebraic integers. The Schur-Siegel-Smyth constant 휌 is defined by

휌 := lim inf tr(훼). 훼∈TP

The following conjecture is known as the Schur-Siegel-Smyth trace problem; see [12, Chapter 10].

Conjecture 4.1.4. With notation as above, 휌 = 2.

Suitably modified Chebyshev polynomials give an infinite family of totally positive algebraic integers with normalized trace 2, so 휌 6 2. The current best lower bound for 휌 is 1.79193; see [36].

The following proposition is implicit in [8].

Proposition 4.1.5. We have

lim (︀(푞 + 1)2 − 퐴(푞2))︀ 휌. 푞→∞ >

80 2 2 −2 For every 푞 we have (푞 + 1) 6 퐴(푞 ) + 2 + 푞 .

Assuming Conjecture 4.1.4, from Proposition 4.1.5 we get 퐴(푞2) = (푞+1)2−2+표(1) as 푞 → ∞. Even though the exact values of 푎(푞), 퐴(푞) seem hard to determine, the following proposition shows that they are not far from the Weil bounds, as the following proposition shows.

Proposition 4.1.6. For every prime power 푞, the following inequalities hold:

√ 2 √ 2 ( 푞 − 1) 6 푎(푞) 6 ⌈( 푞 − 1) ⌉ + 2,

√ 2 −1 √ 2 ⌊( 푞 + 1) ⌋ − 2 − 푞 6 퐴(푞) 6 ( 푞 + 1) .

Even though the Weil bounds cannot be significantly strengthened for large values of 푞, it is still interesting to get some improvements. The following theorem of Aubry, Haloui and Lachaud gives one such improvement.

Theorem 4.1.7. [8, Corollaries 2.2 and 2.14] For every prime power 푞, the following inequalities hold:

√ 2 √ 2 푎(푞) > ⌈( 푞 − 1) ⌉, 퐴(푞) 6 ⌊( 푞 + 1) ⌋.

We will derive improved estimates for 푎(푞) and 퐴(푞) without using metric properties of traces. Our method works especially well for small values of 푞, and we mostly focus on this case. As a demonstration of the method for arbitrary 푞 we give a simple proof of a stronger version of Theorem 4.1.7.

Theorem 4.1.8. For any prime power 푞, the following inequalities hold:

√ 2 √ 2 푎(푞) > ⌊( 푞 − 1) ⌋ + 1, 퐴(푞) 6 ⌈( 푞 + 1) ⌉ − 1.

Theorem 4.1.8 is equivalent to Theorem 4.1.7 when 푞 is not a square. For small values of 푞 we obtain the following result.

81 Theorem 4.1.9. For 푞 = 2, 3, 4, 5, 7, 8, 9 the upper and lower bounds on 푎(푞) and 퐴(푞) are given in Table 4.1.

푞 푎(푞) 퐴(푞) 2 1 4.035 3 1.359 5.634 4 2.275 7.382 5 2.7 8.835 7 3.978 11.734 8 4.635 13.05 9 5.47 14.303

Table 4.1: Lower and upper bounds on 푎(푞) and 퐴(푞), respectively.

Madan and Sät [39] give an explicit list of all isogeny classes of simple abelian

varieties over F2 with #퐴(F2) = 1. We do not know if for some 푘 > 1 there are

infinitely many simple abelian varieties with #퐴(F2) = 푘.

4.2 Abelian varieties over large fields

We use an explicit description of the set 풜푞 of isogeny classes of simple abelian varieties provided by the Honda–Tate correspondence (see, for example [57]). Recall

that by Honda-Tate, the set 풜푞 is in bijection with the set 풲푞 of Galois orbits of 푞- Weil numbers. For our purposes it is more convenient to use an equivalent description

of the set 풜푞 in terms of certain totally real algebraic integers.

′ Proposition 4.2.1. Let 풜푞 denote the set of Galois orbits of totally real algebraic in- √ √ tegers 훼 such that 훼 and all of its Galois conjugates lie in the interval [︀( 푞 − 1)2, ( 푞 + 1)2]︀.

′ Then there is a bijection 휑 : 풜푞 → 풲푞 such that for every 훼 ∈ Q, with Gal(Q/Q)훼 ∈ ′ 풜푞 the abelian variety 퐴 corresponding to 휑(Gal(Q/Q)훼) under the Honda-Tate correspondence satisfies

1/ dim 퐴 1/ deg 훼 (︁ )︁ (Norm 훼) = #퐴(F푞)

82 Proof. Given a 푞-Weil number 훾, define the algebraic integer 훼 by 훼 := (1 − 훾)(1 − √ 훾) = 1 + 푞 − 훾 − 푞/훾. The integer 훼 is totally positive. Since |훾| = 푞 and 훼 = 1 + 푞 + 2Re(훾), we conclude that 훼 and all of its conjugates belong to the segment √ √ [︀1 + 푞 − 2 푞, 1 + 푞 + 2 푞]︀.

′ Let 휓 : 풲푞 → 풜푞 be the map 훾 ↦→ (1 − 훾)(1 − 훾). We claim that 휓 is a bijection. ′ Given 훼 ∈ Q, with Gal(Q/Q)훼 ∈ 풜푞, a root of 푥+푞/푥 = 1+푞 −훼 is a 푞-Weil number 2 (the discriminant of this quadratic equation is (훼 − 1 − 푞) − 4푞 6 0, and the product ′ of its roots is 푞). This defines a map 휑 : 풜푞 → 풲푞 which is inverse to 휓.

If 퐴 is an abelian variety corresponding to the 푞-Weil number 훾, and 훼 ∈ 휑(Gal(Q/Q)훾), 1/2푔 1/ deg 훾 1/2 deg 훼 then #퐴(F푞) = (Norm(1 − 훾)) = (Norm 훼) .

Proposition 4.2.1 shows that we need to understand the possibilities for the norm of √ √ a totally real algebraic integer whose conjugates lie in the interval [( 푞−1)2, ( 푞+1)2].

√ 2 deg 훼 √ 2 deg 훼 The trivial inequalities ( 푞 − 1) 6 Norm(훼) 6 ( 푞 + 1) are equivalent to the Weil estimates under the correspondence of Proposition 4.2.1. We will produce totally real integers with almost extremal norms in Proposition 4.2.3 by utilizing shifted Chebyshev polynomials of Lemma 4.2.2; Proposition 4.2.3 combined with Theorem 4.1.1 gives Proposition 4.1.6.

Lemma 4.2.2. Let 푇푛 denote the Chebyshev polynomial of degree 푛 > 1, and let 푃푛

be the integer monic polynomial defined by 푃푛 := 푇푛(푥/2 − 1). Then for 푁 > 1 we have:

1/푛 푁 + 2 − 1/푁 lim |푃푛(−푁)| 푁 + 2. 6 푛→∞ 6

Proof. We use the formula

1 (︁(︁ √︀ )︁푛 (︁ √︀ )︁푛)︁ 푇 (푦) = 푦 + 푦2 − 1 + 푦 − 푦2 − 1 , 푛 2

1/푛 see [40, Equation 1.49]. By calculus, for 푀 > 1 we have lim푛→∞ |푇푛(−푀)| = 푀 + √ 2 1/푛 푀 − 1. Substituting 푀 = 1+푁/2 gives an explicit formula for lim푛→∞ |푃푛(−푁)| , and the conclusion follows.

83 Proposition 4.2.3. The numbers 푎(푞), 퐴(푞) satisfy

√ 2 √ 2 −1 푎(푞) 6 ⌈( 푞 − 1) ⌉ + 2, 퐴(푞) > ⌊( 푞 + 1) ⌋ − 2 − 푞 .

Proof. To prove either inequality it suffices to construct infinitely many algebraic

′ integers 훼 ∈ 풜푞 with geometric mean of the conjugates of 훼 close to the corresponding √ √ end of the interval [︀(1 − 푞)2, (1 + 푞)2]︀. Let 푁 be a positive integer. For a prime

number ℓ let 푃ℓ be the polynomial of Lemma 4.2.2. Recall that the roots of 푇ℓ belong

to the segment [−1, 1], so the roots of 푃ℓ belong to [0, 4]. By irreducibility of the

cyclotomic polynomial, the monic integer polynomial 푃ℓ factors as 푃ℓ = (푥 − 2)푅ℓ,

where 푅ℓ is irreducible. Let 훼ℓ denote a root of 푅ℓ(푥 − 푁), then 훼ℓ and all of its

conjugates belong to the interval [푁, 푁 + 4]. The norm of 훼ℓ satisfies

1/ deg 훼ℓ 1/(ℓ−1) 1/ℓ lim Norm(훼ℓ) = lim |푅ℓ(−푁)| = lim |푃ℓ(−푁)| . ℓ→∞ ℓ→∞ ℓ→∞

√ The right hand side is within 1/푁 of 푁 +2 by Lemma 4.2.2. Taking 푁 = ⌈(1− 푞)2⌉ √ √ produces infinitely many algebraic integers on [︀(1 − 푞)2, (1 + 푞)2]︀ with geometric

√ 2 mean of the conjugates asymptotically less than ⌈(1 − 푞) ⌉ + 2. Therefore 푎(푞) 6 √ √ ⌈( 푞−1)2⌉+2. Similarly, taking 푁 = ⌊( 푞−1)2⌋−4, gives the estimate on 퐴(푞).

Proposition 4.2.3 shows that the bounds of Theorem 4.1.1 are almost tight. We now give a simple proof of a more precise version of Proposition 4.1.5 (the original claim is recovered by taking limits as 푞 → ∞).

Proposition 4.2.4. Let 휌 be as in Definition 4.1.3. Then the following inequalities hold:

2 −2 2 2 −1 (푞 + 1) − 2 − 푞 6 퐴(푞 ) 6 (푞 + 1) − 휌 + 푂(푞 ).

Proof. The first inequality follows from Proposition 4.2.3. For the second inequality,

′ 2 2 fix a prime power 푞 and an element 훼 ∈ 풜푞. For 푥 ∈ [(푞 − 1) , (푞 + 1) ] Taylor

84 expansion of log at (푞 + 1)2 gives

푥 − (푞 + 1)2 log 푥 2 log(푞 + 1) + + 푂(푞−3) (4.1) 6 (푞 + 1)2

Applying 4.1 over the conjugates of 훼 and averaging gives the following inequality, where tr denotes the normalized trace (trace divided by the degree) and the implied constants do not depend on 훼 or 푞:

tr (훼 − (푞 + 1)2) log(Norm(훼)1/ deg 훼) 2 log(푞 + 1) + + 푂(푞−3). 6 (푞 + 1)2

Exponentiating and using Taylor expansion of the exponent we get

2 1/ deg 훼 2 tr(훼−(푞+1)2)/(푞+1)2 −1 퐴(푞 ) = lim sup Norm(훼) 6 lim sup(푞 + 1) 푒 + 푂(푞 ) ′ ′ 훼∈풜푞 훼∈풜푞

2 2 −1 6 (푞 + 1) + lim sup tr(훼 − (푞 + 1) ) + 푂(푞 ). ′ 훼∈풜푞

2 2 Since (푞 + 1) − 훼 is a totally positive algebraic integer, we have 퐴(푞 ) 6 (푞 + 1)2 − 휌 + 푂(푞−1).

The following theorem combined with Proposition 4.2.3 show that 푎(푞), 퐴(푞) can be determined up to an error of 1 + 푞−1.

Theorem 4.2.5. For all abelian varieties 퐴 ∈ 풜푞 one of the following holds

′ √ 2 √ 2 1. The element of 풜푞 corresponding to 퐴 is ⌊( 푞 − 1) ⌋ or ⌈( 푞 + 1) ⌉,

√ 2 1/푔 √ 2 2. ⌊( 푞 − 1) ⌋ + 1 6 #퐴(F푞) 6 ⌈( 푞 + 1) ⌉ − 1.

′ Proof. We want to estimate the norm of an algebraic integer 훼 ∈ 풜푞. First we derive the lower bound. By calculus, for every integer 푛 > 0 the function

푥 푥 ↦→ (푥 − 푛)1/(푛+1)

85 on the open interval (푛, +∞) has minimum 푛 + 1 at the point 푥 = 푛 + 1. Let 푛 :=

√ 2 ′ ⌊( 푞 − 1) ⌋ and let 훼 ∈ Q ∖ {푛} be an algebraic integer such that Gal(Q/Q)훼 ∈ 풜푞.

Let 훼1, 훼2, ..., 훼푑 denote the conjugates of 훼. We have

Norm 훼 ∏︁ 훼푖 = (푛 + 1)푑. ∏︀ (훼 − 푛)1/(푛+1) (훼 − 푛)1/(푛+1) > 푖 푖 푖 푖 ∏︀ Since 훼 is an algebraic integer, the product 푖(훼푖 − 푛) is a rational integer and ∏︀ therefore | 푖(훼푖 − 푛)| > 1. Hence if 퐴 ∈ 풜푞 is an abelian variety corresponding to 1/ dim 퐴 1/푑 훼, then #퐴(F푞) = (Norm 훼) > (푛 + 1). To get an upper bound, let 푁 = √ ⌈( 푞 + 1)2⌉. The function 푥(푁 − 푥)1/(푁−1) has a maximum of 푁 − 1 at 푥 = 푁 − 1.A

′ 푑 similar argument shows that for every 훼 ∈ 풜푞, 훼 ̸= 푁 we have Norm 훼 6 (푁 −1) .

Theorem 4.2.5 improves on Theorem 4.1.7 when 푞 is a square. In particular, combining Theorem 4.2.5 with Proposition 4.2.3 determines 푎(푞2), 퐴(푞2) up to an error of 1/2 + 푞−2.

It may be possible to improve Theorem 4.2.5 by replacing the function 푥/(푥 − 푛)1/(푛+1) by a different auxiliary function. We do not know if this can be donefor large 푞. However, in the next section we find better auxiliary functions for small 푞 and use them to improve the Weil estimates.

4.3 Abelian varieties over small fields

The following lemma gives a general form of the auxiliary function method used in the proof of Theorem 4.2.5.

Lemma 4.3.1. Suppose that for some positive 퐴, 퐵, 푚, 푀 ∈ R, some monic integer

polynomials 푃1, ..., 푃푛, 푄1, ..., 푄푚 ∈ Z[푥], and some positive 훾1, ..., 훾푛, 훽1, ..., 훽푚 ∈ R, the following inequalities hold for all 푥 ∈ [퐴, 퐵]:

푥 푚 ∏︀ 훾푖 > 푖 |푃푖(푥)|

86 ∏︁ 훽푗 푥 |푄푗(푥)| 6 푀. 푗 Suppose that 훼 is an algebraic integer whose conjugates lie in [퐴, 퐵] and such that

푃푖(훼), 푄푗(훼) ̸= 0 for all 푖, 푗. Then

1/ deg 훼 푚 6 Norm(훼) 6 푀.

Proof. We will prove the lower bound, the upper bound can be derived similarly. Let

{훼1, ..., 훼푑} be the Galois orbit of 훼 = 훼1. Since 푃푖 is a monic integer polynomial, the value of the product Π푗|푃푖(훼푗)| is a nonzero integer for every 푖. Therefore

(︂ )︂ Norm 훼 ∏︁ 훼푗 푑 Norm 훼 > = > 푚 . ∏︀ ∏︀ |푃 (훼 )|훾푖 ∏︀ 푃 (훼 )훾푖 푖 푗 푖 푗 푗 푖 푖 푗

We apply Lemma 4.3.1 to give bounds for 푎(푞) and 퐴(푞) for small values of 푞.

Theorem 4.3.2. For all but finitely many simple abelian varieties of dimension 푔 over

1/푔 F푞 the inequalities 푚(푞) 6 #퐴(F푞) 6 푀(푞) hold, where the values of 푚(푞), 푀(푞) ′ are given in Table 4.2. The minimal polynomials 푃푖, 푄푗 of elements of 풜푞 that do not satisfy the lower and the upper bound, respectively, are exactly the polynomials of Table 4.3 marked with an asterisk.

푞 푚(푞) 푀(푞) 2 1 4.035 3 1.359 5.634 4 2.275 7.382 5 2.7 8.835 7 3.978 11.734 8 4.635 13.05 9 5.47 14.303

1/푔 Table 4.2: Lower and upper bounds on #퐴(F푞)

Proof. For every 푞 we obtain the bounds by applying Lemma 4.3.1 to the segment

87 √ 2 √ 2 [( 푞−1) , ( 푞+1) ] and using the auxiliary polynomials 푃푖, 푄푗 listed in Table 4.3; the

corresponding parameters 훾푖, 훽푗 are in Table 4.4. We know explain how the auxiliary

functions 푃푖 and corresponding parameters 훾푖 were found, the values of 푄푗, 훽푗 were obtained similarly.

′ The auxiliary functions were found by searching for algebraic integers in 풜푞 with

small degree and small norm and taking 푃푖 to be equal to the corresponding minimal polynomials. The parameters 훾푖 were then chosen by solving a linear programming problem of maximizing the minimum of the auxiliary function of Lemma 4.3.1 on a

√ 2 √ 2 fine mesh 푆 ⊂ [( 푞 − 1) , ( 푞 + 1) ]. Explicitly, after 푆 and 푃푖 are fixed, the values

of 훾푖 were found at which

(︁ ∑︁ )︁ max min log 푥 − 훾푖 log 푃푖(푥) 훾푖 푥∈푆

is attained. Finally, the values of 훾푖 were used to create the auxiliary function 푥 ·

∏︀ −훾푖 푖 |푃푖(푥)| and find its minimum using calculus.

′ ′ 푞 푃푖 ∈ 풜푞 푄푗 ∈ 풜푞 2 n/a 푥−5*, 푥2 −9푥+19, 푥3 −13푥2 +54푥−71 3 푥 − 1*, 푥2 − 4푥 + 2, 푥3 − 7푥2 + 12푥 − 5 푥 − 7*, 푥 − 6*, 푥2 − 12푥 + 34 4 푥 − 1*, 푥 − 3, 푥2 − 5푥 + 5*, 푥3 − 8푥2 + 푥 − 9*, 푥 − 8*, 푥2 − 15푥 + 55*, 푥3 − 19푥 − 13 22푥2 + 159푥 − 377* 5 푥−2*, 푥2−6푥+7*, 푥3−10푥2+28푥−23, 푥 − 10*, 푥 − 9*, 푥2 − 18푥 + 79, 푥2 − 푥3 − 10푥2 + 30푥 − 26 17푥 + 71, 푥2 − 17푥 + 69 7 푥 − 3*, 푥2 − 10푥 + 23, 푥3 − 13푥2 + 푥 − 13*, 푥 − 12*, 푥2 − 23푥 + 131, 54푥 − 71, 푥3 − 14푥2 + 61푥 − 83 푥2 −22푥+119, 푥3 −35푥2 +406푥−1561, 푥3 − 34푥2 + 381푥 − 1405, 푥3 − 34푥2 + 379푥 − 1379 8 푥 − 3*, 푥 − 4*, 푥2 − 9푥 + 19*, 푥2 − 푥 − 14*, 푥2 − 27푥 + 181* 10푥 + 23 푥3 − 15푥2 + 68푥 − 97*, 푥3 − 15푥2 + 71푥 − 107 9 푥 − 4*, 푥 − 5*, 푥 − 6, 푥2 − 11푥 + 29*, 푥 − 16*, 푥 − 15*, 푥2 − 129푥 + 209*, 푥2 − 12푥 + 33, 푥2 − 12푥 + 34 푥3 − 43푥2 + 614푥 − 2911

Table 4.3: Auxiliary polynomials for Theorem 4.3.2.

88 푞 훾푖 훽푗 2 n/a 0.141, 0.23, 0.09 3 0.306, 0.199, 0.019, 0.05, 0.108 0.1445, 0.155, 0.099 4 0.37, 0.12, 0.065, 0.01 0.054, 0.112, 0.02, 0.08 5 0.323, 0.063, 0.062, 0.007 0.11, 0.08, 0.066, 0.001, 0.003 7 0.289, 0.0048, 0.0457, 0.0178 0.055, 0.033, 0.026, 0.003, 0.035, 0.009, 0.006 8 0.044, 0.13, 0.09, 0.01, 0.02 0.08, 0.04 9 0.15, 0.08, 0.02, 0.03, 0.002, 0.033, 0.037, 0.033, 0.02 0.003

Table 4.4: Auxiliary parameters for Theorem 4.3.2

We apply the results of Theorem 4.3.2 to bound #퐴(F푞)[2], the number of rational 2-torsion points on a simple abelian variety 퐴; a similar result for Jacobians is [11, Theorem 7.1].

Corollary 4.3.3. Suppose 푞 = 2 or 푞 = 3. Then for all but finitely many simple

abelian varieties 퐴 over F푞 we have

⎧ ⎨⎪2.717푔, for 푞 = 2 퐴(F푞)[2] 6 . ⎩⎪3.782푔, for 푞 = 3

Proof. Let Frob denote the 푞-power Frobenius endomorphism of 퐴. A rational 2-

torsion point is in the kernel of both Frob +1 and Frob −1. Therefore #퐴(F푞)[2] 6 √︀ 2 1/2 1/2 deg(Frob −1) deg(Frob +1) = deg(Frob −1) = #퐴(F푞2 ) . Applying Theorem 4.3.2 to the right hand side proves the claimed inequalities.

Given a field extension 퐿/퐾 and a scheme 푋 over 퐾, a point 푥 ∈ 푋(퐿) is called new if 푥 ̸∈ 푋(퐹 ) for all fields 퐹 with 퐾 ⊆ 퐹 ( 퐿.

Corollary 4.3.4. Let 퐴/F푞 be a simple abelian variety. Then 퐴 has no new points over F푞푟 if and only if one of the following holds:

1. 푟 = 2, 푞 ∈ {2, 3, 4}, and 퐴 is the quadratic twist of an abelian variety 퐴′ with

′ 퐴 (F푞) = 0 (these are described in Theorem 4.3.2 for 푞 = 3, 4, and in [39] for 푞 = 2),

89 ′ 2. 푟 = 3, 푞 = 2, and the element of 풜푞 corresponding to 퐴 is 4 or 5.

Proof. Suppose 푟 = 2. The equality 퐴(F푞2 ) = 퐴(F푞) is equivalent to the assertion ′ that the quadratic twist 퐴 of 퐴 has a unique rational point over F푞. From now on suppose 푟 > 2.

The Weil conjectures imply that 퐴 has a new point when 푞 and 푟 are sufficiently

large, as we will now show. Suppose that for some abelian variety 퐴/F푞 of dimension

푔 the set 퐴(F푞푟 ) has no new points. Then the following inequalities hold

(︀ 푟/2 )︀2푔 ∑︁ ∑︁ (︀ 푑/2 )︀2푔 √ (︀ 푟/4 )︀2푔 푞 − 1 6 #퐴(F푞푟 ) 6 #퐴(F푞푑 ) 6 푞 + 1 6 2 푟 푞 + 1 , 푑|푟, 푑<푟 푑|푟, 푑<푟

√ 푟/2 2푔 √ 푟/4 2푔 because the number of divisors of 푟 is at most 2 푟. So (푞 − 1) 6 2 푟(푞 + 1) , 푟/4 2푔 √ which implies (푞 − 1) 6 2 푟. The last inequality together with the condition 푟 > 2 imply that the pair (푞, 푟) is equal to one of the following (2, 3), (2, 4), (2, 5), (2, 6), (3, 3), (3, 4), or (4, 3). Out of these only the pairs (2, 3), (2, 4) satisfy the in- 푟/2 2푔 ∑︀ 푑/2 2푔 equality (푞 − 1) 6 푑|푟,푑<푟(푞 + 1) for some 푔 > 1. Suppose 푞 = 2, 푟 = 3, 푒 then 퐴(F8) = 퐴(F2). Since 퐴 is simple, 퐴F8 is isotypic: 퐴F8 ∼ 퐵 with 푒 6 3. There- fore, by Theorem 4.3.2 the following inequality holds with finitely many exceptions

1/푔 1/푔 #퐴(F2) 6 4.04 < 4.635 6 #퐴(F8) . In each of the exceptional cases we test

if 퐴(F2) = 퐴(F8) by a direct computation; the resulting exceptions are listed in the

statement of Case 2. Suppose that 푞 = 2 and 푟 = 4, which means 퐴(F16) = 퐴(F2). ′ Then 퐴(F4) = 퐴(F16), and so 퐴F4 is the quadratic twist of an abelian variety 퐴 with ′ ′ ′ 퐴 (F4) = 0. Theorem 4.3.2 implies that the element of 풜4 corresponding to 퐴 is 1. ′ A direct computation shows that the element of 풜2 corresponding to 퐴 is 3 and that for this abelian variety 퐴(F2) ̸= 퐴(F16).

90 Appendix A

Tables

A.1 Cycle types of Mathieu groups

The following table describes cycle types of Mathieu groups in their standard multiply transitive actions; see [59]. The expression (32, 93, 5) means a product of two 3-cycles, three 9-cycles, and one 5-cycle, all of which are disjoint.

Group Number of points Cycle types 4 3 2 2 푀11 11 2 , 3 , 4 , 5 , (2, 3, 6), (2, 8), 11 4 3 2 2 2 푀11 12 2 , 3 , (2 , 4 ), 5 , (2, 3, 6), (4, 8), 11 6 4 3 4 2 2 2 2 2 푀12 12 2 , 2 , 3 , 3 , (2 , 4 ), 4 , 5 , 6 , (2, 3, 6), (4, 8), (2, 8), (2, 10), 11 8 6 2 4 4 2 2 2 3 2 2 푀22 22 2 , 3 , (2 , 4 ), 5 , (2 , 3 , 6 ), 7 , (2, 4, 8 ), 11 7 8 11 4 3 4 6 2 4 4 Aut(푀22) 22 2 , 2 , 2 , (2, 4 ), (2 , 4 ), 3 , (2 , 4 ), 5 , (2, 32, 62), (22, 32, 62), 73, (2, 4, 82), (4, 82), (2, 102), 112, (4, 6, 12), (7, 14) 8 6 2 4 4 2 2 2 3 2 2 푀23 23 2 , 3 , (2 , 4 ), 5 , (2 , 3 , 6 ), 7 , (2, 4, 8 ), 11 , (2, 7, 14), (3, 5, 15), 23 8 12 6 8 4 4 2 4 6 4 2 2 2 푀24 24 2 , 2 , 3 , 3 , (2 , 4 ), (2 , 4 ), 4 , 5 , (2 , 3 , 6 ), 64, 73, (2, 4, 82), 22, 102, 112, (2, 4, 6, 12), 122, (2, 7, 14), (3, 5, 15), (2, 21), 23

Table A.1: Cycle types of Mathieu groups

91 A.2 Factors of trinomials over finite fields

2 In the following table 푐 ∈ F4 is a root of 푥 + 푥 + 1.

Polynomial Field Degrees of irreducible factors 11 푥 + 푥 + 1 F2 2 × 9 11 푥 − 푥 − 1 F5 1 × 3 × 7 11 3 푥 + 푥 + 1 F2 5 × 6 11 4 푥 + 푥 + 1 F7 1 × 1 × 2 × 7 11 5 푥 + 푥 + 1 F3 1 × 3 × 7 11 5 푥 + 푥 + 1 F2 3 × 8 23 푥 + 푥 + 1 F2 2 × 8 × 13 23 푥 + 푥 + 1 F11 1 × 2 × 5 × 15 23 2 푥 + 푥 + 1 F3 1 × 2 × 20 23 2 푥 + 푥 + 1 F7 7 × 16 23 3 푥 + 푥 + 1 F5 1 × 22 23 4 푥 + 푥 + 1 F19 1 × 1 × 1 × 4 × 7 × 9 23 5 푥 + 푥 + 1 F3 1 × 2 × 5 × 7 × 8 23 5 푥 + 푐푥 + 1 F4 1 × 9 × 13 23 6 푥 + 푥 + 1 F17 2 × 3 × 9 × 9 23 7 푥 + 푥 + 1 F2 2 × 10 × 11 23 8 푥 + 푥 + 1 F3 1 × 3 × 19 23 8 푥 + 푥 + 1 F5 1 × 4 × 5 × 13 23 9 푥 + 푥 + 푐 F4 1 × 2 × 20 23 9 푥 + 푥 + 1 F7 4 × 19 23 10 푥 + 푥 + 1 F13 1 × 1 × 4 × 6 × 11 23 11 푥 + 푥 + 1 F2 5 × 6 × 12 23 11 푥 + 푥 + 1 F3 1 × 3 × 19 Table A.2: Factorization of some trinomials

92 Bibliography

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