Arboreal Representations, Sectional Monodromy Groups
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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 rep- resentations 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 K, a separable polynomial f 2 K[x], and an element t 2 K, the full − S ∘−k backward orbit O (t) := k f (t) has a natural action of the Galois group GalK . For a fixed f 2 K[x] with deg f = d and for most choices of t, the orbit O−(t) has − the structure of complete rooted d-ary tree T1. The Galois action on O (t) thus defines a homomorphism 휑 = 휑f;t : GalK ! Aut T1. The map 휑f;t is the arboreal representation attached to f and t. In analogy with Serre’s open image theorem, one expects im 휑 = Aut T1 to hold for most f; t, but until very recently for most degrees d not a single example of a degree d polynomial f 2 Q[x] with surjective 휑f;t 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 n geometrically integral proper curve X ⊂ PK , consider the generic hyperplane H 2 nV PK(t1;:::;tn). The intersection H \X is the spectrum of a finite separable field extension L=K(t1; :::; tn) of degree d := deg X. The Galois group GX := Gal(L=K(t1; :::; tn)) is known as the sectional monodromy group of X. When char K = 0, the group GX equals Sd for all curves X. This result has numerous applications in algebraic geometry, in particular to the degree-genus problem. However, when char K > 0, the sectional monodromy groups can be smaller. We classify all nonstrange nondegenerate n curves X ⊂ P , for n > 3 such that GX 6= Ad;Sd. 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 Fq-points on simple abelian varieties over finite fields; these improve upon the Weil bounds. For example, when q = 3; 4 the Weil bound gives #A(Fq) > 1 for all abelian varieties A. We prove 3 dim A dim A that A(F3) > 1:359 , and A(F4) > 2:275 hold for all but finitely many simple abelian varieties A (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 a(q) and A(q), respectively. 82 1=g 4.2 Lower and upper bounds on #A(Fq) ................. 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 d topological covering of manifolds f : Y ! X gives rise to a homo- −1 1 morphism 휑: 휋1(X; x) ! Aut(f (x)) = Sd for any choice of basepoint x 2 X. In this way, the finite covering f produces the group Mon(f) := 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 fun- damental 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 mon- odromy groups. Section 1.1 contains a brief introduction to the theory of étale fun- damental 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 mon- odromy 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 ': Y ! X is called étale if it is smooth of relative dimension 0. In other words, locally on Y the morphism ' corresponds to an extension of rings A ! B of the form B = A[x1; : : : ; xn]=(f1; : : : ; fn) such that the Jacobian determinant det(@fi=@xj) is invertible in B. A morphism of C-varieties X ! Y is étale if and only if it is biholomorphic. Example 1.1.2. The following are examples of étale morphisms. 1. An open immersion Y ! X. 2. A finite separable extension of fields Spec L ! Spec K. 1 1 p 3. The Artin-Schreier cover ': AK ! AK given by '(x) = x − x, where p = char K > 0. Suppose X is a connected scheme. Fix a geometric point x of X, that is a mor- phism x: Spec K ! X for some algebraically closed field K. Let FÉtX denote the category of finite étale coverings of X. Let Fx : FÉtX ! Sets denote the functor sending a covering ': Y ! X to the fiber '−1(x). Fact 1.1.3 ([1] Section V.7). The category FÉtX is a Galois category and Fx is its fiber functor, so Fx defines an equivalence of categories between FÉtX and the category of Aut(Fx)-sets. In other words, there is a Galois correspondence between finite étale coverings of X and open subgroups of the profinite group Aut(Fx). 14 et The group 휋1 (X; x) := Aut(Fx) is called the étale fundamental group of X with base point x. The following theorem supports the claim that étale fundmental groups are similar to fundamental groups in topology. Recall that for a C-variety X the set X(C) inherits the natural analytic topology from C. Theorem 1.1.4 (“Riemann’s existence theorem”, [1] Theorem XII.5.1). Suppose X is a connected separated scheme of finite type over C and x 2 X(C) is a point. There et ∼ top is a canonical isomorphism 휋1 (X; x) ! 휋b1(X(C) ; x) between the étale fundamental group of X, and the profinite completion of the topological fundamental group of X(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 X(C). When X 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 K. Qn Finite étale coverings of Spec K are of the form Spec L for L = i=1 Li a finite product of finite separable field extensions Li=K. So the fundamental group of Spec K is just et sep the absolute Galois group of K, 휋1 (Spec K) = Gal(K =K). For a general variety, the étale fundamental group is an extension of the Galois group of the ground field by the geometric fundamental group.