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Hilbert Class Fields of Imaginary Quadratic Fields and Reflex Fields of Certain Sextic CM Fields

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University of Vermont ScholarWorks @ UVM Graduate College Dissertations and Theses Dissertations and Theses 2017 Hilbert Class Fields of Imaginary Quadratic Fields and Reflex ieldF s of Certain Sextic CM Fields Garvin Gaston University of Vermont Follow this and additional works at: https://scholarworks.uvm.edu/graddis Part of the Mathematics Commons Recommended Citation Gaston, Garvin, "Hilbert Class Fields of Imaginary Quadratic Fields and Reflex ieF lds of Certain Sextic CM Fields" (2017). Graduate College Dissertations and Theses. 808. https://scholarworks.uvm.edu/graddis/808 This Thesis is brought to you for free and open access by the Dissertations and Theses at ScholarWorks @ UVM. It has been accepted for inclusion in Graduate College Dissertations and Theses by an authorized administrator of ScholarWorks @ UVM. For more information, please contact [email protected]. Hilbert Class Fields of Imaginary Quadratic Fields and Reflex Fields of Certain Sextic CM Fields A Thesis Presented by Garvin Gaston to The Faculty of the Graduate College of The University of Vermont In Partial Fulfillment of the Requirements for the Degree of Master of Science Specializing in Mathematics October, 2017 Defense Date: August 2, 2017 Thesis Examination Committee: Christelle Vincent, Ph.D., Advisor Christian Skalka, Ph.D., Chairperson Richard Foote, Ph.D. Cynthia J. Forehand, Ph.D., Dean of Graduate College Abstract In this thesis we look at particular details of class field theory for complex multipli- cation fields. We begin by giving some background on fields, abelian varieties, and complex multiplication. We then turn to the first task of this thesis and give an implementation in Sage of a classical algorithm to compute the Hilbert class field of a quadratic complex multiplication field using the j-invariant of elliptic curves with complex multiplication by the ring of integers of the field, and we include three explicit examples to illustrate the algorithm. The second part of this thesis contains new results: Let K be a sextic complex multiplication field with Galois closure L such that the Galois group of L over Q is isomorphic to D12, the dihedral group with twelve elements. For each complex multiplication type Φ of K, we compute the reflex field and reflex type of the pair (K, Φ) explicitly. We then illustrate our results with the case of K = Q[x]/(x6 − 2x5 + 2x4 + 2x3 + 4x2 − 4x + 2). Acknowledgements First and foremost, I’d like to express my overwhelming gratitude to my advisor, Professor Christelle Vincent, for her unwavering guidance, her obvious love and en- thusiasm for mathematics, her talent for teaching, and perhaps above all her sense of humor and her patience throughout this process. She has gone above and be- yond what I expected from an advisor, and I will forever be grateful to her for her dedication. I would also like to thank my family: my parents Gaynor and Tony Banham and John Gaston; my sister and brother-in-law Caroline and Noush Haghpeykar and my nephew Johnny; and my dogs Juno and Martin. I depended on their emotional support like I never have before over the past two years. Thank you. Finally, I have been so fortunate to have incredible teachers throughout my life. I thank them all, and I hope to follow in their footsteps. ii Table of Contents Acknowledgements............................... ii 1 Introduction1 1.1 Explicit class field theory.........................1 1.2 Overview of contents...........................4 2 Field preliminaries7 2.1 Field theory................................7 2.1.1 Fields and field extensions....................7 2.1.2 Finding the minimal polynomial.................8 2.1.3 Number fields........................... 10 2.1.4 Galois extensions......................... 11 2.2 Rings of integers............................. 13 2.2.1 Ideals in a ring.......................... 13 2.2.2 The ring of integers of a number field.............. 14 2.2.3 Dirichlet’s Unit Theorem..................... 16 2.2.4 The different of a number field.................. 17 2.3 Class field theory............................. 17 2.4 Fixed fields................................ 19 3 Abelian varieties 22 3.1 Elliptic curves............................... 22 3.1.1 Preliminaries........................... 22 3.1.2 Analytic Theory......................... 25 3.2 General case................................ 29 3.2.1 Preliminaries........................... 30 3.2.2 Analytic theory.......................... 31 4 Complex Multiplication 36 4.1 Elliptic curves............................... 36 4.2 Generating fK (x) ............................. 38 4.3 CM-types................................. 39 4.4 Construction of dimension 3 abelian varieties.............. 42 5 Elliptic curves and Sage 44 5.1 Computing the j-invariant........................ 44 5.2 Examples................................. 45 iii 6 Reflex fields 52 6.1 Galois group preliminaries........................ 52 6.2 Matching an embedding to a Galois element.............. 53 6.3 Equivalence classes of CM-types..................... 55 6.4 Finding reflex types............................ 58 6.4.1 Reflex of Φ1 ............................ 59 6.4.2 Reflex of Φ2 ............................ 60 6.4.3 Reflex of Φ3 ............................ 61 6.4.4 Reflex of Φ4 ............................ 61 6.4.5 Reflex of Φ1 ............................ 62 6.4.6 Reflex of Φ2 ............................ 62 6.4.7 Reflex of Φ3 ............................ 63 6.4.8 Reflex of Φ4 ............................ 64 7 Example 65 7.1 Preliminaries............................... 65 7.2 Computation of Reflex Fields and Reflex Types............ 70 7.2.1 CM-types Φ1 and Φ1 ....................... 71 7.2.2 CM-types Φ2 and Φ2 ....................... 73 7.2.3 CM-types Φ3 and Φ3 ....................... 74 7.2.4 CM-types Φ4 and Φ4 ....................... 75 Bibliography 78 A Sage code 80 iv Chapter 1 Introduction Given a number field K, it is of interest to consider its abelian field extensions, by which we mean the Galois extensions of K with abelian Galois group. The study of these extensions is called class field theory, which more generally is a branch of algebraic number theory that looks at abelian extensions of local and global fields, and the arithmetic properties of these extensions. Of particular interest is the Hilbert class field, which is the maximal abelian unramified extension of a field. If K is a number field, its Hilbert class field exists and has finite degree over K, as shown by Furtwängler [6]. 1.1 Explicit class field theory In [4], Daberkow and Pohst give an algorithm to explicitly compute the Hilbert class field of arbitrary number fields. However, the algorithm is not practical, as it requires computing a large number of Kummer extensions of the base field. In any case, their algorithm shows that the problem is decidable. 1 If one is willing to restrict the class of number fields considered, much more can be said. If K = Q, then all of the abelian Galois extensions of K are contained in some th Q(ζn), where ζn is a primitive n root of unity. In other words, for any extension L/Q that is abelian, there exists an n such that L ⊆ Q(ζn). It is important to note that not every Galois extension of Q is contained in some Q(ζn), just the abelian extensions. However, all of these extensions are ramified, as Q does not have any unramified extensions. The next case for which we have an explicit construction of abelian extensions is that of K, an imaginary quadratic field, i.e., a quadratic CM field. For example, we can construct the Hilbert class field of K, which we denote HK , explicitly: Theorem 1.1.1. Let K be an imaginary quadratic field. Then its Hilbert class field is K(j) for j the j-invariant of an elliptic curve with endomorphism ring OK , where OK is the ring of integers of K. In this thesis, we provide code that, given any imaginary quadratic field K, gives a monic polynomial fK with integer coefficients such that HK = Q[x]/(fK (x)). The algorithm that we give is not new, and is not an optimal implementation. It is merely a toy problem to grapple with the higher degree CM fields that form the real heart of this thesis. For an optimal implementation, we direct the reader to Cohen’s book A course in computational algebraic number theory [2]. Although we won’t consider this here, we note that more general abelian extensions of such K can be generated using the torsion points of the elliptic curves appearing in Theorem 1.1.1. The next case for which we have partial explicit constructions is that of K, a general CM field. When K is of degree four or more over Q, we must consider a few differences. First, instead of considering the field of moduli of the abelian variety 2 (which is generated by the j-invariant when the abelian variety has dimension 1), we must give the abelian variety a polarization, and consider the field of moduli of the polarized abelian variety. Secondly, given a polarized abelian variety with endomorphism ring isomorphic to OK , its field of moduli is an extension of the reflex field, which we denote KR, and not necessarily of the field of complex multiplication. Thirdly, this field of moduli will be an unramified abelian extension of KR, but in general it will not be the maximal such extension. In other words, the field of moduli of the polarized abelian variety will not necessarily give us the Hilbert class field of KR. In this thesis, we introduce the definitions and notions necessary to more fully explain the concepts of the previous paragraph. We then consider the situation of K a sextic CM field whose Galois closure L has Galois group isomorphic to D12, the dihedral group with twelve elements. For this class of fields and any CM-type attached to K, we compute the corresponding reflex field and reflex type. In addition, we choose an explicit such field to carry out the computations, as an example of our results.
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