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Class Field Towers, Solvable Galois Representations and Noether's Class field towers, solvable Galois representations and Noether's problem in Galois theory by Jonah Leshin B.A., Northwestern University; Evanston, IL, 2008 M.A., University of Cambridge; Cambridge, UK, 2009 A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics at Brown University PROVIDENCE, RHODE ISLAND May 2014 c Copyright 2014 by Jonah Leshin This dissertation by Jonah Leshin is accepted in its present form by Mathematics as satisfying the dissertation requirement for the degree of Doctor of Philosophy. Date Joseph Silverman, Ph.D., Advisor Recommended to the Graduate Council Date Michael Rosen, Ph.D., Reader Date Stephen Lichtenbaum, Ph.D., Reader Approved by the Graduate Council Date Peter Weber, Dean of the Graduate School iii Vitae The author grew up in Newton, Massachusetts. He received his B.A. in Mathematics in 2008 from Northwestern University. The following year, he attended the University of Cambridge on a Fulbright Scholarship, where he earned an M.S. in Mathematics. He enrolled in the Ph.D. program at Brown University in the fall of 2009 and completed his Ph.D. in the spring of 2014. iv Acknowledgements First and foremost, I would like to thank my advisor, Joe Silverman. Whenever I was stuck, Joe was there with a multitude of suggestions. I am grateful for his guidance, encouragement, and ultimately his investment in my development as a mathematician. I am also thankful to Mike Rosen, who was always happy to speak with me and so often suggested exactly the right reference. I am grateful to have spent the past five years at Brown. I am indebted to many fellow and former graduate students for their helpful conversations. In particular, I would like to acknowledge E. Mehmet Kiral, Florian Sprung, and Wade Hindes. In the course of all the reading, writing, and problem solving that goes with a Ph.D., I have also managed to make several friendships that I hope will be life long. I have truly enjoyed being a part of the welcoming, collegial, Kabob and Curry frequenting math graduate student community. Finally, it is my pleasure to thank my family { my parents Rosalyn and Michael, and my sisters Miriam and Rachel { for their unconditional love and support. My largest debt of gratitude is owed to my wife, Dahlia, for being right beside me through the trials and tribulations of everyday life. v Abstract of \ Class field towers, solvable Galois representations and Noether's problem in Galois theory " by Jonah Leshin, Ph.D., Brown University, May 2014 We begin by investigating the class field tower problem for Kummer extensions of cyclo- tomic fields. Specifically, given primes l and p satisfying certain mild hypotheses, we show pl a b the existence of infinitely many primes q for which the field Q(ζl; p q ) has infinite class field tower, where a and b are appropriately chosen. As an application, we show that the p 3 field Q(ζ3; 79 · 97) has infinite class field tower, giving an example of a \small" field with infinite class field tower. Motivated by the class field tower problem, we next study the behavior of the root discriminant in a solvable number field extension. In particular, we use class field theory to show that for any fixed number field K, there are only finitely many extensions of K of a given solvable length and bounded root discriminant. This theorem should be viewed in light of the fact that all fields in a class field tower have the same root discriminant. In the next part of the thesis, we analyze the possibilities for the fields cut out by a solvable three-dimensional representation of GQ ramified at a single finite prime. In doing so, we give bounds for the number of such representations with given Artin conductor. This work is motivated by similar results in the case of two-dimensional Galois representations, where Langlands-type techniques are available. We then continue our study of Galois representations in the context of torsion points on abelian varieties over number fields. We prove several facts about the image of the representation of Galois acting on the l-torsion points of an abelian variety subject to certain constraints. Lastly, we study a variant of Noether's problem in Galois theory. Building on work of Lenstra [35], we give an upper bound for the degree of irrationality of fields of the form A K(x1; : : : ; xjAj) , where A is a finite abelian group, K is a field, and A acts on the variables xi via the regular representation. Contents Vitae iv Acknowledgments v 1 Class Field Towers 3 1.1 Introduction . 4 1.2 The embedding problem for Kummer extensions of cyclotomic fields . 7 1.2.1 Proof of Theorem 3 . 8 1.2.2 The case l =3 .............................. 13 1.2.3 Some other fields with infinite 3-class field tower . 13 1.3 p-Principal fields . 14 1.4 Solvable number field extensions of bounded root discriminant . 16 1.4.1 Discriminants and ramification groups . 17 1.4.2 Proof of Theorem 7 . 18 1.4.3 Further questions . 30 2 Artin representations 32 2.1 Introduction . 33 2.2 An application of Serre's conjecture . 33 2.3 Counting Artin representations . 35 2.4 Definitions and preliminaries . 37 2.5 Imprimitive representations . 39 2.6 Primitive representations . 43 2.6.1 Representations with projective image isomorphic to P1 . 44 2.6.2 Representations with projective image isomorphic to P3 . 56 2.7 Comparison to the two-dimensional case . 61 3 Torsion points on elliptic curves and abelian surfaces 63 3.1 Introduction . 64 3.2 3-Torsion . 65 3.3 Roots of unity in torsion fields . 66 3.4 A local-global property . 67 vi 4 Noether's Problem in Galois Theory 69 4.1 Introduction . 70 4.2 Notation and Lenstra's Setup . 74 4.3 Bounding the Degree of Irrationality from Above . 76 4.3.1 The General Case . 79 4.3.2 An Example of Theorem 23 . 80 vii 1 Commonly used notation • ζl Primitive lth root of unity. • µl Group of lth roots of unity. • OK Ring of integers of a number field K. ab • K Maximal subfield of K that is abelian over Q. • ClK Ideal class group of a number field K. • hK Class number jClK j of K. • Km Ray class field of K modulo m. m • ClK Ray class group of K modulo m. • dK=F Relative discriminant from K to F . • dK j dK=Q j. • DK=F Relative different from K to F . • G(L=K) Galois group of L over K. • GK G(K=K). • Frobp Frobenius element at p in GK , where p is a prime of K. n • vp(I) Maximal integer n such that p jI. • e(P=p) vP(p). • f(P=p) Residue degree of P over p. • A[l] Kernel of the multiplication by l map on an abelian variety A. • ρA;l Representation of GK acting on A[l], where A is defined over K. • Sn Symmetric group on n letters. 2 • Cn Cyclic group of order n. • Z(G) Center of a group G. Chapter One Class Field Towers 4 1.1 Introduction Let K be a number field. In general, the ring of integers OK of K is not a unique factoriza- tion domain (UFD). One way to rectify this problem is to look instead at factorization of ideals in OK into prime ideals. Since OK is a Dedekind domain, any ideal factors uniquely into a product of prime ideals. We can measure the extent to which OK fails to have unique factorization with the ideal class group ClK := IK =PK of K. Here IK denotes the group of fractional ideals of K and PK those fractional ideals that are principal. (From hereon, we may abuse notation and use language such as \ideals of K" as shorthand for ideals of OK ). Since OK is a Dedekind domain, it is a UFD if and only if it is a principal ideal domain, i.e., ClK = 1. Suppose that ClK 6= 1, but we insist on unique factorization on the level of prime elements for all elements of OK . Although this is impossible in K, we might hope that K can be embedded in some larger number field L with unique factorization, so that every x 2 K can at least be factored uniquely as an element of L (up to units). This question{the existence of such an L{ is known as the \embedding problem," and has an equivalent reformulation in the language of class field theory, which we now describe. Let H denote the maximal abelian extension of K that is unramified at all primes of K. By class field theory, H is a finite extension of K, called the Hilbert class field of K. The Hilbert class field enjoys several nice properties, including the following two. Theorem 1. [44] Let K be a number field and H the Hilbert class field of K. 1. There is an isomorphism ClK ! G(H=K). 2. Every ideal of OK becomes principal in OH . From the first statement of Theorem 1, we see that OK has unique factorization if and only if H = K. Now set K = K(0) and let K(1) denote the Hilbert class field of K. For i ≥ 1, we recursively define K(i) to be the Hilbert class field of K(i−1). The tower K(0) ⊆ K(1) ⊆ K(2) ⊆ ::: is called the Hilbert class field tower of K.
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