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UC San Diego UC San Diego Electronic Theses and Dissertations Title On a generalization of the rank one Rubin-Stark conjecture Permalink https://escholarship.org/uc/item/15r5d68s Author Vallières, Daniel Publication Date 2011 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California UNIVERSITY OF CALIFORNIA, SAN DIEGO On a generalization of the rank one Rubin-Stark conjecture A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Mathematics by Daniel Valli`eres Committee in charge: Professor Cristian Popescu, Chair Professor Ivano Caponigro Professor Wee Teck Gan Professor Ronald Graham Professor Harold M. Stark 2011 Copyright Daniel Valli`eres,2011 All rights reserved. The dissertation of Daniel Valli`eresis approved, and it is acceptable in quality and form for publi- cation on microfilm: Chair University of California, San Diego 2011 iii A` mes parents: Michelle et Lucien. iv A une ´epoque o`ules approches du temple du savoir ressemblent quelquefois `acelles de la bourse, o`ul’on y court le risque d’ˆetre ´etourdi par des troupes bruyantes de marchands et de bateleurs, nous avions besoin de nous affermir dans notre croyance qu’une vie donn´eetoute `ala science est encore possible, sans faiblesses ni compromissions ... —Andr´eWeil v TABLE OF CONTENTS Signature Page . iii Dedication . iv Epigraph . v Table of Contents . vi List of Tables . viii Acknowledgements . ix Vita and Publications . xi Abstract . xii Chapter 1 Introduction . 1 1.1 Notations . 2 Chapter 2 Classical results . 11 2.1 Stickelberger’s Theorem . 11 2.1.1 Gauss sums (Lagrange Resolvents) . 13 2.1.2 The m-th power residue symbol and Hasse’s principle for powers . 16 2.1.3 Stickelberger’s Theorem . 22 2.2 Dirichlet L-functions . 25 Chapter 3 The abelian rank one Stark conjecture, the Brumer-Stark conjecture and the Gross conjecture . 31 3.1 The Brumer conjecture . 31 3.2 The abelian rank one Stark conjecture . 34 3.3 Quadratic extensions . 40 3.4 The prime which splits completely is infinite . 40 + 3.4.1 The case of Q(√ζm) /Q ................... 40 3.4.2 The case of Q( d)/Q, d > 0 . 42 3.5 The distinguished prime which splits completely is finite (The Brumer-Stark conjecture) . 45 3.5.1 The case of Q(√ζm)/Q .................... 50 3.5.2 The case of Q( d)/Q, d < 0 . 51 3.6 The case of arbitrary abelian extensions of Q . 53 3.7 A reciprocity law satisfied by the Stark unit when the base field is Q ................................. 58 3.8 Equivariant formulation of the abelian rank one Stark conjecture 61 3.9 The (S, T )-version of the abelian rank one Stark conjecture . 69 3.10 The Gross conjecture . 77 vi 3.11 Current state of knowledge . 78 Chapter 4 The extended abelian rank one Stark conjecture . 81 4.1 Formulation of the extended abelian rank one Stark conjecture 81 4.2 Equivariant reformulation of the extended abelian rank one Stark conjecture . 85 4.3 The (S, T )-version of the extended abelian rank one Stark con- jecture . 89 4.4 Minimal cocyclic subgroups and 1-covers . 90 4.5 An explicit example . 97 Chapter 5 Investigation of the extended abelian rank one Stark conjecture . 106 5.1 A stronger question (the S-version) . 107 5.1.1 The case where v is a finite prime . 109 5.1.2 An extension of the Brumer-Stark conjecture . 111 5.2 The (S, T )-version of the stronger question . 113 5.3 Equivalence between the S and (S, T )-versions . 117 5.4 Functoriality questions . 120 5.5 Some results . 122 5.5.1 The place v ∈ Smin is real infinite . 126 5.5.2 The place p ∈ Smin is finite unramified . 128 5.6 Further results when the base field is Q . 130 Chapter 6 Conclusion . 132 Appendix A Some Kummer theory . 135 Appendix B Numerical examples when the base field is Q . 141 B.1 Full cyclotomic fields Q(ζm). 142 B.1.1 The field Q(ζ20). 142 B.1.2 The field Q(ζ24). 144 B.1.3 The field Q(ζ40). 146 B.1.4 The field Q(ζ48). 148 B.1.5 The field Q(ζ60). 150 B.1.6 The field Q(ζ68). 155 B.1.7 The field Q(ζ80). 157 B.1.8 The field Q(ζ96). 160 B.1.9 The field Q(ζ120). 162 B.1.10 The field Q(ζ171). 165 B.2 General abelian extensions of Q. 170 B.2.1 Example 1 . 170 B.2.2 Example 2 . 171 B.2.3 Example 3 . 173 B.2.4 Example 4 . 176 Bibliography . 179 vii LIST OF TABLES Table 4.1: Pairs of prime numbers (p, q) satisfying properties (4.4). 97 Table B.1: Full cyclotomic fields with finite distinguished subgroups. 142 Table B.2: Data for the field Q(ζ20). 143 Table B.3: The Stickelberger element for the ramified prime 5. 143 Table B.4: Data for the field Q(ζ24). 145 Table B.5: The Stickelberger element for the ramified prime 3. 145 Table B.6: Data for the field Q(ζ40). 146 Table B.7: The Stickelberger element for the ramified prime 5. 147 Table B.8: Data for the field Q(ζ48). 148 Table B.9: The Stickelberger element for the ramified prime 3. 149 Table B.10: Data for the field Q(ζ60) (First example). 150 Table B.11: The Stickelberger element for the ramified prime 3. 151 Table B.12: The Stickelberger element for the ramified prime 5. 151 Table B.13: Data for the field Q(ζ60) (Second example). 153 Table B.14: The Stickelberger element for the ramified prime 2. 153 Table B.15: The Stickelberger element for the ramified prime 5. 154 Table B.16: Data for the field Q(ζ68). 155 Table B.17: Data for the field Q(ζ68). 155 Table B.18: The Stickelberger element for the ramified prime 17. 156 Table B.19: Data for the field Q(ζ80). 158 Table B.20: Data for the field Q(ζ80). 158 Table B.21: The Stickelberger element for the ramified prime 5. 158 Table B.22: Data for the field Q(ζ96). 160 Table B.23: Data for the field Q(ζ96). 160 Table B.24: The Stickelberger element for the ramified prime 3. 161 Table B.25: Data for the field Q(ζ120). 162 Table B.26: Data for the field Q(ζ120). 163 Table B.27: The Stickelberger element for the ramified prime 3. 163 Table B.28: The Stickelberger element for the ramified prime 5. 164 Table B.29: Data for the field Q(ζ171). 165 Table B.30: The Stickelberger element for the ramified prime 19. 166 Table B.31: Data for the field Q(ζ171). 167 Table B.32: Data for example 1. 170 Table B.33: The Stickelberger element for the ramified prime 5. 170 Table B.34: Data for example 2. 172 Table B.35: The Stickelberger element for the ramified prime 43. 172 Table B.36: Data for example 3. 174 Table B.37: Data for example 3. 174 Table B.38: The Stickelberger element for the ramified prime 229. 174 Table B.39: Data for example 4. 177 Table B.40: The Stickelberger element for the ramified prime 109. 177 viii ACKNOWLEDGEMENTS Speaking of Max Dehn, Andr´eWeil wrote in his autobiography: “A humanistic mathematician, who saw mathematics as one chapter – certainly not the least important – in the history of the human mind...” I like to think the same way. It is thus a pleasure for me to take some time in order to thank several people, who have helped me becom- ing what I am as a human being, and whose influence can be seen in my way of doing mathematics. First of all, I am very grateful to my family. The simple house in which I grew up with my siblings and my parents has always been filled with a deep common affection. A warm thanks to all my good friends, particularly Guillaume, Mathieu, Benoit, Pierre, Charlie, the other Guillaume, and Danny. It always strikes me that whenever we are together, we enjoy the company of each other, even though we have different interests. If there is something I learned from all of them is the acceptance of the difference... I am really grateful to my former advisor during my Master’s Thesis, Professor Eyal Goren. I still remember when I saw him for the first time in his office at McGill Univer- sity. I told Professor Goren that I wanted to do number theory, but not only algebra, or analysis... He kept silence for, at least what appeared to me, about twenty minutes, and he finally told me: “I do have a nice project consisting of deep mathematics involving the classical theory of complex multiplication which is beautiful...” At the time, I did not understand a iota of what he said, but this is how I became intrigued with Stark’s conjecture. Some mathematics professors have been quite influential and I would like to thank them for their teaching, work and support: Norbert Schlomiuk, Abraham Broer, Marl`ene Frigon, Eyal Goren, Henri Darmon, Francesco Thaine, Farshid Hadjir, Wee Teck Gan, Harold Stark. Thanks to my good friend here in San Diego, Luke. Before he got married, we used to go every Friday night in a pub, and we just talked about life, i.e. beers, girls, friends, violence, absurdity, happiness, family, religion, and above all mathematics.