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Elliptic Nets and Elliptic Curves

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Elliptic Nets and Elliptic Curves ELLIPTIC NETS AND ELLIPTIC CURVES BY KATHERINE E. STANGE B. MATH., UNIVERSITY OF WATERLOO, 2001 M. SC., BROWN UNIVERSITY, 2003 SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN THE DEPARTMENT OF MATHEMATICS AT BROWN UNIVERSITY PROVIDENCE,RHODE ISLAND MAY 2008 c Copyright 2008 by Katherine E. Stange This dissertation by Katherine E. Stange is accepted in its present form by the Department of Mathematics as satisfying the dissertation requirement for the degree of Doctor of Philosophy. Date Joseph H. Silverman, Director Recommended to the Graduate Council Date Alexander Goncharov, Reader Date Stephen Lichtenbaum, Reader Date Joseph H. Silverman, Reader Approved by the Graduate Council Date Sheila Bonde Dean of the Graduate School iii Vita Katherine E. Stange was assembled in North Bay, Ontario, Canada in 1978. She was further processed at the University of Waterloo, where she was labelled “Bachelor of Mathematics” in 2001. iv Acknowledgements There are a great many people who have helped me along the long road that led, finally, to this thesis. First among these is my advisor, Joseph Silverman. Thank you for your incredible patience and for your enthusiastic telling of wonderful mathematical tales every Wednesday morning. And thank you for your confidence in me. It is hard to put into words exactly what an advisor does, because it happens between the practical matters of correcting papers and answering confusions. It has to do with learning what mathematics is, and it is why we care about our mathematical genealogy. I’m proud to have you as a mathematical father. This work was supported for two years by the National Sciences and Engineering Research Council of Canada in the form of a post-graduate scholarship, for which I am very grateful. I also wish to thank Microsoft Research for providing me with a one-semester internship working with Kristin Lauter in San Diego, California. Thank you, Kristin, for your amazing generosity, infectious curiosity and your confidence in my work. Thank you to Audrey, Carol, Doreen, Larry and Natalie. Without each of you, I wouldn’t have ever found my way long enough to get to the math at all. Doreen and Natalie, you deserve special thanks for your labours on my behalf, and for your love, both of which were above and beyond, and for which I owe you more than I can give. Thank you to my readers, Stephen Lichtenbaum and Alexander Goncharov. Thank you to Michael Rosen for your interest and useful discussions. Thank you to Alf van der Poorten for your support and enthusiasm. To Thomas Banchoff, for teaching me about teaching. To George Daskalopoulos, who was always especially kind. Thank you to everyone in the Brown mathematics department for being a good family to the graduate students. Thank you to Christine Swart, Michael Scott, Graeme Everest, Gary McGuire, Peter Stevenhagen, Nelson Stephens and Noam Elkies for your interest, discussions and support. And to many others who took the time to discuss this work with me. I want to thank my high school math teacher, Mr. Garrett, who is his own grandpa, and who always answered my interest with excitement. It was in your class, 15 years ago, that I decided to do this today. Thank you also to my teachers at the University of Waterloo, especially Peter Hoffman, Ken Davidson, Brian Forrest and Cam Stewart. Thank you also to Murat for your support. Thank you to Donald Knuth for LaTeX, and to Mistress LaSpliffe and the New York Times for providing the necessary procrastination. v These are the people and things who made the mathematics possible. However, many of you I also count among my friends. And I must also thank those, not necessarily mathematically involved, whose friendship was every bit as important to the success of my crazy quest to become a mathematician as the mathematics itself. This includes all of the mathematics department graduate students, who each in their way con- tributed to my upbringing, mathematical and otherwise. Thank you to Panayotis for being a loveably argumentative Greek. To Dan for putting us into song. To Steve for telling us jungle tales. To Mikey G for balls. To Alina for the challenge. To Michelle for her generosity. To Karen. To Graeme. To the Mikes, Ben, Hatice, Steffan (for being Canadian), Ebru, Hyun, and Yu. To Michael I owe more than thanks, for you are family. To Rafe, for wonderful times. I miss you. To those in the wider community of Brown whose paths I crossed and learned from. In particular, thank you to Marie. To Joanna and Anita, for your open and accepting friendship. To Becca, for fascinating conversation. To Kevin, for the Fez. To Lionel, for letting me in. To David, for the experiences, the filters, the words. To all those I rode with. If graduate school was a journey, I did it by bicycle. Thank you to the ECCC for the haybales and portajohns and all the rest. To Providence Bicycle. To Joe for swearing at me when I got the townlines. To Forest for trying to ride me off your wheel while eating ice cream, and putting up with me when I hung on. To the Brown Cycling Club for suffering my dictatorship. A special thank you to Bob and Bikeworks for your incredible support and enthusiasm while I tried my legs at racing. And thank you to Preston, for every time you agreed to go longer. Thank you also to all those, named and nameless, whom I met in my travels abroad. Your strange customs but universal generosity were a gift. To Yousra, Ahmed, Denis, Victor, Vika, Katya, Tatyana Makarovna, Cody, Galina, Alain, Barbara, Patrick, Hanna, Spring. Especially to Yulia and Lera, and to Tenzin, Dukgyal and Nordron. Thank you to Laura, Ruxandra, Dan, Heidrun, Voichitsa, and Helmut, and especially to Oma Oma. And to many others whose names I have since lost, or never knew. Thank you to Turtle Soup: to Meg, Helen, Kirsti, Lisa, Leah and Jessica. I wouldn’t have survived high school without H.O.T. and quinzees and the Eighteenth Ruler of the Universe, without Bump the King of Dorks and the belly button tree, without torso dancing and Psycho-pro-tection, without the nickname Chubbo, without Dr. Niel Paterson or Cookie Dough. These things, ironically, made me sane. And without the freedom and creativity of our strange community, without all of you to convince me that anything was possible, I wouldn’t have believed I could do it. And finally, there are those for whom thanks are simply inadequate. For Jonathan: How you managed to stay convinced that I was a mathematician I don’t know. But it is only thanks to that delusion that I really am. You’ve taught me anew how to love mathematics, and whenever that relationship was going poorly, your bad puns laughed me back from the brink. You’ve been my co-conspirator in a shameless neverending childhood. You’ve given me mathematics, but you’ve given me much more: you’ve given me the world to do it in. I can’t find any words besides vi these1: I love you. For Oma: Your love, your warmth, your interest, and your stories, have given me a cherished sense of connection between what I do now and where I come from. And thank you for the baked apples. For Christiaan, who derailed any attempt I ever had to be serious: My life would have been impov- erished without the Zappa, the sizzling floor, the Hee Haw in the Hole, without you spinning Marley on the linoleum. You taught me to read with raisins and Pavlovian conditioning. And you rode your bike with me, always just a little faster. I think it was those bicycle rides that taught me tenacity, which is still what I credit for anything I may go so far as to say I have earned. And last, for my parents, mom and dad: How does one thank one’s parents? Without you, I wouldn’t have had anyone to count up the myriad of small boxes and wedges I drew with glee all over scraps of paper, to count those filled and those unfilled and pronounce the delightful name of the fraction I had created. Without you, I wouldn’t know that six times eight is forty-eight. (With a little more of you, I might have known that seven times eight is fifty-six.) Without you, I wouldn’t have been drawn wide-eyed into thinking about whether there were more integers than there were even integers. Or whether the universe was infinite, or how many different kinds of tastebuds we had. Without you, I wouldn’t have had the confidence to take ‘contest math.’ And without you, I wouldn’t have been happy and laughing and warm and strong when I did. Hell, without you, I wouldn’t even exist. 1With the possible exception of certain Frank Zappa lyrics. vii Contents List of Tables xii List of Figures xiii List of Examples xiv List of Algorithms xv I Overture 1 1 Introduction 2 1.1 Lucas’ story . 2 1.2 Ward’s story . 3 1.3 Moving to higher rank . 6 1.4 Deeper connections . 10 1.5 Cryptographic applications . 12 1.6 Prerequisites . 13 2 Basic properties of elliptic divisibility sequences 15 2.1 Making the curve-sequence relation explicit . 15 2.2 Relations to the group law on the elliptic curve . 16 2.3 More on division polynomials . 16 2.4 Induction properties . 17 2.5 The integer case . 18 2.6 Periodicity modulo p ....................................
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