Coleman Integration for Hyperelliptic Curves: Algorithms and Applications ARCHNvES by MASSACHUSETS INSTITUE OF TECHNOLOGY Jennifer Sayaka Balakrishnan =SEPP 0 2 2011 A.B., Harvard University (2006) A.M., Harvard University (2006) L-LI BRA-R iEF-s 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 2011 @ Jennifer Sayaka Balakrishnan, MMXI. All rights reserved. The author hereby grants to MIT permission to reproduce and distribute publicly paper and electronic copies of this thesis document in whole or in part. A uthor ... ..... .. .................... Department of Mathematics April 26, 2011 Certified by ........... ..2. .... ... Kiran S. Kedlaya Associate Professor Thesis Supervisor Accepted by ..... .. !. O................................ Bjorn Poonen Chairman, Department Committee on Graduate Theses 2 Coleman Integration for Hyperelliptic Curves: Algorithms and Applications by Jennifer Shyamala Sayaka Balakrishnan Subnitted to the Department of Mathematics on April 26, 2011, in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract The Colemani integral is a p-adice line integral that can be used to encapsulate several quantities relevant, to a study of the arithmetic of varieties. In this thesis, I describe algorithms for computing Coleman integrals on hyperelliptic curves and discuss some immediate applications. I give algorithms to compute single and iterated integrals on odd models of hyperelliptic curves, as well as the necessary modifications to iplemieit these algorithms for even models. Furthermore, I show how these algorithinis can be used in various situations. The first application is the method of Chabatv to find rational points on curves of genus greater than 1. The second is Mlihyong Kim's recent nonabelian analogue of the Chabauty method for elliptic curves. The last two applications concern p-adic heights on Jacobians of hyperelliptic curves. necessary to formulate a p-adic analogue of the Birch and Swinnerton-Dyer conjecture. I conclude by stating the analogue of the Mazur-Tate-Teitelbaum conjecture iii our setting and presenting supporting data. Thesis Supervisor: Kiran S. Kedlaya Title: Associate Professor 4 Acknowledgments It is a pleasure to thank my advisor, Kiran Kedlaya, for his support of the efforts that went into this thesis. He has been incredibly generous with his time and ideas, patiently answering my numerous questions, suggesting many interesting problems, and encouraging several collaborations. One such collaboration led to the results in Chapter 3, which form the backbone of this thesis. I would also like to thank a few others who have had a direct impact on this the- sis. During rny first year in college, I happened to take a seminar with William Stein, who introduced me to computational approaches in number theory. Over the years, we have worked on a few things together - most recently, a portion of the material that forms Chapter 9 of this thesis. Special thanks must also go out to two of my coauthors, Minhyong Kirn and Amnon Besser, who kindly explained their work to me; collaborations with them led to Chapters 7 and 8, respectively, of this thesis. Thanks are also due to Barry Mazur, Bjorn Poonen, Alice Silverberg, Michael Stoll, Drew Sutherland, and Liang Xiao, for several productive discussions, and Robert Bradshaw and David Roe for laying much of the p-adic groundwork in Sage. I must also thank Steffen iller for visiting MIT last summer arid carefully computing (and recomputing) many things in Magma, despite the many ill-posed queries I came up with during the process. It would have been impossible to assen- ble the data in Chapter 9 without his efforts. Repeating names, I would also like to thank Bjorn Poonen, Abhinav Kumar, and Kiran Kedlava for agreeing to serve on rny thesis cornnittee, William Stein for access to his cluster of computers (chief among then {mod, boxen, sage}.math.washington.edu, funded by NSF grant DMS- 0821725), and Kiran Kedlaya for access to dwork.mit . edu. Finally, to my dear friends and family, thank you for your love, laughter, and sup- port throughout the years. Kartik arid Liz, I have thoroughly enjoyed our adventures in baking arid ballet. And above all, Bala, Shizuko, Stephanie, and Vivek, thank you for always cheering me on, regardless of the time or time zone. This research was supported by the National Science Foundation Graduate Re- search Fellowship and the National Defense Science arid Engineering Graduate Fel- lowship. 6i For those in Yamada, especially my grandmother Fum iko Contents 1 Introduction 15 1.0.1 Why hyperelliptic curves?...... .. .. .. .. .. .. 1) 1.0.2 Beyond lyperelliptic curves'? .. .. .. .. .. .. .. .. .. 16 1.0.3 Outline ........... .. ... .. .. .. .. .. 16 2 Hyperelliptic curves and p-adic cohomology 19 2.1 Hyperelliptic curves....... .... .. .. .. .. .. .. .. .. 19 2.2 p-adic cohonology .. .. .. .. .. .. .. .. .. .. .. .. .. 21 2.2.1 Frobenius..... .. .. .. .. .. .. .. .. 23 2.2.2 Kedlaya's algorithm......... .. .. .. ... .. 23 3 Coleman integration: the basic integrals 27 3.1 Coleman's theory of p-adic integration . .. .. .. .. .. .. .. 27 3.2 Explicit integrals for hyperelliptic curves . .. .. .. .. .. .. .. 29 3.2.1 A basis for de Rham cohonology . ............. 30 3.2.2 T iny integrals .. .. .. ... .. .. .. .. .. .. .. .. .. 30 3.2.3 Non-W eierstrass disks .. .. .. .. .. .. .. .. ... .. 31 3.2.4 Weierstrass endpoints of integration ............... 34 3.3 Implementation notes and precision . .. .. .. .. .. .. .. .. 36 3.3.1 Precision estim ates .. .. .. .. .. .. .. .. .. .. .. 36 3.3.2 Complexity analysis . .. .. .. .. .. .. .. .. .. .. .. 37 3.3.3 Numerical examples . .. .. .. .. .. .. .. .. .. 38 4 Coleman integration: even degree models 41 4.1 Introduction .. .. .. .. ... .. .. .. .. .. .. .. .. .. .. 41 4.2 A bit more p-adic cohomology .. .. .. .. .. .. .. .. .. .. 41 4.3 Coleman integration on even models . .. .. .. .. .. .. .. .. 42 4.3.1 Local coordinates . .. .. .. .. .. .. .. .. .. .. 42 4.3.2 Integrals . .. .. .. .. .. .. .. .. .. .. .. .. .. .. 42 4.3.3 Using the linear system . .. .. .. .. .. .. .. .. .. .. 44 5 Coleman integration: iterated integrals 47 5.1 Introduction .. .. .. .. .. .. .. ... .. .. .. .. .. .. .47 5.2 Iterated path integrals . .. .. .. .. .. .. .. .. ... .. .. .. 48 5.3 Tiny iterated integrals....... ... ... .. .. ... .. .. 50 5. 3.1 Examples . .. .. .. .. 5.4 Iterated integrals: linear system..... .. ... .. .. ..... 5.5 Explicit double integrals ... .. .. .. .. .. ........... 5.5.1 Tiny double integrals .. .. .. .. .. .. .. .. .. .. .. 5.5.2 The linear system for double integrals between Teichnffller points 5.5.3 Linking double integrals . .. .. .. .. .. .. .. .. .. 5.5.4 Without Teichinliler points .. .. .. .. .. .. .. .. .. 5.5.5 W eierstrass points . .. ... .. .. .. .. .. .. ... .. .. 5.6 Future w ork . .. .. .. .. .. .. .. .. .. .. .. .. .. .. 5.6.1 Tangential basepoint at infinity .. .. .. .. .. .. .. .. 5.6.2 The fundamental linear system . .. .. ... .. .. .. .. 5.6.3 The region of convergence in a Weierstrass disk .. .. .. .. 6 Explicit computations with the Chabauty method 6.1 Introduction........... .. .. .. .. 6.2 Explicit integrals to a parameter . ........... 6.3 Example: odd model.... .. .. .. .. .. .. 6.4 Future work . .. .. .. .. .. .. .. .. .. 7 Kim's nonabelian Chabauty method 73 7.1 Introduction: the method......... 73 7.1.1 Elliptic curves. .... 7.2 A few moire explicit double integrals . 74 7.2.1 Integrals to a parameter z 7.2.2 Double integrals to a parameter -6 7.3 Carrying out the nonabelian Chabauty met hod 7.3.1 An example with 65A . .. .. 7 7 7.3.2 An example with 37A 78 7.4 Connection to p-adic heights . .. .. 80 7.5 Future work . .. .. .. .. .. .. 81 7.5.1 Tangential basepoint . .. .. .. 81 7.5.2 Tamagawa number hypothesis 81 8 p-adic heights on Jacobians of hyperelliptic curves 83 Q 1 I t d t 83 8.2 The p-adic height pairing .. .. ... ... .. ... .. 84 8.3 HypereIliptic curves. ... ... ... .. ... .. .. 89 8.4 Colem an integrals .. .. .. ... .. .. .. .. ... .. 90 8.4.1 Generalities on Coleman integrals and symbols 90 8.4.2 Integration of forms with poles outside Weierstrass discs 90 8.5 The lo cal height pairing at primes above p .. ... .. 93 8.5.1 Computing cup products ... .. ... .. ... 99 8.5.2 The m ap T . ... ... ... ... ... ... .. 99 8.5.3 Decomposing a divisor D into D' and D"l . .. 100 8.5.4 A form with the required residue divisor . ... 100 8.5.5 Finding wJD for a Weierstrass divisor D... .. .. 101 8.5.6 Finding WD for a non-Weierstrass divisor D....... 101 8.5.7 Integration when D 2 reduces to W'eierstrass points 101 8.5.8 Computation wh en Di is Weierstrass and D 2 is not 101 8.5.9 Computation wh en both divisors are non-Weierstrass 101 8.6 Implementation notes . .... ... ... .. 101 8.6.1 Auxiliary choices.. .......... 102 8.6.2 Precision........ ... .... .. ... .. 102 8.7 Exam ples . .. .. .. ... .. .. .. ... .. .. .. 106 8.7.1 Local heights: genus 2, general divisors..... 106 8.7.2 Local heights: oenus 2 anti-symnetric divisors 107 8.7.3 Global heights: genus I.............. 109 8.7.4 Global heights: genus 2 ............... 110 8.8 Future work...... ... ... ... .. .. .. 110 8.8.1 Global height pairings. .............. 110 8.8.2 Optimizations.. .. .. .. .. .. .. .. 111 8.8.3 Comparison with the work of Mazur-Stein-Tate 111 9 A p-adic Birch and Swinnerton-Dyer conjecture 113 9.1 Introduction