UNIVERSITY OF CALGARY Unconditional Class Group Tabulation to 240 by Anton S. Mosunov A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE DEPARTMENT OF COMPUTER SCIENCE CALGARY, ALBERTA July, 2014 c Anton S. Mosunov 2014 Abstract In this thesis, we aim to tabulate the class groups of binary quadratic forms for all fun- damental discriminants ∆ < 0 satisfying j∆j < 240. Our computations are performed in several stages. We first multiply large polynomials in order to produce class numbers h(∆) for ∆ 6≡ 1 (mod 8). This stage is followed by the resolution of class groups Cl∆ with the Buchmann-Jacobson-Teske algorithm, which can be significantly accelerated when the class numbers are known. In order to determine Cl∆ for ∆ ≡ 1 (mod 8), we use this algorithm in conjunction with Bach's conditional averaging method and the Eichler-Selberg trace formula, required for unconditional verification. Our novel class group tabulation method allowed us to gather unconditional numerical evidence in support of certain hypotheses, such as the Littlewood's bounds on L(1; χ∆) and the Cohen-Lenstra heuristics. i Acknowledgements I am grateful to my supervisor, Dr. Michael J. Jacobson, Jr., who suggested this exciting project to me when I was still a Baccalaureate student. Without him, this project would not have started, and most certainly would not be finished, as his wise advice were of enormous help to my research. I thank Prof. Richard K. Guy and Dr. Renate Scheidler for their comments on my thesis. I would also like to thank \Alberta Innovates Technology Futures" for their generous support of my project over the past two years. I hope that they will find it worth their investments. All of my computations were performed on one of WestGrid's supercomputers, and I would like to acknowledge an outstanding work of their support personnel, namely Douglas Phillips, Masao Fujinaga, and especially Belaid Moa, who spent countless hours over the phone helping to debug my program. Perhaps, the most warm and tender thank you goes to my dear family, who always supported me when I was eight thousand kilometres away from home. They made my arrival to Canada possible, and I will always be thankful for this great opportunity. Finally, I thank all of my friends on both sides of the Atlantic Ocean, especially my fellow graduate students Sebastian Lindner, Longsheng Zhou and Ostap Orobets for making office time pass joyfully. ii Table of Contents Abstract ........................................i Acknowledgements .................................. ii Table of Contents . iii List of Tables . iv List of Figures . .v 1 INTRODUCTION . .1 1.1 Motivation . .2 1.2 Previous Work and Contribution . .3 1.3 Organization of the Thesis . .5 2 NUMBER THEORY BACKGROUND . .7 2.1 Binary Quadratic Forms . .8 2.2 Binary Quadratic Forms with Even Middle Coefficient . 13 2.3 Subdivision of Classes Into Genera . 17 2.4 Ternary Quadratic Forms . 21 3 THE THETA SERIES . 24 3.1 Jacobi Theta Functions . 24 3.2 Relationship Between Theta Series and Class Numbers . 27 3.3 Class Number Tabulation Formulas . 31 4 OUT-OF-CORE MULTIPLICATION . 36 4.1 Out-of-Core Multiplication with Chinese Remainder Theorem . 36 4.2 Algorithms . 41 4.3 Computational Parameters . 50 4.4 Complexity Analysis . 52 5 CLASS GROUP COMPUTATION . 58 5.1 Bach's Bound on the Size of the Group . 59 5.2 The Class Group Computation . 60 5.3 Unconditional Verification . 67 6 IMPLEMENTATION AND NUMERICAL RESULTS . 71 6.1 Implementation . 71 6.2 Performance . 72 6.3 Littlewood's Bounds . 78 6.4 The Cohen-Lenstra Heuristics . 97 6.5 Exotic Groups . 104 7 CONCLUSION . 134 7.1 Future Work . 135 Bibliography . 138 iii List of Tables 2.1 Subdivision of classes of forms with determinant D = −52 ·7 into orders . 14 2.2 Subdivision of classes of forms with determinant D = −32 ·5 · 7 into genera . 19 2.3 Characteristic numbers of forms with determinant D = −32 ·5 · 7 ........... 21 4.1 Extraction of coefficients from H(x)........................... 40 5.1 Suggested values of Q for Bach's averaging method . 59 6.1 Computational parameters . 72 6.2 Timings for the class number tabulation program . 73 6.3 Timings for the class group tabulation program . 75 6.4 Counts of h(∆) with various divisibility properties . 75 6.5 Timings for various class group tabulation implementations . 76 6.6 Timings for the verification program . 77 6.7 Mean values of L(1; χ∆) for various congruence classes of ∆ . 81 6.8 Successive L(1; χ∆) maxima, ∆ ≡ 1 (mod 8) . 82 6.9 Successive ULI∆ maxima, ∆ ≡ 1 (mod 8) . 86 6.10 Successive L(1; χ∆) minima, ∆ ≡ 1 (mod 8) . 88 6.11 Successive LLI∆ minima, ∆ ≡ 1 (mod 8) . 89 6.12 Successive L(1; χ∆) maxima, ∆ ≡ 5 (mod 8) . 90 6.13 Successive ULI∆ maxima, ∆ ≡ 5 (mod 8) . 92 6.14 Successive L(1; χ∆) minima, ∆ ≡ 5 (mod 8) . 93 6.15 Successive LLI∆ minima, ∆ ≡ 5 (mod 8) . 94 6.16 Successive L(1; χ∆) maxima, ∆ ≡ 0 (mod 4) . 95 6.17 Successive ULI maxima, ∆ ≡ 0 (mod 4) . 95 6.18 Successive L(1; χ∆) minima, ∆ ≡ 0 (mod 4) . 96 6.19 Successive LLI minima, ∆ ≡ 0 (mod 4) . 96 6.20 Number of noncyclic odd parts of class groups . 100 6.21 Counts of class numbers divisible by l .......................... 101 6.22 Counts of class numbers divisible by l2 .......................... 102 6.23 Counts of class numbers divisible by l3 .......................... 103 6.24 Counts of class groups with l-rank = r .......................... 104 6.25 Non-cyclic rank 2 2-Sylow squared subgroups . 107 6.26 Non-cyclic rank 2 p-Sylow subgroups . 110 6.27 Non-cyclic rank 3 2-Sylow squared subgroups . 117 6.28 Non-cyclic rank 3 p-Sylow subgroups . 121 6.29 Non-cyclic rank 4 2-Sylow squared subgroups . 123 6.30 Non-cyclic rank 4 p-Sylow subgroups . 125 6.31 Non-cyclic rank 5 2-Sylow squared subgroups . 126 6.32 Doubly non-cyclic class groups . 127 6.33 Trebly non-cyclic class groups . 132 6.34 Quadruply non-cyclic class groups . 133 7.1 Values of R3(n) for squarefree n ≡ 7 (mod 8) . 136 iv List of Figures and Illustrations 6.1 Program performance when N varies, B = 211, T = 26, m = 212 ....... 73 6.2 Program performance when B varies, N = 232, T = 26, m = 212 ....... 73 6.3 Program performance when T varies, N = 232, B = 28, m = 28 ........ 74 6.4 Scalability of the verification algorithm, LHS ................. 77 6.5 Scalability of the verification algorithm, RHS ................. 77 28 28 6.6 Local ULI∆ maxima for x ≤ j∆j < x + 2 ; x 2 2 Z .............. 96 28 28 6.7 Local LLI∆ minima for x ≤ j∆j < x + 2 ; x 2 2 Z .............. 97 6.8 Ratios of non-cyclic odd parts of class groups . 100 6.9 Values of pl(x)................................... 101 6.10 Values of pl2 (x).................................. 102 6.11 Values of pl3 (x).................................. 103 6.12 Values of pl;2(x).................................. 104 6.13 Values of pl;3(x).................................. 104 v List of Algorithms 4.1 Initialization [HTW10, Algorithm 1 : Phase 1] . 43 4.2 Multiplication [HTW10, Algorithm 1 : Phase 2] . 44 4.3 Restoration [HTW10, Algorithm 1 : Phase 3] . 45 4.4 Initialization of a block of #3(q)......................... 46 4.5 Initialization of a block of r(q)......................... 47 4.6 Initialization of a block of r(q2)......................... 47 2 4.7 Initialization of a block of #3(q)......................... 48 4.8 Initialization of a block of r2(q)......................... 49 4.9 Initialization of a block of r2(q2)........................ 50 5.1 Bach's approximation of log L(1; χ∆) [Bac95] . 60 5.2 BJT algorithm [Ram06, Algorithm 3.1] . 65 5.3 Verification algorithm [Ram06, Algorithm 4.1] . 70 vi Chapter 1 INTRODUCTION The class group of binary quadratic forms is a mathematical object that was thoroughly studied by many mathematicians over the past two centuries. First discovered by Gauß in 1798, this group consists of equivalence classes of binary quadratic forms, i.e. integer variable functions of the form ax2 + bxy + cy2 with a; b; c 2 Z. Quadratic forms in every equivalence class of a certain class group possess the same discriminant ∆ = b2 − 4ac. For that reason, this finite abelian group is denoted by Cl∆, in order to indicate that the structure of a group varies depending on ∆. Many things are known about the class group. For example, if we know its class number h(∆), which is merely the size of Cl∆, we can find a non-trivial factor of ∆. Also, from the prime factorization of ∆ we can determine the parity of h(∆), as well as the rank of the 2-Sylow subgroup. Reduction and composition algorithms, such as NUCOMP developed by Shanks [Sha89], allow us to efficiently manipulate the elements of Cl∆. However, the number of open questions about Cl∆ most certainly exceeds the number of answered. For example, the structure computation of Cl∆ is believed to be hard. The computation of the class number h(∆) is considered to be a difficult problem as well. However, the heuristics of Cohen and Lenstra allow us to make certain predictions regarding the structure of the class group, and divisibility properties of h(∆) [CL83, CL84]. Also, Littlewood's bounds allow to deduce an estimate on the size of h(∆) [Lit28]. However, Littlewood's result depends on the Extended Riemann Hypothesis, and might potentially be incorrect. Both Cohen-Lenstra heuristics and the Littlewood's bounds require careful examination, which we do in this thesis by providing an extensive computational evidence in their support.