Ranks of Elliptic Curves and Random Matrix Theory (London Mathematical Society Lecture Note Series, Series Number 341)

Random Matrix Theory Is An Area Of Mathematics First Developed By Physicists Interested In The Energy Levels Of Atomic Nuclei, But It Can Also Be Used To Describe Some Exotic Phenomena In The Number Theory Of Elliptic Curves. The Purpose Of This Book Is To Illustrate This Interplay Of Number Theory And Random Matrices. It Begins With An Introduction To Elliptic Curves And The Fundamentals Of Modelling By A Family Of Random Matrices, And Moves On To Highlight The Latest Research. There Are Expositions Of Current Research On Ranks Of Elliptic Curves, Statistical Properties Of Families Of Elliptic Curves And Their Associated L-functions And The Emerging Uses Of Random Matrix Theory In This Field. Most Of The Material Here Had Its Origin In A Clay Mathematics Institute Workshop On This Topic At The Newton Institute In Cambridge And Together These Contributions Provide A Unique In-depth Treatment Of The Subject. Edited By J.b. Conrey ... [et Al.]. Some Of The Papers Originated In A Workshop Held At The Isaac Newton Institute, Feb. 2004. Includes Bibliographical References And Index. Cover......Page 1 Ranks of Elliptic Curves and Random Matrix Theory......Page 4 ISBN-13 978 0521699648......Page 5 Contents......Page 6 Introduction......Page 8 References......Page 12 1 A concrete introduction to elliptic curves......Page 14 1.1 Elliptic curves as algebraic curves, complex tori and the link between the two......Page 15 1.2 The group law on elliptic curves and maps between elliptic curves......Page 18 1.3 The arithmetic of elliptic curves: the Mordell-Weil group......Page 21 1.4 Reduction modulo primes and the Hasse-Weil Lfunction of an elliptic curve......Page 23 1.5 The Birch and Swinnerton-Dyer conjecture......Page 27 1.6 The Tate-Shafarevitch group......Page 29 1.7 Enter random matrices.........Page 33 2.1 Families and invariants......Page 34 2.2 Conjectures and heuristics......Page 38 2.3 Random matrix models......Page 39 2.4 Theoretical results......Page 42 2.5 Basic analytic tools......Page 46 2.6 The Delta-symbol for a family......Page 48 2.7 Sketch of proof of Theorem 2.8......Page 50 2.8 Some discussion of numerical evidence......Page 53 References......Page 56 1 Introduction......Page 60 1.1 A quick history of families......Page 61 2.1 L-functions......Page 62 2.2 Families of characters......Page 63 2.3 Families of L-functions......Page 64 2.4 Modeling a family of L-functions......Page 65 2.5 Summary of modeling......Page 66 3.1 Families with a given rank......Page 67 3.2 Two models for two kinds of families......Page 68 3.4 Imposing zeros......Page 70 3.5 Some issues......Page 71 4 Refined modeling......Page 72 References......Page 74 1.2 Notation......Page 78 2.1 The tools......Page 79 2.2 The setup......Page 82 2.3 Completing the sum......Page 83 2.4 Sketching the method for the family of all elliptic curves......Page 85 2.6 Open problems and directions for improvement......Page 89 3.1 Methods and results......Page 90 A.1 The family of all elliptic curves......Page 94 A.2 The large positive rank family......Page 95 References......Page 96 1.1 Random matrix theory and number theory......Page 100 1.2 Random matrix theory and elliptic curves......Page 102 2 Random matrix calculations......Page 107 3 Discussion......Page 111 References......Page 112 Function fields and random matrices......Page 116 1.2 Function fields over finite fields......Page 117 1.3 Curves over finite fields......Page 119 1.4 Morphisms and rational functions......Page 120 1.5 The function eld/curve dictionary......Page 123 1.6 Points, prime divisors, and places......Page 125 1.7 The Riemann-Roch theorem......Page 126 1.8 Extensions, coverings, and splitting......Page 127 1.9 Frobenius elements......Page 129 2 -functions and L-functions......Page 130 2.1 The L-function of a curve......Page 131 2.2 Spectral interpretation of L-functions......Page 133 2.3 Examples of L-functions......Page 134 2.4 L-functions attached to Galois representations......Page 136 2.5 Spectral interpretation of L-functions......Page 137 2.6 Symmetries......Page 138 3.1 Arithmetic and geometric families......Page 139 3.2 Variation of L-functions......Page 140 3.3 Other families......Page 141 3.4 Idea of proofs......Page 143 3.5 Large N limits......Page 144 3.6 Applications......Page 145 4 Further reading......Page 146 References......Page 147 1 Introduction......Page 150 2.1 Unitary Group......Page 151 2.2 Weyl Integration Formula......Page 152 2.3 High Powers of Unitary Matrices......Page 155 3 Schur functions as characters of the unitary group......Page 156 4 Frobenius-Schur duality & moments of traces......Page 159 5 Moments of characteristic polynomials......Page 162 6 Averages of ratios of characteristic polynomials......Page 164 7 Combinatorial de nition of Schur functions and some of its consequences......Page 167 7.1 Relationship to plane partitions......Page 168 7.2 Random matrices and magic squares......Page 169 7.3 Pseudomoments of the Riemann zeta-function and pseudomagic squares......Page 173 References......Page 174 The distribution of ranks in families of quadratic twists of elliptic curves......Page 178 References......Page 181 1 Introduction......Page 184 2 The general method......Page 185 3 Rank 4......Page 186 4 Root numbers......Page 192 5 Rank 5......Page 193 References......Page 194 1 Introduction......Page 196 2 Discussion......Page 197 3 Tamagawa numbers......Page 198 5 Acknowledgments......Page 199 References......Page 200 1 Notations and introduction......Page 202 2 The example......Page 203 References......Page 206 1 Introduction......Page 208 2 A model from Heegner points (largely due to Birch)......Page 210 3 Alternative ideas......Page 211 4 Data......Page 213 4.1 Quadratic twists in arithmetic progressions......Page 215 4.2 Beyond twists......Page 216 5 Conclusion......Page 218 References......Page 219 1 Introduction......Page 222 2 Moments of LE(1; d)......Page 224 3 Vanishings of LE(1; d) in progressions......Page 225 4 Evaluating the rst two terms of ME (X; k; q; )......Page 227 5 Conjecture for the rst two terms in Rq (X)......Page 230 6 Numerical Data......Page 231 References......Page 234 Fudge Factors in the Birch and Swinnerton-Dyer Conjecture......Page 240 References......Page 243 1 Introduction......Page 244 2 Numerical data......Page 245 4 Random matrix models......Page 248 References......Page 251 1 Introduction......Page 254 2 Special values and modular symbols......Page 256 3 Discretisation......Page 258 4 Unitary Random Matrices......Page 261 5 Numerical Evidence......Page 264 References......Page 265 Computing central values of L-functions......Page 267 1 Introduction......Page 280 2 Quaternion algebras and Brandt matrices......Page 281 3 Construction of g and g+......Page 283 3.2 Weight functions and l......Page 284 3.3 Odd weight functions and l......Page 285 4.1 11A......Page 287 4.2 37 A......Page 289 4.3 43A......Page 291 4.4 389A......Page 293 References......Page 295 1 Introduction......Page 296 2 Quaternion algebras and Shimura correspondence......Page 297 2.1 Modular forms of weight 3/2......Page 298 4 On certain non-maximal orders and level p2......Page 299 5 An Algorithm for the Real Quadratic Twists......Page 300 6 Example: level 7^2......Page 301 6.1 f49A......Page 302 7.1 f11A......Page 304 7.2 f121A......Page 306 7.4 f121C......Page 309 7.5 f121D......Page 311 8.1 f17A......Page 312 9 Example: level 19^2......Page 314 9.2 f361A......Page 316 References......Page 320 1 Introduction......Page 322 2 Spherical polynomials and f......Page 323 3.1 Imaginary twists......Page 325 3.2 Real twists......Page 326 References......Page 328 Heuristics on class groups and on Tate-Shafarevich groups: The magic of the Cohen-Lenstra heuristics......Page 330 1 Analogy between number fields and elliptic curves de ned over Q......Page 331 2 Heuristics on class groups of quadratic number fields......Page 333 2.1 Imaginary quadratic number fields......Page 334 2.2 Real quadratic fields......Page 336 2.3 Examples......Page 337 3.1 Rank 0 case......Page 339 3.2 Rank 1 case......Page 341 3.3 Examples......Page 342 4 Quadratic twist families......Page 344 References......Page 347 A Note on the 2-Part of III for the Congruent Number Curves......Page 348 References......Page 351 2-Descent Through the Ages......Page 352 References......Page 363 Index......Page 364 Random matrix theory is an area of mathematics first developed by physicists interested in the energy levels of atomic nuclei, but it can also be used to describe some exotic phenomena in the number theory of elliptic curves. This book illustrates this interplay of number theory and random matrices. It begins with an introduction to elliptic curves and the fundamentals of modeling by a family of random matrices, and moves on to highlight the latest research. There are expositions of current research on ranks of elliptic curves, statistical properties of families of elliptic curves and their associated L-functions and the emerging uses of random matrix theory in this field. Most of the material here had its origin in a Clay Mathematics Institute workshop on this topic at the Newton Institute in Cambridge and together these contributions provide a unique in-depth treatment of the subject.