Elliptic Curve Handbook Ian Connell February, 1999 http://www.math.mcgill.ca/connell/ Foreword The first version of this handbook was a set of notes of about 100 pages handed out to the class of an introductory course on elliptic curves given in the 1990 fall semester at McGill University in Montreal. Since then I have added to the notes, holding to the principle: If I look up a certain topic a year from now I want all the details right at hand, not in an “exercise”, so if I’ve forgotten something I won’t waste time. Thus there is much that an ordinary text would either condense, or relegate to an exercise. But at the same time I have maintained a solid mathematical style with the thought of sharing the handbook. Montreal, August, 1996. 1 Contents 1 Introduction to Elliptic Curves 101 1.1 The a,b,c’s and ∆,j ...................... 101 1.2 Quartic to Weierstrass ..................... 105 1.3 Projective coordinates. ..................... 111 1.4 Cubic to Weierstrass: Nagell’s algorithm ........... 115 1.4.1 Example 1: Selmer curves ............... 117 1.4.2 Example 2: Desboves curves .............. 121 1.4.3 Example 3: Intersection of quadric surfaces ..... 123 1.5 Singular points. ......................... 125 − 1.5.1 Example: No E/Z has ∆ = 1 or 1 .......... 130 1.6 Affine coordinate ring, function field, generic points ..... 132 1.7 The group law: nonsingular case ............... 133 1.7.1 Halving points ..................... 140 1.7.2 The division polynomials ............... 145 1.7.3 Remarks on the group of division points ....... 151 1.8 The group law: singular case ................. 152 1.8.1 Examples over finite fields ............... 156 Appendix: introduction to apecs .................. 160 2 Formal Groups 201 2.1 Discrete valuations ....................... 202 2.1.1 Examples ........................ 206 2.1.2 The filtration Em(K) ................. 208 2.1.3 Finite extensions .................... 210 2.1.4 Gauss’s lemma ..................... 212 2.2 Krull domains .......................... 212 2.2.1 Dedekind domains ................... 216 2.2.2 One variable function fields .............. 217 2.2.3 Elliptic function fields ................. 223 2.3 The group of reversible power series ............. 225 3 4 CONTENTS 2.4 Hensel’s lemma ......................... 228 2.4.1 An application to P -adically reversible series .... 232 2.5 Applications to elliptic curves ................. 234 2.5.1 Infinitesimal shifts ................... 234 2.5.2 Reduction mod π : a first look ............ 235 2.5.3 Local expansions .................... 240 2.6 Formal groups .......................... 244 2.6.1 The additive and multiplicative formal groups .... 248 2.6.2 The formal group of an elliptic curve ......... 251 2.7 The invariant differential of a formal group ......... 253 2.7.1 The elliptic curve case ................. 258 2.8 Formal groups in characteristic p............... 261 2.9 Formal groups in characteristic 0 ............... 263 2.9.1 The formal logarithm ................. 263 2.9.2 Formal groups over discrete valuation rings ..... 264 2.10 The Nagell-Lutz theorem for Krull domains ......... 268 2.10.1 Nagell-Lutz for Z .................... 273 2.10.2 Nagell-Lutz for quadratic fields ............ 277 3 The Mordell-Weil theorem 301 3.1 F2-Krull domains ........................ 303 3.2 The weak Mordell-Weil theorm ................ 305 3.3 Heights ............................. 307 3.3.1 Heights in number fields ................ 308 3.3.2 Heights in function fields ............... 311 3.4 Completion of the proof of Mordell-Weil ........... 314 3.4.1 Function fields in characteristic 0 ........... 318 3.5 The canonical height ...................... 321 3.5.1 Calculating the canonical height: a first look .... 329 3.5.2 The successive minima ................. 333 3.6 Algorithms for Mordell-Weil bases: a first look ....... 336 3.6.1 Simple 2-descent .................... 340 3.6.2 Simple 2-descent over UFD’s ............. 345 3.6.3 Examples over Q .................... 348 3.6.4 The Hilbert norm residue symbol ........... 351 3.6.5 Continuation of examples over Q ........... 355 3.6.6 Second descent ..................... 366 3.6.7 A transcendental example ............... 371 3.7 Billing’s upper bound for the rank .............. 372 3.7.1 Examples ........................ 377 3.7.2 An example of Selmer ................. 381 CONTENTS 5 4 Twists 401 4.1 Isomorphisms of elliptic curves ................ 401 4.2 Simplified Weierstrass equations ............... 405 4.3 Twists, quadratic and otherwise ................ 408 4.4 The isomorphism algorithm .................. 414 4.5 Automorphisms and fields of definition ............ 417 4.6 Legendre and Deuring forms .................. 420 4.7 Finite fields ........................... 423 4.7.1 The trace of Frobenius: preliminaries ......... 423 4.7.2 An application of Burnside’s formula ......... 426 4.7.3 The trace of Frobenius: continuation ......... 431 4.7.4 Examples ........................ 436 4.7.5 A preview of some future topics ............ 439 5 Minimal Weierstrass Equations 501 5.1 Some definitions ........................ 502 5.2 Kraus’s theorem ........................ 503 5.2.1 The case v(p) = 1 ................... 507 5.2.2 The globalization of Kraus’s theorem to number fields 510 5.3 Local minimal forms and δv .................. 513 5.3.1 The special cases v(3) = 1 and v(2) = 1 ....... 516 5.4 Unramified base change .................... 518 5.5 Global minimal equations and AE .............. 523 5.6 The Laska-Kraus algorithm .................. 525 5.6.1 The case Z ....................... 525 5.6.2 The general number field case ............. 530 5.7 How twisting affects minimal models ............. 534 5.7.1 The local case ...................... 534 5.7.2 The global case ..................... 537 5.7.3 Minimal twists ..................... 540 Chapter 1 Introduction to Elliptic Curves. 1.1 The a, b, c ’s and ∆, j, . We begin with a series of definitions of elliptic curve in order of increasing generality and sophistication. These definitions involve technical terms which will be defined at some point in what follows. The most concrete definition is that of a curve E given by a nonsingular Weierstrass equation: 2 3 2 y + a1xy + a3y = x + a2x + a4x + a6. (1) The coefficients ai are in a field K and E(K) denotes the set of all solutions (x, y) K K, together with the point O “at infinity” — to be explained in 1.3. We∈ will× see later why the a’s are numbered in this way; to remember the Weierstrass§ equation think of the terms as being in a graded ring with weight of x = 2 "" y = 3 "" ai = i so that each term in the equation has weight 6. (This also “explains” the absence of a5.) A slightly more general definition is: a plane nonsingular cubic with a ra- tional point (rational means the coordinates are in the designated field K and does not refer to the rational field Q, unless of course K = Q). An example of such a curve that is not a Weierstrass equation is the Fermat curve x3 + y3 = 1, with points (x, y) = (1, 0), (0, 1), 101 102 CHAPTER 1. INTRODUCTION TO ELLIPTIC CURVES. assuming the characteristic of K, denoted char K, is not 3. In Corollary 1.4.2 we will see how to transform such an equation into Weierstrass form. More general still: a nonsingular curve of genus 1 with a rational point. (As we will explain later, conic sections — circles, ellipses, parabolas, and hyperbolas — have genus 0 which implies that they are not elliptic curves.) An example that is not encompassed by the previous definitions is y2 = 3x4 2, with points (x, y) = ( 1, 1), − ± ± assuming char K = 2, 3. Proposition 1.2.1 below explains how to transform such quartic equations6 into Weierstrass form (without using √4 3 or √ 2 !). Alternative terminology which emphasizes the algebraic group− structure: abelian variety of dimension 1. More abstractly: E is a scheme over a base scheme S (e.g. spec K) which is proper, flat and finitely presented, equipped with a section ... : there is little point to state all the technicalities at this time. Suffice it to say that the work of Tate, Mazur and many others makes it plain that it is essential to know the language of schemes to understand the deeper arithmetic properties of elliptic curves. (More easily said than done!) Now let us begin to fill in some details. Consider a Weierstrass equation (1), which we denote as E. If char K = 2 we can complete the square by defining 6 η = y + (a1x + a3)/2: b b b η2 = x3 + 2 x2 + 4 x + 6 (2) 4 2 4 2 2 where b2 = a1 + 4a2, b4 = a1a3 + 2a4, b6 = a3 + 4a6. (3) If char K = 3 we can complete the cube by setting ξ = x + b /12: 6 2 c c η2 = ξ3 4 ξ 6 (4) − 48 − 864 2 3 where c4 = b 24b4, c6 = b + 36b2b4 216b6. (5) 2 − − 2 − One then defines b = a2a a a a + 4a a + a a2 a2, (6) 8 1 6 − 1 3 4 2 6 2 3 − 4 2 3 2 and ∆ = b b8 8b 27b + 9b2b4b6. (7) − 2 − 4 − 6 The subscripts on the b’s and c’s are their weights. We refer to (1), (2) and (4) as the a-form, b-form and c-form respectively. The definitions (3) and (5)–(7) are made for all E, regardless of the characteristic of K, and the condition that the curve be nonsingular, and so define an elliptic curve, is that ∆ = 0, as we will explain in 1.5. Then one defines j = c3/∆. For example 6 § 4 when char K = 2, ∆ = 0 = a and a are not both zero. 6 ⇒ 1 3 1.1. THE A, B, C ’S AND ∆, J, . 103 Thus κ := 2y + a1x + a3 is nonzero† for every elliptic curve E in any characteristic. When char K = 2 we have κ = 2η and κ is determined up to sign by x.