Table Of ContentElliptic 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
handedouttotheclassofanintroductorycourseonellipticcurvesgivenin
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 has ∆=1 or −1 . . . . . . . . . . 130
/Z
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 E (K) . . . . . . . . . . . . . . . . . 208
m
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:
y2+a xy+a y =x3+a x2+a x+a . (1)
1 3 2 4 6
The coefficients a are in a field K and E(K) denotes the set of all solutions
i
(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
" " a = i
i
sothateachtermintheequationhasweight6. (Thisalso“explains”theabsence
of a .)
5
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 charK, 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 curveof genus1 with a rational point. (As
wewillexplainlater,conicsections—circles,ellipses,parabolas,andhyperbolas
— 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),
− ± ±
assumingcharK =2,3. Proposition1.2.1belowexplainshowtotransformsuch
quartic equations6 into Weierstrass form (without using √43 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!)
Nowletusbegintofillinsomedetails. ConsideraWeierstrassequation(1),
which we denote as E. If charK = 2 we can complete the square by defining
6
η =y+(a x+a )/2:
1 3
b b b
η2 =x3+ 2x2+ 4x+ 6 (2)
4 2 4
where b2 =a21+4a2, b4 =a1a3+2a4, b6 =a23+4a6. (3)
If charK =3 we can complete the cube by setting ξ =x+b /12:
2
6
c c
η2 =ξ3 4ξ 6 (4)
− 48 − 864
where c4 =b22−24b4, c6 =−b32+36b2b4−216b6. (5)
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
and ∆=−b22b8−8b34−27b26+9b2b4b6. (7)
The subscripts on the b’s and c’s are their weights. We refer to (1), (2) and (4)
asthea-form,b-formandc-formrespectively. Thedefinitions(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
6
will explain in 1.5. Then one defines j =c3/∆. For example
§ 4
when charK =2, ∆=0= a and a are not both zero.
1 3
6 ⇒
1.1. THE A,B,C ’S AND ∆,J,... 103
Thus κ:=2y+a x+a
1 3
is nonzero† for every elliptic curve E in any characteristic. When charK = 2
6
we have κ=2η and κ is determined up to sign by x. Note that
κ2 =4x3+b x2+2b x+b
2 4 6
is valid in all characteristics.
The covariants c ,c and the discriminant ∆ have weights 4,6,12 respec-
4 6
tively. The quantity j defined above when ∆ = 0 is called the j-invariant, or
6
simply the invariant of E; its weight is 0.
It is often convenient to include ∆ as a third covariant. Thus we say that
y2+y =x3 x2 (A11)
−
has covariants 16, 152, 11, meaning that c =16, c = 152 and ∆= 11.
4 6
− − − −
The label A11 is the standard catalog name of this elliptic curve as in [AntIV];
weputtheletterfirst, ratherthan11A,sothatA11canbeusedasthenameof
thiscurveincomputerprogramssuchasapecs;seetheappendixtothischapter.
In[Cre92],whichextendsthecatalogof[AntIV],thelabellinghasbeenmodified
(with the former notation given in parentheses) — this curve is denoted A 11;
by force of habit, we will use the notation of [AntIV] for curves contained in
that catalog, and then use Cremona’s notation for curves that are only in the
larger catalog.
For convenience of reference, we collect these various definitions in a box:
b = a2+4a ,
2 1 2
b = a a +2a ,
4 1 3 4
b = a2+4a ,
6 3 6
b = a2a a a a +4a a +a a2 a2,
8 1 6− 1 3 4 2 6 2 3− 4
c = b2 24b ,
4 2− 4
c = b3+36b b 216b ,
6 − 2 2 4− 6
∆ = b2b 8b3 27b2+9b b b ,
− 2 8− 4− 6 2 4 6
κ = 2y+a x+a ,
1 3
4b = b b b2,
8 2 6− 4
1728∆ = c3 c2,
4− 6
j = c3/∆ = 1728+c2/∆.
4 6
Thelastthreelinesintheboxareidentitiesthatonecanverifyonthecomputer.
†asanelementofthefieldL=K(x,y)obtainedasaquadraticextensionK(x)(y)ofthe
transcendental extension K(x), where y is defined by equation (1). As will be discussed in
1.6,Liscalledthefunction fieldofE,andP =(x,y) E(L)iscalledageneric point.
§ ∈
104 CHAPTER 1. INTRODUCTION TO ELLIPTIC CURVES.
Examples:
1. Suppose charK = 2. Then ∆ is 16 times the polynomial discriminant†
6
of the cubic on the right side of the b-form (2):
Dis x3+(b /4)x2+(b /2)x+b /4 =∆/16.
2 4 6
¡ ¢
Hence ∆=0 iff the cubic has a multiple root.
2. If charK =2 or 3, an alternative to the c-form is
6
η02 =ξ03 27c¯ ξ0 54c¯ , η0 =63η, ξ0 =62ξ.
4 6
− −
Caution: We have put bars on the c’s because with the displayed values for
the Weierstrass coefficients a = 0,...,a = 54c¯ the formulas give c =
1 6 6 4
−
64c¯ , c = 66c¯ . In the case of (4), bars are not necessary: the calculated c’s
4 6 6
are the same as the c’s in the equation.
3.
y2 =x3+bx+c
has covariants 48b, 864c, 16(4b3+27c2).
− − −
Thus provided ∆= 2433c2 =0, y2 =x3+c
− 6
has c =0 and j =0;
4
and provided ∆= 64b3 =0, y2 =x3+bx
− 6
has c =0 and j =1728=123.
6
4. “Generic j”: provided j =0,1728,
6
36 1
y2+xy =x3 x
− j 1728 − j 1728
− −
has j-invariant =j; the covariants are
j j2
c = c = , and ∆= .
4 − 6 j 1728 (j 1728)3
− −
5. WhenKistherealfieldRwecantaketheequationinc-formη2 =ξ3+ .
···
Thecubichaseither1or3realrootsaccordingasthediscriminant∆isnegative
or positive; thus as a real manifold there are 1 or 2 components. We will see in
1.3 that the addition of the point O at will compactify the curve.
§ ∞
On the following interleaving sheet there are plots of three examples (the
same ones used in [Sil86,p.47]).
†intheusualsenseDis(f)=( 1)n(n−1)/2Resultant(f,f0)wheren=deg(f):
−
Dis(X2+aX+b) = a2 4b,
−
Dis(X3+aX2+bX+c) = 4a3c+a2b2+18abc 4b3 27c2,
− − −
inparticular, Dis(X3+bX+c) = 4b3 27c2, and
− −
Dis(X4+bX2+cX+d) = 16b4d 4b3c2 128b2d2
− −
+144bc2d 27c4+256d3.
−
