COMPLEX MULTIPLICATION OF ABELIAN SURFACES

197 Pages·2010·1.59 MB·English

COMPLEX MULTIPLICATION OF ABELIAN SURFACES
Proefschrift
ter verkrijging van
de graad van Doctor aan de Universiteit Leiden,
op gezag van Rector Magnificus prof. mr. P.F. van der Heijden,
volgens besluit van het College voor Promoties
te verdedigen op dinsdag 1 juni 2010
klokke 15:00 uur
door
Theodorus Cornelis Streng
geboren te IJsselstein
in 1982
Samenstelling van de promotiecommissie:
Promotor
prof. dr. Peter Stevenhagen
Overige leden
prof. dr. Gunther Cornelissen (Universiteit Utrecht)
prof. dr. Bas Edixhoven
prof. dr. David R. Kohel (Universit´e de la M´editerran´ee)
prof. dr. Hendrik W. Lenstra Jr.
dr. Ronald van Luijk
Complex multiplication of abelian surfaces
Marco Streng
Marco Streng
Complex multiplication of abelian surfaces
ISBN-13 / EAN: 978-90-5335-291-5
AMS subj. class.: 11G15, 14K22
NUR: 921
(cid:13)c Marco Streng, Leiden 2010
[email protected]
Typeset using LaTeX
Printed by Ridderprint, Ridderkerk
Asteroids, of which a screen shot is shown on page 188, is due to
Atari, 1979.
The cover illustration shows the complex curve C : y2 = x5+1 in the
coordinates (Rex,Imx,Rey). Its Jacobian J(C) is an abelian surface
with complex multiplication by Z[ζ ] induced by the curve automor-
5
phism ζ :(x,y)(cid:55)→(ζ x,y). The colored curves are the real locus of C
5 5
and its images under (cid:104)ζ (cid:105). The illustration was created using Sage [70]
5
and Tachyon.
Contents
Contents 5
Introduction 9
I Complex multiplication 17
1 Kronecker’s Jugendtraum . . . . . . . . . . . . . . . . . 17
2 CM-fields . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3 CM-types . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4 Complex multiplication . . . . . . . . . . . . . . . . . . 21
5 Complex abelian varieties . . . . . . . . . . . . . . . . . 23
5.1 Complex tori and polarizations . . . . . . . . . . 23
5.2 Ideals and polarizations . . . . . . . . . . . . . . 24
5.3 Another representation of the ideals . . . . . . . 27
6 Jacobians of curves . . . . . . . . . . . . . . . . . . . . 28
7 The reflex of a CM-type . . . . . . . . . . . . . . . . . 30
8 The type norm . . . . . . . . . . . . . . . . . . . . . . . 32
9 The main theorem of complex multplication . . . . . . 33
10 The class fields of quartic CM-fields . . . . . . . . . . . 35
II Computing Igusa class polynomials 39
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 39
2 Igusa class polynomials . . . . . . . . . . . . . . . . . . 41
2.1 Igusa invariants . . . . . . . . . . . . . . . . . . . 42
2.2 Alternative definitions . . . . . . . . . . . . . . . 43
3 Abelian varieties with CM . . . . . . . . . . . . . . . . 44
3.1 The general algorithm . . . . . . . . . . . . . . . 45
3.2 Quartic CM-fields . . . . . . . . . . . . . . . . . 46
3.3 Implementation details. . . . . . . . . . . . . . . 47
4 Symplectic bases . . . . . . . . . . . . . . . . . . . . . . 49
4.1 A symplectic basis for Φ(a) . . . . . . . . . . . . 49
6 Contents
4.2 A symplectic basis for (z,b) . . . . . . . . . . . . 51
5 The fundamental domain of the Siegel upper half space 52
5.1 The genus-1 case . . . . . . . . . . . . . . . . . . 52
5.2 The fundamental domain for genus two . . . . . 55
5.3 The reduction algorithm for genus 2 . . . . . . . 57
5.4 Identifying points on the boundary . . . . . . . . 62
6 Bounds on the period matrices . . . . . . . . . . . . . . 64
6.1 The bound on the period matrix . . . . . . . . . 64
6.2 A good pair (z,b) . . . . . . . . . . . . . . . . . 65
7 Theta constants . . . . . . . . . . . . . . . . . . . . . . 67
7.1 Igusa invariants in terms of theta constants . . . 68
7.2 Bounds on the theta constants . . . . . . . . . . 70
7.3 Evaluating Igusa invariants . . . . . . . . . . . . 72
7.4 Evaluating theta constants . . . . . . . . . . . . 74
8 The degree of the class polynomials . . . . . . . . . . . 76
9 Denominators . . . . . . . . . . . . . . . . . . . . . . . 76
9.1 The bounds of Goren and Lauter . . . . . . . . . 77
9.2 The bounds of Bruinier and Yang. . . . . . . . . 80
9.3 Counterexample to a conjectured formula . . . . 82
10 Recovering a polynomial from its roots . . . . . . . . . 82
10.1 Polynomial multiplication . . . . . . . . . . . . . 82
10.2 Recovering a polynomial from its roots. . . . . . 84
10.3 Recognizing rational coefficients . . . . . . . . . 86
11 The algorithm . . . . . . . . . . . . . . . . . . . . . . . 87
III The irreducible components of the CM locus 91
1 The moduli space of CM-by-K points . . . . . . . . . . 92
2 The irreducible components of CM . . . . . . . . . 92
K,Φ
3 Computing the irreducible components . . . . . . . . . 94
4 The CM method . . . . . . . . . . . . . . . . . . . . . . 98
5 Double roots . . . . . . . . . . . . . . . . . . . . . . . . 101
IV Abelian varieties with prescribed embedding degree 105
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 105
2 Weil numbers yielding prescribed embedding degrees . 107
3 Performance of the algorithm. . . . . . . . . . . . . . . 112
4 Constructing abelian varieties with given Weil numbers 117
5 Numerical examples . . . . . . . . . . . . . . . . . . . . 119
Contents 7
V Abelian surfaces with p-rank 1 123
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 123
2 Characterization of abelian surfaces with p-rank one . . 125
3 Existence of suitable Weil numbers . . . . . . . . . . . 127
4 The algorithms. . . . . . . . . . . . . . . . . . . . . . . 130
5 Constructing curves with given Weil numbers . . . . . 136
6 A sufficient and necessary condition . . . . . . . . . . . 137
7 Factorization of class polynomials mod p . . . . . . . . 141
8 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 143
Appendix 145
1 The Fourier expansion of Igusa invariants . . . . . . . . 147
2 An alternative algorithm for enumerating CM varieties 151
2.1 Reduced pairs (z,b) . . . . . . . . . . . . . . . . 151
2.2 Real quadratic fields . . . . . . . . . . . . . . . . 153
2.3 Analysis of Algorithm 2.5 . . . . . . . . . . . . . 156
2.4 Generalization of Spallek’s formula . . . . . . . . 157
3 Experimental results . . . . . . . . . . . . . . . . . . . 159
3.1 Good absolute Igusa invariants . . . . . . . . . . 159
3.2 Asymptotics of bit sizes . . . . . . . . . . . . . . 163
Bibliography 167
List of notation 177
Index of terms 179
Index of people 182
Nederlandse samenvatting 185
1 Priemgetallen . . . . . . . . . . . . . . . . . . . . . . . 185
2 Een probleem uit de getaltheorie . . . . . . . . . . . . . 185
3 De oplossing . . . . . . . . . . . . . . . . . . . . . . . . 186
4 Een variant op het probleem . . . . . . . . . . . . . . . 187
5 Fietsbanden . . . . . . . . . . . . . . . . . . . . . . . . 188
6 Elliptische krommen . . . . . . . . . . . . . . . . . . . . 189
7 Pinpassen en slimme prijskaartjes . . . . . . . . . . . . 190
8 Dubbele donuts . . . . . . . . . . . . . . . . . . . . . . 192
9 Wat staat er in dit proefschrift? . . . . . . . . . . . . . 192
Dankwoord / Acknowledgements 195
Curriculum vitae 197

Introduction
The theory of complex multiplication makes it possible to construct
certain class fields and abelian varieties. The main theme of this thesis
is making these constructions explicit for the case where the abelian
varieties have dimension 2.
Elliptic curves over finite fields
One-dimensionalabelianvarietiesareknownaselliptic curves,whichin
most cases can be represented as a curve in the (x,y)-plane given by
y2 =x3+ax+b (0.1)
for some choice of parameters a, b in a field k. Elliptic curves come
with a natural (abelian) group law, which can be described completely
geometrically.
Intherepresentation(0.1),theunitelementofthegroupisanextra
point O at infinity, and three points P,Q,R satisfy P +Q+R = O in
the group if and only if they are collinear. For k = R, this looks as
follows.
P P +Q
Q
R
The group law can be given by algebraic equations, and we can define
elliptic curves over any field k. If k has characteristic different from 2
and3,whichweassumefromnowonforsimplicity,thenthisisdoneby
takingaandbink. Ifwedothisforafinite fieldk,thenthegroupE(k)
10 Introduction
ofpointsdefinedoverk isfinite. Indeed,thenumberofelements#E(k)
of E(k) can be computed simply by testing for every x-coordinate in k
whether x3+ax+b is a square in k.
If the order q = #k of k gets large, then this method of point
countingtakestoomuchtime. However,therearefastermethodsbased
on the properties of the Frobenius endomorphism F : (x,y) (cid:55)→ (xq,yq)
of E. The points in E(k) are exactly those points over an algebraic
closure of k that are left invariant by F. In particular, they are the
points in the kernel of the endomorphism (F −id), where subtraction
takesplaceintheringofendomorphisms End(E)ofE. Itisknownthat
F is (as an element of the endomorphism ring) a root of a quadratic
Weil polynomial
f =X2−tX+q ∈Z[X], (0.2)
and that we have
#E(k)=deg(F −id)=f(1)=q+1−t.
√
ThetraceofFrobenius tisboundedinsizeby|t|≤2 q,andindicatesto
whichextent#E(k)differsfromthenumberq+1ofpointsonastraight
line. Schoof realized in 1985 that the reductions (tmodl) at small
primesl canbecomputedbylookingattheactionofF onthel-torsion
points of E, and that this allows one to compute the number t, and
therefore #E(k), efficiently. This yields a polynomial time algorithm
that, for large q, is much faster than the exponential time method of
direct point counting.
Cryptography
Suppose one has a finite group G in which the group operation can be
efficiently implemented, but the discrete logarithm problem is thought
to be hard. This means that given x,y ∈ G, finding an integer m
such that y =xm holds is hard. Then the Diffie-Hellman key exchange
protocolfrom1976allowsonetoagreeuponacryptographickeyinsuch
a way that eavesdroppers, who intercept the entire communication, are
believed to be unable to derive the key from it. The original example
of such a group G is the unit group G=k∗ for a prime finite field k =
F . Index calculus methods like the number field sieve provide a sub-
p
exponential method for solving the discrete logarithm problem in k∗.
To protect the protocol against this algorithm until the year 2030, it is
generally recommended to use primes p of over 3000 bits.
As for G = k∗, the group order of G = E(k) for an elliptic curve
E is of size approximately #k = q. However, it seems that the dis-
crete logarithm problem for the group E(k) is harder, as 35 years of