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The handle http://hdl.handle.net/1887/62814 holds various files of this Leiden University dissertation.
Author: Martindale, C.R.
Title: Isogeny graphs, modular polynomials, and applications
Issue Date: 2018-06-14
Isogeny graphs, modular polynomials, and applications
Proefschrift ter verkrijging van
de graad van Doctor aan de Universiteit Leiden
op gezag van Rector Magnificus prof. mr. C.J.J.M. Stolker, volgens besluit van het College voor Promoties
te verdedigen op donderdag 14 juni 2018 klokke 11:15 uur
door
Chloe Martindale
geboren te Huntingdon, Verenigd Koninrijk
in 1990
Promotor: Prof. dr. Peter Stevenhagen
Promotor: Prof. dr. Andreas Enge (Unversit´ e de Bordeaux) Copromotor: Dr. Marco Streng
Samenstelling van de promotiecommissie:
Prof. dr. Aad van der Vaart (voorzitter) Prof. dr. Bart de Smit (secretaris)
Prof. dr. David Kohel (Universit´ e de Aix-Marseille)
Prof. dr. Dimitar Jetchev (´ Ecole polytechnique f´ ed´ erale de Lausanne) Dr. Peter Bruin
Dit werk werd gefinancierd door Algant-Doc Erasmus Action en werd uitgevoerd aan de
Universiteit Leiden en de Universit´ e de Bordeaux.
TH` ESE
pr´ esent´ ee ` a
L’UNIVERSIT´ E DE BORDEAUX
ECOLE DOCTORALE DE MATH´ ´ EMATIQUES ET INFORMATIQUE
par Chloe Martindale POUR OBTENIR LE GRADE DE
DOCTEUR
SPECIALIT´ E: Math´ ematiques Pures
Isogeny Graphs, Modular Polynomials, and Applications
Soutenue le : 14 juin 2018 ` a Leiden
Devant la commission d’examen form´ ee de :
ENGE, Andreas Professeur Universit´e de Bordeaux Directeur
STRENG, Marco Docteur Universiteit Leiden Directeur
KOHEL, David Professeur Universit´e de Aix-Marseille Rapporteur JETCHEV, Dimitar Docteur Ecole polytechnique f´´ ed´erale de Lausanne Rapporteur
Ce travail a ´ et´ e financ´ e par Algant-Doc Erasmus Action et a ´ et´ e r´ ealis´ e
`
a l’Universiteit Leiden et ` a l’Universit´ e de Bordeaux.
Contents
Introduction x
1 The theory of canonical lifts and other preliminaries 1
1.1 Principally polarised abelian varieties . . . 1
1.2 Lifting ordinary abelian varieties over Fq to ideals . . . 2
1.3 The Fixed Frobenius Lifting Theorem . . . 3
1.4 The theory of canonical lifts . . . 6
1.4.1 Serre-Tate lifts of ordinary abelian varieties . . . 7
1.4.2 Deligne lifts of ordinary abelian varieties . . . 8
1.4.3 Howe lifts of polarised ordinary abelian varieties . . . 9
1.4.4 Proof of the Fixed Frobenius Lifting Theorem . . . 11
1.5 Maximal real multiplication . . . 13
1.6 Hilbert modular forms . . . 14
1.7 A normalisation lemma for principally polarised ideals . . . 17
2 Hilbert modular polynomials 19 2.1 Introduction and statement of the results . . . 19
2.2 Defining RM isomorphism invariants . . . 21
2.3 Algorithm to compute a set of Hilbert modular polynomials . . . 23
2.4 Computing the RM isomorphism invariants for a given genus 2 curve . . . 28
2.4.1 The algorithm . . . 31
2.5 Complexity and simplifications for genus 2 . . . 34
3 The structure of µ-isogeny graphs 38 3.1 The Volcano Theorem . . . 38
3.2 Parametrising orders by their real conductors . . . 46
3.3 All µ-isogenies are ascending, descending or horizontal . . . 48
3.4 Principally polarised ideals are invertible . . . 50
3.5 The action of the Shimura class group . . . 53
3.6 Counting horizontal µ-isogenies . . . 55
3.7 A construction of ascending µ-isogenies . . . 58
3.8 Counting the degree of vertices in the µ-isogeny graph . . . 59
3.9 The order of the Shimura class group . . . 60
3.10 Example computation of a µ-isogeny graph . . . 66
4 Isogenies for point counting on genus two hyperelliptic curves with maximal real multiplication 51 4.1 Introduction . . . 52
4.1.1 The state of the art . . . 53
4.1.2 Our contributions, and beyond . . . 53
4.1.3 Vanilla abelian varieties . . . 54
4.2 Genus one curves: elliptic curve point counting . . . 55
4.2.1 Schoof’s algorithm . . . 56
4.2.2 Frobenius eigenvalues and subgroups . . . 57
4.2.3 Modular polynomials and isogenies . . . 57
4.2.4 Elkies, Atkin, and volcanic primes . . . 58
iv
4.2.5 Computing the type of a prime . . . 59
4.2.6 Atkin’s improvement . . . 59
4.2.7 Elkies’ improvement . . . 60
4.3 The genus 2 setting . . . 61
4.3.1 The Jacobian . . . 61
4.3.2 Frobenius and endomorphisms of JC . . . 62
4.3.3 Real multiplication . . . 62
4.3.4 From Schoof to Pila . . . 62
4.3.5 The Gaudry–Schost approach . . . 63
4.3.6 Point counting with efficiently computable RM . . . 64
4.3.7 Generalizing Elkies’ and Atkin’s improvements to genus 2 . . . 65
4.3.8 µ-isogenies . . . 66
4.4 Invariants . . . 66
4.4.1 Invariants for RM abelian surfaces . . . 67
4.4.2 Hilbert modular polynomials for RM abelian surfaces . . . 67
4.4.3 Invariants for curves and abelian surfaces . . . 68
4.4.4 Pulling back curve invariants to RM invariants . . . 70
4.5 Atkin theorems in genus 2 . . . 70
4.5.1 Roots of Gµ and the order of Frobenius . . . 70
4.5.2 The factorization of Gµ . . . 72
4.5.3 The characteristic polynomial of Frobenius . . . 73
4.5.4 Prime types for real multiplication by OF . . . 74
4.5.5 The parity of the number of factors of Gµ . . . 75
4.6 The case F = Q(√ 5): Gundlach–M¨uller invariants . . . 75
4.7 Experimental results . . . 77
A The notions of dual and polarisation in equivalent categories 80
Bibliography 89
Index 92
Summary 92
Samenvatting 97
R´esum´e 103
Acknowledgements 109
Curriculum Vitae 110
v
