Isomorphism classes of abelian varieties over finite fields

68 Pages·2016·0.44 MB·English

Isomorphismclassesofabelianvarietiesoverfinitefields
StefanoMarseglia

Isomorphism classes of abelian
varieties over finite fields
Stefano Marseglia
©StefanoMarseglia,Stockholm,2016
Address:MatematiskaInstitutionen,StockholmsUniversitet,10691Stockholm
E-mailaddress:[email protected]
ISBN978-91-7649-444-8
PrintedinSwedenbyE-PrintAB2015,Stockholm,2016
Distributor:DepartmentofMathematics,StockholmUniversity
Abstract
Following[Del69]and[How95], itispossibletodescribepolarizedabelian
varietiesoverfinitefieldsintermsoffinitelygeneratedfree(cid:90)-modulessat-
isfying a list of easy to state axioms. In this thesis we address the prob-
lem ofdevelopinganeffective algorithm to compute isomorphism classes
of (principally) polarized abelian varieties over a finite field, together with
their automorphism groups. Performing such computations requires the
knowledge of the ideal classes (both invertible and non-invertible) of cer-
tainordersinnumberfields. Hencewedescribeamethodtocomputethe
ideal class monoid of an order and we produce concrete computations in
dimension2,3and4.

Sammanfattning
Deligne [Del69] och Howe [How95] har visat att det är möjligt att beskriva
ordinära polariserade abelska varieteter över ändliga kroppar i termer av
ändligtgenereradefriaheltalsmodulersomuppfyllerenserieaxiom,vilkaär
enklaattformulera. Idennaavhandlinganvändervidennabeskrivningför
attutvecklaeneffektivalgoritmförattberäknaisomorfiklasseravprincip-
ielltpolariseradeabelskavarieteteröverändligakroppar,tillsammansmed
deras automorfigrupper. En sådan beräkning kräver kunskap om idealk-
lasserna (både de inverterbara och de icke-inverterbara) av vissa talringar
ialgebraiskatal-kroppar. Därförbeskrivervienmetodförattberäknaide-
alklassmonoidenavtalringarochviredovisarkonkretaexempelidimension
två,treochfyra.

Contents
Abstract v
Sammanfattning vii
Acknowledgements xi
1 Introduction 13
2 AbelianVarieties 15
2.1 AbelianVarieties . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Isogenies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3 Dualabelianvarietiesandpolarizations. . . . . . . . . . . . . . 17
2.4 Endomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.5 TheRosatiinvolution . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.6 Abelianvarietiesover(cid:67) . . . . . . . . . . . . . . . . . . . . . . . 21
2.7 Abelianvarietiesoverafinitefield . . . . . . . . . . . . . . . . . 22
2.8 Weilconjectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.9 TheTatemoduleandcharacteristicpolynomialsofFrobenius 24
2.10 Honda-Tatetheory . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3 Theidealclassmonoid 27
3.1 Orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 TheIdealClassMonoidandthePicardGroup . . . . . . . . . . 29
3.3 GorensteinOrdersandCliffordmonoids . . . . . . . . . . . . . 31
3.4 ComputationofthePicardGroup . . . . . . . . . . . . . . . . . 34
3.5 Weakequivalenceclasses . . . . . . . . . . . . . . . . . . . . . . 36
3.6 ComputationofW(R) . . . . . . . . . . . . . . . . . . . . . . . . 37
3.7 ComputationoftheICM . . . . . . . . . . . . . . . . . . . . . . . 38
3.8 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.9 Conjugacyclassesofintegralmatrices . . . . . . . . . . . . . . . 41
4 TheDelignecategory 43
4.1 TheDelignecategoryL . . . . . . . . . . . . . . . . . . . . . . . 43
q
4.2 DualabelianvarietyandpolarizationsinL . . . . . . . . . . . 45
q
4.3 Isomorphismclassesinaordinaryisogenyclass . . . . . . . . . 46
5 Computing isomorphism classes of principally polarized abelian
varieties 51
5.1 Computationalremarks . . . . . . . . . . . . . . . . . . . . . . . 51
5.2 Ellipticcurves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.3 Abeliansurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.4 Abelianvarietiesofhigherdimension . . . . . . . . . . . . . . . 58
References lxv