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The handle http://hdl.handle.net/1887/38431 holds various files of this Leiden University dissertation
Author: Gunawan, Albert
Title: Gauss's theorem on sums of 3 squares, sheaves, and Gauss composition Issue Date: 2016-03-08
Gauss’s theorem on sums of 3 squares, sheaves, and Gauss composition
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 dinsdag 8 maart 2016 klokke 16:15 uur
door
Albert Gunawan
geboren te Temanggung in 1988
Promotor: Prof. dr. Bas Edixhoven
Promotor: Prof. dr. Qing Liu (Universit´e de Bordeaux)
Samenstelling van de promotiecommissie:
Prof. dr. Philippe Gille (CNRS, Universit´e Lyon) Prof. dr. Hendrik Lenstra (secretaris)
Prof. dr. Aad van der Vaart (voorzitter)
Prof. dr. Don Zagier (Max Planck Institute for Mathematics, Bonn)
This work was funded by Algant-Doc Erasmus-Mundus and was carried out at Universiteit Leiden and Universit´e de Bordeaux.
2
TH` ESE EN COTUTELLE PR´ ESENT´ EE POUR OBTENIR LE GRADE DE
DOCTEUR DE
L’UNIVERSIT´ E DE BORDEAUX ET DE UNIVERSITEIT LEIDEN
ECOLE DOCTORALE MATH´ ´ EMATIQUES ET INFORMATIQUE
MATHEMATISCH INSTITUUT LEIDEN SPECIALIT´ E : Math´ ematiques Pures
Par Albert GUNAWAN
GAUSS’S THEOREM ON SUMS OF 3 SQUARES, SHEAVES, AND
GAUSS COMPOSITION
Sous la direction de : Bas EDIXHOVEN et Qing LIU Soutenue le : 8 Mars 2016 ` a Leiden
Membres du jury :
M LENSTRA, Hendrik Prof. Universiteit Leiden Pr´esident M GILLE, Philippe Prof. CNRS, Universit´e Lyon Rapporteur
M ZAGIER, Don Prof. MPIM, Bonn Rapporteur
Mme LORENZO, Elisa Dr. Universiteit Leiden Examinateur
Contents
1 Introduction 1
1.1 Motivation . . . . 1
1.2 Cohomological interpretation . . . . 2
1.2.2 Examples . . . . 3
1.2.3 Sheaves of groups . . . . 4
1.3 Gauss composition on the sphere . . . . 5
1.3.1 Explicit description by lattices, and a computation . 5 2 Tools 9 2.1 Presheaves . . . . 9
2.2 Sheaves . . . . 11
2.2.5 Sheafification . . . . 14
2.3 Sheaves of groups acting on sheaves of sets and quotients . . 18
2.4 Torsors . . . . 22
2.5 Twisting by a torsor . . . . 27
2.6 A transitive action . . . . 31
2.7 The Zariski topology on the spectrum of a ring . . . . 36
2.8 Cohomology groups and Picard groups . . . . 41
2.9 Bilinear forms and symmetries . . . . 45 i
2.9.4 Minkowski’s theorem . . . . 49
2.10 Descent . . . . 49
2.11 Schemes . . . . 52
2.12 Grothendieck (pre)topologies and sites . . . . 59
2.13 Group schemes . . . . 62
2.13.8 Affine group schemes . . . . 65
3 Cohomological interpretation 67 3.1 Gauss’s theorem . . . . 67
3.2 The sheaf SO3 acts transitively on spheres . . . . 69
3.3 Triviality of the first cohomology set of SO3 . . . . 73
3.4 Existence of integral solutions . . . . 76
3.4.2 Existence of a rational solution . . . . 77
3.4.5 Existence of a solution overZ(p) . . . . 78
3.4.8 The proof of Legendre’s theorem by sheaf theory . . 81
3.5 The stabilizer in Gauss’s theorem . . . . 83
3.5.4 The orthogonal complement P⊥ of P in Z3. . . . 84
3.5.7 The embedding of H in N . . . . 86
3.5.12 The automorphism group scheme of P⊥ . . . . 89
3.5.17 Determination of H overZ[1/2] . . . . 95
3.6 The group H1(S, T ) as Picard group . . . . 99
3.7 The proof of Gauss’s theorem . . . 101
4 Gauss composition on the 2-sphere 107 4.1 The general situation . . . 107
4.1.1 A more direct description . . . 109
4.2 Gauss composition: the case of the 2-sphere . . . 109
4.2.1 Description in terms of lattices in Q3 . . . 110
4.2.5 Summary of the method . . . 114 ii
4.3 Finding an orthonormal basis for M explicitly . . . 115 4.3.3 The quotient of ts−1M +Z3 by Z3 . . . 117 4.3.6 Explicit computation for ts−1M +Z3⊂Q3 continued 122 4.3.7 Getting a basis for ts−1M given one for ts−1M +Z3 124 4.4 Some explicit computation . . . 129 4.4.2 An example of Gauss composition for 770 . . . 132 4.4.3 Another example for 770 . . . 135
Bibliography 139
Summary 143
Samenvatting 145
R´esum´e 147
Acknowledgments 149
Curriculum Vitae 151
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