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Applications of Homogeneous Dynamics to Quadratic Forms : 동질 동역학의 이차 형식들로의 응용

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dc.contributor.advisor임선희-
dc.contributor.author한지영-
dc.date.accessioned2017-07-14T00:42:02Z-
dc.date.available2017-07-14T00:42:02Z-
dc.date.issued2016-02-
dc.identifier.other000000132400-
dc.identifier.urihttps://hdl.handle.net/10371/121303-
dc.description학위논문 (박사)-- 서울대학교 대학원 : 수리과학부, 2016. 2. 임선희.-
dc.description.abstractWe study the geometry of quadratic forms using equidistribution theorems in homogeneous dynamics. First we study the mean square limit of exponential sums associated to a rational ellipsoid of arbitrary center. We obtain a lower bound for arbitrary center and that lower bound turns out to be the upper bound as well for ellipsoids with the center of certain diophantine type(see theorem 1.0.4). This result generalizes a work of Marklof.

The second topic is the quantitative Oppenheim conjecture for $S$-arithmetic quadratic forms. For an arbitrary open set $I$ in $\mathbb Q_S$, we show that the number of $S$-integral vectors of norm at most $T$, whose values of an irrational quadratic form are $Q$ in $I$, is asymptotically $c(Q, I)T^{n-2}$ as $T$ goes to infinity.
This is a generalization of a work of Eskin-Margulis-Mozes for real case.		-
dc.description.tableofcontentsChapter 1 Introduction 1
I. Distribution of Integral Lattice Points in an Ellipsoid with a Diophantine Center 1
II. Quantitative Oppenheim Conjecture for S-arithmetic case 5

Part I. Distribution of Integral Lattice Points in an Ellipsoid with a Diophantine Center8
Chapter 2.Two Representations and Jacobi Theta Sums 9
2.1 Schr\"{o}dinger representation 10
2.2 Shale-Weil representation 12
2.3 Maslov index and the cocycle of $ R$ 15
2.4 The subgroup $\mathrm {SL}_2( R)^n$ and notations 17
2.5 Jacobi's theta sum 20
2.6 Relation between Jacobi's theta sums and the mean square value of exponential sums 26
Chapter 3. Dynamics on $\mathrm {SL}_2(\mathbb R)^n\ltimes {\mathbb {R}}^{2n}/\Gamma $ 29
3.1 Equidistribution of closed orbits 29
3.2 Proof of Theorem 1.0.4 38

Part II. Quantitative Oppenheim Conjecture for $S$-arithmetic case 44
Chapter 4. Preliminaries 45
4.1 Geometry of $\mathrm {SL}_n({\mathbb {Q}}_S)/\mathrm {SL}_n({\mathbb {Z}}_S)$ 45
4.2 Quadratic forms in ${\mathbb {Q}}_S^n$ and orthogonal groups 48
4.2.1 Quadratic forms over ${\mathbb {Q}}_p$ 48
4.2.2 Orthogonal groups 52
4.3 Integration on ${\mathbb {Q}}_S$ 57
4.3.1 Measure on ${\mathbb {Q}}_p^n$ 58
4.3.2 Norm of $\wedge ^i({\mathbb {Q}}_S^n)$ 61
4.3.3 Integration of submanifolds in ${\mathbb {Q}}_p^n$ 62
Chapter 5 $\alpha $-function 64
5.1 The rational subspace and the $\alpha $-function 64
5.2 The limit of $\mathrm K$-orbit in $\wedge ^i({\mathbb {Q}}_p^n)$ 67
5.3 Behavior of the $\alpha $-function 76
Chapter 6 $J_f$-function and results 85
6.1 p-adic analogue of the $J_f$ function 85
6.2 $S$-arithmetic Result 91
6.3 Proof of Main Theorem 97

Bibliography 100

Abstract (in Korean) 103		-
dc.formatapplication/pdf-
dc.format.extent2337319 bytes-
dc.format.mediumapplication/pdf-
dc.language.isoen-
dc.publisher서울대학교 대학원-
dc.subjecthomogeneous dynamics-
dc.subjectJacobi theta sum-
dc.subjectquantitative Oppenheim conjecture-
dc.subjectequidstribution of unbounded functions-
dc.subject.ddc510-
dc.titleApplications of Homogeneous Dynamics to Quadratic Forms-
dc.title.alternative동질 동역학의 이차 형식들로의 응용-
dc.typeThesis-
dc.contributor.AlternativeAuthorHan, Jiyoung-
dc.description.degreeDoctor-
dc.citation.pages103-
dc.contributor.affiliation자연과학대학 수리과학부-
dc.date.awarded2016-02-
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