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Quantitative Aspects of Arakelov Theory in Arithmetic Dynamics
Quantitative Aspects of Arakelov Theory in Arithmetic Dynamics
상세정보
- 자료유형
- 학위논문 서양
- 최종처리일시
- 20260202103104
- ISBN
- 9798280718739
- DDC
- 510
- 저자명
- Yap, Jit Wu.
- 서명/저자
- Quantitative Aspects of Arakelov Theory in Arithmetic Dynamics
- 발행사항
- [Sl] : Harvard University, 2025
- 발행사항
- Ann Arbor : ProQuest Dissertations & Theses, 2025
- 형태사항
- 140 p
- 주기사항
- Source: Dissertations Abstracts International, Volume: 86-12, Section: B.
- 주기사항
- Advisor: DeMarco, Laura.
- 학위논문주기
- Thesis (Ph.D.)--Harvard University, 2025.
- 초록/해제
- 요약Let K be a number field and φ : ℙm → ℙm be an endomorphism of degre d ≥ 2 that is defined over K. Let hφ : ℙm (K)→ℝ≥0 be the canonical height associated to φ. Given a sequence (xn) ∈ ℙm(K), we say that it is generic if no hypersurface Z contains infinitely many xn's. Yuan [Yua08], using Arakelov theory, proves that given a generic sequence of points (xn) with hφ(xn) → 0 and a place v ∈ MK, the Galois orbits of xn will equidistribute to the equilibrium measure μυ.The aim of this thesis is to prove a quantitative version of Yuan's theorem for archimedean places. Given a smooth function f: ℙm (ℂ) → ℝ and an ε 0, we bound the degree of a hypersurface Z(f, e) and a constant δ 0 such thatholds for all x ∉ Z(f, ε) and hφ(x) δ, where Fx = Gal(K/K) · x is the Galois orbit of x. This upper bound on deg Z(f, e) tells us how generic x has to be. There are two main new ingredients in the proof which follows Yuan's approach. The first is a quantitative form of the asymptotic expansion of the Bergman kernel, first established by Tian, and the second is a construction of a "dynamical" basis of polynomials due to Looper. As an application, for ℙ2 or smooth projective surfaces in general, we are able to deduce an exponential rate of convergence of n-periodic points Pern, to the equilibrium measure. A more arithmetic application is that we are able to deduce an exponential growth of the degree [K(Pern) : K] in terms of n, generalizing results due to Baker in dimension one.[Equation omitted].
- 일반주제명
- Mathematics
- 일반주제명
- Theoretical mathematics
- 키워드
- Arakelov theory
- 키워드
- Equidistribution
- 키워드
- Small points
- 기타저자
- Harvard University Mathematics
- 기본자료저록
- Dissertations Abstracts International. 86-12B.
- 전자적 위치 및 접속
- 로그인 후 원문을 볼 수 있습니다.
MARC
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■00520260202103104
■006m o d
■007cr#unu||||||||
■020 ▼a9798280718739
■035 ▼a(MiAaPQ)AAI31935213
■040 ▼aMiAaPQ▼cMiAaPQ
■0820 ▼a510
■1001 ▼aYap, Jit Wu.▼0(orcid)0000-0002-3375-2490
■24510▼aQuantitative Aspects of Arakelov Theory in Arithmetic Dynamics
■260 ▼a[Sl]▼bHarvard University▼c2025
■260 1▼aAnn Arbor▼bProQuest Dissertations & Theses▼c2025
■300 ▼a140 p
■500 ▼aSource: Dissertations Abstracts International, Volume: 86-12, Section: B.
■500 ▼aAdvisor: DeMarco, Laura.
■5021 ▼aThesis (Ph.D.)--Harvard University, 2025.
■520 ▼aLet K be a number field and φ : ℙm → ℙm be an endomorphism of degre d ≥ 2 that is defined over K. Let hφ : ℙm (K)→ℝ≥0 be the canonical height associated to φ. Given a sequence (xn) ∈ ℙm(K), we say that it is generic if no hypersurface Z contains infinitely many xn's. Yuan [Yua08], using Arakelov theory, proves that given a generic sequence of points (xn) with hφ(xn) → 0 and a place v ∈ MK, the Galois orbits of xn will equidistribute to the equilibrium measure μυ.The aim of this thesis is to prove a quantitative version of Yuan's theorem for archimedean places. Given a smooth function f: ℙm (ℂ) → ℝ and an ε 0, we bound the degree of a hypersurface Z(f, e) and a constant δ 0 such thatholds for all x ∉ Z(f, ε) and hφ(x) δ, where Fx = Gal(K/K) · x is the Galois orbit of x. This upper bound on deg Z(f, e) tells us how generic x has to be. There are two main new ingredients in the proof which follows Yuan's approach. The first is a quantitative form of the asymptotic expansion of the Bergman kernel, first established by Tian, and the second is a construction of a "dynamical" basis of polynomials due to Looper. As an application, for ℙ2 or smooth projective surfaces in general, we are able to deduce an exponential rate of convergence of n-periodic points Pern, to the equilibrium measure. A more arithmetic application is that we are able to deduce an exponential growth of the degree [K(Pern) : K] in terms of n, generalizing results due to Baker in dimension one.[Equation omitted].
■590 ▼aSchool code: 0084.
■650 4▼aMathematics
■650 4▼aTheoretical mathematics
■653 ▼aArithmetic dynamics
■653 ▼aArakelov theory
■653 ▼aEquidistribution
■653 ▼aSmall points
■653 ▼aHigher dimensions
■690 ▼a0405
■690 ▼a0642
■71020▼aHarvard University▼bMathematics.
■7730 ▼tDissertations Abstracts International▼g86-12B.
■790 ▼a0084
■791 ▼aPh.D.
■792 ▼a2025
■793 ▼aEnglish
■85640▼uhttp://www.riss.kr/pdu/ddodLink.do?id=T17356938▼nKERIS▼z이 자료의 원문은 한국교육학술정보원에서 제공합니다.
