[過去ﾛｸﾞ] Inter-universal geometry と ABC予想 (応援スレ) 45 (1002ﾚｽ)
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33(1): 現代数学の系譜 雑談 ◆e.a0E5TtKE 2020/05/04(月)18:08 ID:ncpDqOGk(25/40) AAS
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In mathematics, Arakelov theory (or Arakelov geometry) is an approach to Diophantine geometry, named for Suren Arakelov. It is used to study Diophantine equations in higher dimensions.
Contents
1 Background
2 Results
Background
Arakelov geometry studies a scheme X over the ring of integers Z, by putting Hermitian metrics on holomorphic vector bundles over X(C), the complex points of X. This extra Hermitian structure is applied as a substitute for the failure of the scheme Spec(Z) to be a complete variety.
Results
Arakelov (1974, 1975) defined an intersection theory on the arithmetic surfaces attached to smooth projective curves over number fields, with the aim of proving certain results, known in the case of function fields, in the case of number fields.
Gerd Faltings (1984) extended Arakelov's work by establishing results such as a Riemann-Roch theorem, a Noether formula, a Hodge index theorem and the nonnegativity of the self-intersection of the dualizing sheaf in this context.
Arakelov theory was used by Paul Vojta (1991) to give a new proof of the Mordell conjecture, and by Gerd Faltings (1991) in his proof of Serge Lang's generalization of the Mordell conjecture.
Pierre Deligne (1987) developed a more general framework to define the intersection pairing defined on an arithmetic surface over the spectrum of a ring of integers by Arakelov.
Arakelov's theory was generalized by Henri Gillet and Christophe Soule to higher dimensions. That is, Gillet and Soule defined an intersection pairing on an arithmetic variety. One of the main results of Gillet and Soule is the arithmetic Riemann?Roch theorem of Gillet & Soule (1992), an extension of the Grothendieck?Riemann?Roch theorem to arithmetic varieties.
The arithmetic Riemann?Roch theorem then describes how the Chern class behaves under pushforward of vector bundles under a proper map of arithmetic varieties.
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