Numerical equivalence of R-divisors and Shioda-Tate formula for arithmetic varieties
Let X be an arithmetic variety over the ring of integers of a number field K, with smooth generic fiber X K . We give a formula that relates the dimension of the first Arakelov–Chow vector space of X with the Mordell–Weil rank of the Albanese variety of X K and the rank of the Néron–Severi group of...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2022 |
| País: | España |
| Institución: | Universitat Politècnica de Catalunya (UPC) |
| Repositorio: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglés |
| OAI Identifier: | oai:upcommons.upc.edu:2117/406861 |
| Acceso en línea: | https://hdl.handle.net/2117/406861 https://dx.doi.org/10.1515/crelle-2021-0081 |
| Access Level: | acceso abierto |
| Palabra clave: | Arithmetical algebraic geometry Geometry, Algebraic Diophantine geometry Geometria algèbrica--Aritmètica Geometria algèbrica Aritmètica Classificació AMS::11 Number theory::11G Arithmetic algebraic geometry (Diophantine geometry) Classificació AMS::14 Algebraic geometry::14C Cycles and subschemes Classificació AMS::14 Algebraic geometry::14G Arithmetic problems. Diophantine geometry Àrees temàtiques de la UPC::Matemàtiques i estadística::Àlgebra::Aritmètica Àrees temàtiques de la UPC::Matemàtiques i estadística::Geometria::Geometria algebraica |
| Sumario: | Let X be an arithmetic variety over the ring of integers of a number field K, with smooth generic fiber X K . We give a formula that relates the dimension of the first Arakelov–Chow vector space of X with the Mordell–Weil rank of the Albanese variety of X K and the rank of the Néron–Severi group of X K . This is a higher-dimensional and arithmetic version of the classical Shioda–Tate formula for elliptic surfaces. Such an analogy is strengthened by the fact that we show that the numerically trivial arithmetic R -divisors on X are exactly the linear combinations of principal ones. This result is equivalent to the non-degeneracy of the arithmetic intersection pairing in the argument of divisors, partially confirming a conjecture by H. Gillet and C. Soulé. |
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