Limit heights and special values of the Riemann zeta function
We study the distribution of the height of the intersection between the projective line defined by the linear polynomial $x_0+x_1+x_2$ and its translate by a torsion point. We show that for a strict sequence of torsion points, the corresponding heights converge to a real number that is a rational mu...
| Autores: | , |
|---|---|
| Tipo de recurso: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 2025 |
| País: | España |
| Institución: | Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya) |
| Repositorio: | Recercat. Dipósit de la Recerca de Catalunya |
| OAI Identifier: | oai:recercat.cat:2445/227963 |
| Acceso en línea: | https://hdl.handle.net/2445/227963 |
| Access Level: | acceso abierto |
| Palabra clave: | Teoria de torsió (Àlgebra) Geometria algebraica aritmètica Funcions zeta Torsion theory (Algebra) Arithmetical algebraic geometry Zeta functions |
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Limit heights and special values of the Riemann zeta functionGualdi, RobertoSombra, MartínTeoria de torsió (Àlgebra)Geometria algebraica aritmèticaFuncions zetaTorsion theory (Algebra)Arithmetical algebraic geometryZeta functionsWe study the distribution of the height of the intersection between the projective line defined by the linear polynomial $x_0+x_1+x_2$ and its translate by a torsion point. We show that for a strict sequence of torsion points, the corresponding heights converge to a real number that is a rational multiple of a quotient of special values of the Riemann zeta function. We also determine the range of these heights, characterize the extremal cases, and study their limit for sequences of torsion points that are strict in proper algebraic subgroups. In addition, we interpret our main result from the viewpoint of Arakelov geometry, showing that for a strict sequence of torsion points the limit of the corresponding heights coincides with an Arakelov height of the cycle of the projective plane over the integers defined by the same linear polynomial. This is a particular case of a conjectural asymptotic version of the arithmetic Bézout theorem. Using the interplay between arithmetic and convex objects from the Arakelov geometry of toric varieties, we show that this Arakelov height can be expressed as the mean of a piecewise linear function on the amoeba of the projective line, which in turn can be computed as the aforementioned real number.Association for Mathematical Research2026202620252026info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersion55 p.application/pdfhttps://hdl.handle.net/2445/227963Articles publicats en revistes (Matemàtiques i Informàtica)reponame:Recercat. Dipósit de la Recerca de Catalunyainstname:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)InglésReproducció del document publicat a: https://doi.org/10.56994/JXM.001.002.008Journal of Experimental Mathematics, 2025, vol. 1, num.2, p. 322-374https://doi.org/10.56994/JXM.001.002.008cc-by-nc (c) Gualdi, R. et al., 2025http://creativecommons.org/licenses/by-nc/4.0/info:eu-repo/semantics/openAccessoai:recercat.cat:2445/2279632026-05-29T05:05:01Z |
| dc.title.none.fl_str_mv |
Limit heights and special values of the Riemann zeta function |
| title |
Limit heights and special values of the Riemann zeta function |
| spellingShingle |
Limit heights and special values of the Riemann zeta function Gualdi, Roberto Teoria de torsió (Àlgebra) Geometria algebraica aritmètica Funcions zeta Torsion theory (Algebra) Arithmetical algebraic geometry Zeta functions |
| title_short |
Limit heights and special values of the Riemann zeta function |
| title_full |
Limit heights and special values of the Riemann zeta function |
| title_fullStr |
Limit heights and special values of the Riemann zeta function |
| title_full_unstemmed |
Limit heights and special values of the Riemann zeta function |
| title_sort |
Limit heights and special values of the Riemann zeta function |
| dc.creator.none.fl_str_mv |
Gualdi, Roberto Sombra, Martín |
| author |
Gualdi, Roberto |
| author_facet |
Gualdi, Roberto Sombra, Martín |
| author_role |
author |
| author2 |
Sombra, Martín |
| author2_role |
author |
| dc.subject.none.fl_str_mv |
Teoria de torsió (Àlgebra) Geometria algebraica aritmètica Funcions zeta Torsion theory (Algebra) Arithmetical algebraic geometry Zeta functions |
| topic |
Teoria de torsió (Àlgebra) Geometria algebraica aritmètica Funcions zeta Torsion theory (Algebra) Arithmetical algebraic geometry Zeta functions |
| description |
We study the distribution of the height of the intersection between the projective line defined by the linear polynomial $x_0+x_1+x_2$ and its translate by a torsion point. We show that for a strict sequence of torsion points, the corresponding heights converge to a real number that is a rational multiple of a quotient of special values of the Riemann zeta function. We also determine the range of these heights, characterize the extremal cases, and study their limit for sequences of torsion points that are strict in proper algebraic subgroups. In addition, we interpret our main result from the viewpoint of Arakelov geometry, showing that for a strict sequence of torsion points the limit of the corresponding heights coincides with an Arakelov height of the cycle of the projective plane over the integers defined by the same linear polynomial. This is a particular case of a conjectural asymptotic version of the arithmetic Bézout theorem. Using the interplay between arithmetic and convex objects from the Arakelov geometry of toric varieties, we show that this Arakelov height can be expressed as the mean of a piecewise linear function on the amoeba of the projective line, which in turn can be computed as the aforementioned real number. |
| publishDate |
2025 |
| dc.date.none.fl_str_mv |
2025 2026 2026 2026 |
| dc.type.none.fl_str_mv |
info:eu-repo/semantics/article info:eu-repo/semantics/publishedVersion |
| format |
article |
| status_str |
publishedVersion |
| dc.identifier.none.fl_str_mv |
https://hdl.handle.net/2445/227963 |
| url |
https://hdl.handle.net/2445/227963 |
| dc.language.none.fl_str_mv |
Inglés |
| language_invalid_str_mv |
Inglés |
| dc.relation.none.fl_str_mv |
Reproducció del document publicat a: https://doi.org/10.56994/JXM.001.002.008 Journal of Experimental Mathematics, 2025, vol. 1, num.2, p. 322-374 https://doi.org/10.56994/JXM.001.002.008 |
| dc.rights.none.fl_str_mv |
cc-by-nc (c) Gualdi, R. et al., 2025 http://creativecommons.org/licenses/by-nc/4.0/ info:eu-repo/semantics/openAccess |
| rights_invalid_str_mv |
cc-by-nc (c) Gualdi, R. et al., 2025 http://creativecommons.org/licenses/by-nc/4.0/ |
| eu_rights_str_mv |
openAccess |
| dc.format.none.fl_str_mv |
55 p. application/pdf |
| dc.publisher.none.fl_str_mv |
Association for Mathematical Research |
| publisher.none.fl_str_mv |
Association for Mathematical Research |
| dc.source.none.fl_str_mv |
Articles publicats en revistes (Matemàtiques i Informàtica) reponame:Recercat. Dipósit de la Recerca de Catalunya instname:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya) |
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Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya) |
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Recercat. Dipósit de la Recerca de Catalunya |
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Recercat. Dipósit de la Recerca de Catalunya |
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