Eigenvalues of Hecke operators on Hilbert modular groups
- Autores
- Bruggeman, Roelof W.; Miatello, Roberto Jorge
- Año de publicación
- 2013
- Idioma
- inglés
- Tipo de recurso
- artículo
- Estado
- versión publicada
- Descripción
- Abstract. Let F be a totally real field, let I be a nonzero ideal of the ring of integers OF Q of F, let 0(I) be the congruence subgroup of Hecke type of G = dj =1 SL2(R) embedded diagonally in G, and let be a character of 0(I) of the form ac b d = (d), where d 7! (d) is a character of OF modulo I. For a finite subset P of prime ideals p not dividing I, we consider the ring HI , generated by the Hecke operators T(p2), p 2 P (see x3.2) acting on (; )- automorphic forms on G. Given the cuspidal space L2;cusp 0(I)nG; , we let V$ run through an orthogonal system of irreducible G-invariant subspaces so that each V$ is invariant under HI . For each 1 j d, let $ = ($; j) be the vector formed by the eigenvalues of the Casimir operators of the d factors of G on V$, and for each p 2 P, we take $;p 0 so that 2 $;p N(p) is the eigenvalue on V$ of the Hecke operator T(p2) For each family of expanding boxes t 7! t , as in (3) in Rd, and fixed an interval Jp in [0;1), for each p 2 P, we consider the counting function N( t; (Jp)p2P) := X $; $2 t : $;p2Jp ;8p2P jcr($)j2 : Here cr($) denotes the normalized Fourier coecient of order r at 1 for the elements of V$, with r 2 O0 F r pO0 F for every p 2 P. In the main result in this paper, Theorem 1.1, we give, under some mild conditions on the t , the asymptotic distribution of the function N( t; (Jp)p2P), as t ! 1. We show that at the finite places outside I the Hecke eigenvalues are equidistributed with respect to the Sato-Tate measure, whereas at the archimedean places the eigenvalues $ are equidistributed with respect to the Plancherel measure. As a consequence, if we fix an infinite place l and we prescribe $; j 2 j for all infinite places j , l and$;p 2 Jp for all finite places p in P (for fixed intervals j and Jp) and then allow j$;lj to grow to 1, then there are infinitely many such $, and their positive density is as described in Theorem 1.1.
Fil: Bruggeman, Roelof W.. Utrecht University; Países Bajos
Fil: Miatello, Roberto Jorge. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Córdoba. Centro de Investigación y Estudios de Matemática de Córdoba(p); Argentina - Materia
-
automorphic representations
Hecke operators
Hilbert modular group
Plancherel measure - Nivel de accesibilidad
- acceso abierto
- Condiciones de uso
- https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
- Repositorio
- Institución
- Consejo Nacional de Investigaciones Científicas y Técnicas
- OAI Identificador
- oai:ri.conicet.gov.ar:11336/11335
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Eigenvalues of Hecke operators on Hilbert modular groupsBruggeman, Roelof W.Miatello, Roberto Jorgeautomorphic representationsHecke operatorsHilbert modular groupPlancherel measurehttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1Abstract. Let F be a totally real field, let I be a nonzero ideal of the ring of integers OF Q of F, let 0(I) be the congruence subgroup of Hecke type of G = dj =1 SL2(R) embedded diagonally in G, and let be a character of 0(I) of the form ac b d = (d), where d 7! (d) is a character of OF modulo I. For a finite subset P of prime ideals p not dividing I, we consider the ring HI , generated by the Hecke operators T(p2), p 2 P (see x3.2) acting on (; )- automorphic forms on G. Given the cuspidal space L2;cusp 0(I)nG; , we let V$ run through an orthogonal system of irreducible G-invariant subspaces so that each V$ is invariant under HI . For each 1 j d, let $ = ($; j) be the vector formed by the eigenvalues of the Casimir operators of the d factors of G on V$, and for each p 2 P, we take $;p 0 so that 2 $;p N(p) is the eigenvalue on V$ of the Hecke operator T(p2) For each family of expanding boxes t 7! t , as in (3) in Rd, and fixed an interval Jp in [0;1), for each p 2 P, we consider the counting function N( t; (Jp)p2P) := X $; $2 t : $;p2Jp ;8p2P jcr($)j2 : Here cr($) denotes the normalized Fourier coecient of order r at 1 for the elements of V$, with r 2 O0 F r pO0 F for every p 2 P. In the main result in this paper, Theorem 1.1, we give, under some mild conditions on the t , the asymptotic distribution of the function N( t; (Jp)p2P), as t ! 1. We show that at the finite places outside I the Hecke eigenvalues are equidistributed with respect to the Sato-Tate measure, whereas at the archimedean places the eigenvalues $ are equidistributed with respect to the Plancherel measure. As a consequence, if we fix an infinite place l and we prescribe $; j 2 j for all infinite places j , l and$;p 2 Jp for all finite places p in P (for fixed intervals j and Jp) and then allow j$;lj to grow to 1, then there are infinitely many such $, and their positive density is as described in Theorem 1.1.Fil: Bruggeman, Roelof W.. Utrecht University; Países BajosFil: Miatello, Roberto Jorge. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Córdoba. Centro de Investigación y Estudios de Matemática de Córdoba(p); ArgentinaInternational Press Boston2013-12info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/11335Bruggeman, Roelof W.; Miatello, Roberto Jorge; Eigenvalues of Hecke operators on Hilbert modular groups; International Press Boston; Asian Journal of Mathematics; 17; 4; 12-2013; 729-7571093-6106enginfo:eu-repo/semantics/altIdentifier/url/http://intlpress.com/site/pub/pages/journals/items/ajm/content/vols/0017/0004/a010/index.htmlinfo:eu-repo/semantics/altIdentifier/doi/10.4310/AJM.2013.v17.n4.a10info:eu-repo/semantics/altIdentifier/url/https://arxiv.org/abs/0912.1692info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-03T10:00:41Zoai:ri.conicet.gov.ar:11336/11335instacron:CONICETInstitucionalhttp://ri.conicet.gov.ar/Organismo científico-tecnológicoNo correspondehttp://ri.conicet.gov.ar/oai/requestdasensio@conicet.gov.ar; lcarlino@conicet.gov.arArgentinaNo correspondeNo correspondeNo correspondeopendoar:34982025-09-03 10:00:42.258CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse |
dc.title.none.fl_str_mv |
Eigenvalues of Hecke operators on Hilbert modular groups |
title |
Eigenvalues of Hecke operators on Hilbert modular groups |
spellingShingle |
Eigenvalues of Hecke operators on Hilbert modular groups Bruggeman, Roelof W. automorphic representations Hecke operators Hilbert modular group Plancherel measure |
title_short |
Eigenvalues of Hecke operators on Hilbert modular groups |
title_full |
Eigenvalues of Hecke operators on Hilbert modular groups |
title_fullStr |
Eigenvalues of Hecke operators on Hilbert modular groups |
title_full_unstemmed |
Eigenvalues of Hecke operators on Hilbert modular groups |
title_sort |
Eigenvalues of Hecke operators on Hilbert modular groups |
dc.creator.none.fl_str_mv |
Bruggeman, Roelof W. Miatello, Roberto Jorge |
author |
Bruggeman, Roelof W. |
author_facet |
Bruggeman, Roelof W. Miatello, Roberto Jorge |
author_role |
author |
author2 |
Miatello, Roberto Jorge |
author2_role |
author |
dc.subject.none.fl_str_mv |
automorphic representations Hecke operators Hilbert modular group Plancherel measure |
topic |
automorphic representations Hecke operators Hilbert modular group Plancherel measure |
purl_subject.fl_str_mv |
https://purl.org/becyt/ford/1.1 https://purl.org/becyt/ford/1 |
dc.description.none.fl_txt_mv |
Abstract. Let F be a totally real field, let I be a nonzero ideal of the ring of integers OF Q of F, let 0(I) be the congruence subgroup of Hecke type of G = dj =1 SL2(R) embedded diagonally in G, and let be a character of 0(I) of the form ac b d = (d), where d 7! (d) is a character of OF modulo I. For a finite subset P of prime ideals p not dividing I, we consider the ring HI , generated by the Hecke operators T(p2), p 2 P (see x3.2) acting on (; )- automorphic forms on G. Given the cuspidal space L2;cusp 0(I)nG; , we let V$ run through an orthogonal system of irreducible G-invariant subspaces so that each V$ is invariant under HI . For each 1 j d, let $ = ($; j) be the vector formed by the eigenvalues of the Casimir operators of the d factors of G on V$, and for each p 2 P, we take $;p 0 so that 2 $;p N(p) is the eigenvalue on V$ of the Hecke operator T(p2) For each family of expanding boxes t 7! t , as in (3) in Rd, and fixed an interval Jp in [0;1), for each p 2 P, we consider the counting function N( t; (Jp)p2P) := X $; $2 t : $;p2Jp ;8p2P jcr($)j2 : Here cr($) denotes the normalized Fourier coecient of order r at 1 for the elements of V$, with r 2 O0 F r pO0 F for every p 2 P. In the main result in this paper, Theorem 1.1, we give, under some mild conditions on the t , the asymptotic distribution of the function N( t; (Jp)p2P), as t ! 1. We show that at the finite places outside I the Hecke eigenvalues are equidistributed with respect to the Sato-Tate measure, whereas at the archimedean places the eigenvalues $ are equidistributed with respect to the Plancherel measure. As a consequence, if we fix an infinite place l and we prescribe $; j 2 j for all infinite places j , l and$;p 2 Jp for all finite places p in P (for fixed intervals j and Jp) and then allow j$;lj to grow to 1, then there are infinitely many such $, and their positive density is as described in Theorem 1.1. Fil: Bruggeman, Roelof W.. Utrecht University; Países Bajos Fil: Miatello, Roberto Jorge. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Córdoba. Centro de Investigación y Estudios de Matemática de Córdoba(p); Argentina |
description |
Abstract. Let F be a totally real field, let I be a nonzero ideal of the ring of integers OF Q of F, let 0(I) be the congruence subgroup of Hecke type of G = dj =1 SL2(R) embedded diagonally in G, and let be a character of 0(I) of the form ac b d = (d), where d 7! (d) is a character of OF modulo I. For a finite subset P of prime ideals p not dividing I, we consider the ring HI , generated by the Hecke operators T(p2), p 2 P (see x3.2) acting on (; )- automorphic forms on G. Given the cuspidal space L2;cusp 0(I)nG; , we let V$ run through an orthogonal system of irreducible G-invariant subspaces so that each V$ is invariant under HI . For each 1 j d, let $ = ($; j) be the vector formed by the eigenvalues of the Casimir operators of the d factors of G on V$, and for each p 2 P, we take $;p 0 so that 2 $;p N(p) is the eigenvalue on V$ of the Hecke operator T(p2) For each family of expanding boxes t 7! t , as in (3) in Rd, and fixed an interval Jp in [0;1), for each p 2 P, we consider the counting function N( t; (Jp)p2P) := X $; $2 t : $;p2Jp ;8p2P jcr($)j2 : Here cr($) denotes the normalized Fourier coecient of order r at 1 for the elements of V$, with r 2 O0 F r pO0 F for every p 2 P. In the main result in this paper, Theorem 1.1, we give, under some mild conditions on the t , the asymptotic distribution of the function N( t; (Jp)p2P), as t ! 1. We show that at the finite places outside I the Hecke eigenvalues are equidistributed with respect to the Sato-Tate measure, whereas at the archimedean places the eigenvalues $ are equidistributed with respect to the Plancherel measure. As a consequence, if we fix an infinite place l and we prescribe $; j 2 j for all infinite places j , l and$;p 2 Jp for all finite places p in P (for fixed intervals j and Jp) and then allow j$;lj to grow to 1, then there are infinitely many such $, and their positive density is as described in Theorem 1.1. |
publishDate |
2013 |
dc.date.none.fl_str_mv |
2013-12 |
dc.type.none.fl_str_mv |
info:eu-repo/semantics/article info:eu-repo/semantics/publishedVersion http://purl.org/coar/resource_type/c_6501 info:ar-repo/semantics/articulo |
format |
article |
status_str |
publishedVersion |
dc.identifier.none.fl_str_mv |
http://hdl.handle.net/11336/11335 Bruggeman, Roelof W.; Miatello, Roberto Jorge; Eigenvalues of Hecke operators on Hilbert modular groups; International Press Boston; Asian Journal of Mathematics; 17; 4; 12-2013; 729-757 1093-6106 |
url |
http://hdl.handle.net/11336/11335 |
identifier_str_mv |
Bruggeman, Roelof W.; Miatello, Roberto Jorge; Eigenvalues of Hecke operators on Hilbert modular groups; International Press Boston; Asian Journal of Mathematics; 17; 4; 12-2013; 729-757 1093-6106 |
dc.language.none.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
info:eu-repo/semantics/altIdentifier/url/http://intlpress.com/site/pub/pages/journals/items/ajm/content/vols/0017/0004/a010/index.html info:eu-repo/semantics/altIdentifier/doi/10.4310/AJM.2013.v17.n4.a10 info:eu-repo/semantics/altIdentifier/url/https://arxiv.org/abs/0912.1692 |
dc.rights.none.fl_str_mv |
info:eu-repo/semantics/openAccess https://creativecommons.org/licenses/by-nc-sa/2.5/ar/ |
eu_rights_str_mv |
openAccess |
rights_invalid_str_mv |
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/ |
dc.format.none.fl_str_mv |
application/pdf application/pdf |
dc.publisher.none.fl_str_mv |
International Press Boston |
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International Press Boston |
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CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicas |
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dasensio@conicet.gov.ar; lcarlino@conicet.gov.ar |
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