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
CONICET Digital (CONICET)
Institución
Consejo Nacional de Investigaciones Científicas y Técnicas
OAI Identificador
oai:ri.conicet.gov.ar:11336/11335

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network_name_str CONICET Digital (CONICET)
spelling 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
publisher.none.fl_str_mv International Press Boston
dc.source.none.fl_str_mv reponame:CONICET Digital (CONICET)
instname:Consejo Nacional de Investigaciones Científicas y Técnicas
reponame_str CONICET Digital (CONICET)
collection CONICET Digital (CONICET)
instname_str Consejo Nacional de Investigaciones Científicas y Técnicas
repository.name.fl_str_mv CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicas
repository.mail.fl_str_mv dasensio@conicet.gov.ar; lcarlino@conicet.gov.ar
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