{"id":{"repo_id":"aachen","oai_identifier":"oai:publications.rwth-aachen.de:62415"},"canonical_url":"https://search.dev.ndltd.org/etd/aachen/oai:publications.rwth-aachen.de:62415","repository":{"repo_id":"aachen","name":"RWTH Aachen University","base_url":"https://publications.rwth-aachen.de/oai2d"},"display":{"title":"Hilbert modular forms for the fields Q( 5), Q( 13) and Q( 17)","abstract":"More than a 100 years ago David Hilbert gave his drafts of a new type of functions of several complex variables to his student Ludwig Otto Blumenthal, who made them the foundation of his Habilschrift \"Über Modulfunktionen von mehreren Veränderlichen\". Hilbert had investigated both the theory of elliptic modular functions and the theory of algebraic number fields and came up with the idea to link these areas. Elliptic modular functions are meromorphic maps from the upper half plane H into the Riemannsphere, which are invariant under the Moebiustransformations in SL(2, Z) and can be suitably extended to infinity. Hilbert replaced Z by the ring of integers of a totally real number field and obtained meromorphic modular functions on H^n, which form, similarly to modular functions, a finitely generated function field. Next to modular functions Hilbert considered also modular forms, which are holomorphic functions no longer invariant under SL(2,Z) but satisfying a certain functional equation. By taking the quotient of suitable modular forms one receives modular functions again. Hilbert had found in Blumenthal, a later professor in Aachen, the right one to work out his ideas. It took some time before further results were obtained, since on the one hand algebraic geometry and the theory of complex functions had to evolve further and on the other hand politics was directing almost all scientific efforts towards military purposes and both world wars aggravated the situation. Since that time however some progress has been made. The concrete computation of the associated rings of all Hilbert' modular forms of an algebraic number field succeeded for Q(sqrt 5), for other number fields subrings have been calculated. We extend the number of the known rings using Eisenstein series and Borcherds products, as they were formulated in Bruinier, Bundschuh, \"On Borcherds products associated with lattices of prime discriminant\", Ramanujan J., 7, 49-61 (2003) for the case of Hilbert modular forms. We compute the rings of modular forms for Q(sqrt 13) and Q(sqrt 17) for Hilbert's modular group with trivial character. We consider the well-known case Q(sqrt 5) as a benchmark for the procedure and to express the generators in Borcherds products. Those are lifts of suitable elliptic modular forms and can be rewritten as a locally uniformly convergent product of elementary factors. Their zeros including multiplicities can be computed explicitely. In addition the behavior of Hilbert modular forms under various transformations can be predicted in a way such that in many cases we can deduce obligatory zeros. There exists a map descending from the ring of Hilbert modular forms to the ring of elliptic modular forms induced by the embedding of H as diagonal in H^n. In the cases regarded in this work, Q(sqrt 5), Q(sqrt 13) and Q(sqrt 17), it suffices to find a sufficient number of generators of the ring of Hilbert modular forms. The proof that a sufficient number of generators has been determined works in each case according to the following pattern: The descent function maps a Hilbert modular form f of positive weight on an elliptic modular form. From the given generators we construct a Hilbert modular form with same weight and same image in the ring of elliptic modular forms and subtract it from f. The resulting modular form vanishes on the diagonal in H^n, so it is a multiple of the most simple of the Borcherds products which vanishes of first order on the diagonal and all equivalent points modulo Hilbert's modular group and is zero-free everywhere else. The quotient of the resulting modular form over the Borcherds product has lower weight than f and the statement follows by induction, since all modular forms of weight 0 are constant and there are no modular forms of negative weight. The work is partitioned into into several parts. We begin with the comparison of the different definitions of Hilbert modular forms. Then we present some special Hilbert modular forms, like Eisenstein series and Poincare series, and examine different elliptic modular forms wich will be needed later on. Subsequently, we introduce Borcherds products in our case and develope general characteristics of Hilbert modular forms. Finally the reduction method supplies the desired results, the generators of the rings of Hilbert modular forms for symmetric multiplier systems as well as for trivial character. An overview over related questions follows.","abstract_html":"More than a 100 years ago David Hilbert gave his drafts of a new type of functions of several complex variables to his student Ludwig Otto Blumenthal, who made them the foundation of his Habilschrift "Über Modulfunktionen von mehreren Veränderlichen". Hilbert had investigated both the theory of elliptic modular functions and the theory of algebraic number fields and came up with the idea to link these areas. Elliptic modular functions are meromorphic maps from the upper half plane H into the Riemannsphere, which are invariant under the Moebiustransformations in SL(2, Z) and can be suitably extended to infinity. Hilbert replaced Z by the ring of integers of a totally real number field and obtained meromorphic modular functions on H^n, which form, similarly to modular functions, a finitely generated function field. Next to modular functions Hilbert considered also modular forms, which are holomorphic functions no longer invariant under SL(2,Z) but satisfying a certain functional equation. By taking the quotient of suitable modular forms one receives modular functions again. Hilbert had found in Blumenthal, a later professor in Aachen, the right one to work out his ideas. It took some time before further results were obtained, since on the one hand algebraic geometry and the theory of complex functions had to evolve further and on the other hand politics was directing almost all scientific efforts towards military purposes and both world wars aggravated the situation. Since that time however some progress has been made. The concrete computation of the associated rings of all Hilbert' modular forms of an algebraic number field succeeded for Q(sqrt 5), for other number fields subrings have been calculated. We extend the number of the known rings using Eisenstein series and Borcherds products, as they were formulated in Bruinier, Bundschuh, "On Borcherds products associated with lattices of prime discriminant", Ramanujan J., 7, 49-61 (2003) for the case of Hilbert modular forms. We compute the rings of modular forms for Q(sqrt 13) and Q(sqrt 17) for Hilbert's modular group with trivial character. We consider the well-known case Q(sqrt 5) as a benchmark for the procedure and to express the generators in Borcherds products. Those are lifts of suitable elliptic modular forms and can be rewritten as a locally uniformly convergent product of elementary factors. Their zeros including multiplicities can be computed explicitely. In addition the behavior of Hilbert modular forms under various transformations can be predicted in a way such that in many cases we can deduce obligatory zeros. There exists a map descending from the ring of Hilbert modular forms to the ring of elliptic modular forms induced by the embedding of H as diagonal in H^n. In the cases regarded in this work, Q(sqrt 5), Q(sqrt 13) and Q(sqrt 17), it suffices to find a sufficient number of generators of the ring of Hilbert modular forms. The proof that a sufficient number of generators has been determined works in each case according to the following pattern: The descent function maps a Hilbert modular form f of positive weight on an elliptic modular form. From the given generators we construct a Hilbert modular form with same weight and same image in the ring of elliptic modular forms and subtract it from f. The resulting modular form vanishes on the diagonal in H^n, so it is a multiple of the most simple of the Borcherds products which vanishes of first order on the diagonal and all equivalent points modulo Hilbert's modular group and is zero-free everywhere else. The quotient of the resulting modular form over the Borcherds product has lower weight than f and the statement follows by induction, since all modular forms of weight 0 are constant and there are no modular forms of negative weight. The work is partitioned into into several parts. We begin with the comparison of the different definitions of Hilbert modular forms. Then we present some special Hilbert modular forms, like Eisenstein series and Poincare series, and examine different elliptic modular forms wich will be needed later on. Subsequently, we introduce Borcherds products in our case and develope general characteristics of Hilbert modular forms. Finally the reduction method supplies the desired results, the generators of the rings of Hilbert modular forms for symmetric multiplier systems as well as for trivial character. An overview over related questions follows.","abstract_has_math":false,"creators":["Mayer, Sebastian"],"institution":"Publikationsserver der RWTH Aachen University","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":["Krieg, Aloys"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2007,"date_issued":"2007","date_published":"2007","updated_at":"2026-07-30T19:43:28Z","subjects":["info:eu-repo/classification/ddc/510","Hilbertsche Modulform","Modulform","Hilbertsche Modulfläche","Zahlentheorie","Borcherds-Produkt","Mathematik","Hilbert modular surface","Hilbert modular form","modular form","Borcherds product","number theory"],"languages":["eng"],"rights":["info:eu-repo/semantics/openAccess"],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["https://publications.rwth-aachen.de/search?p=id:%22RWTH-CONV-123986%22"],"render_values":[{"text":"https://publications.rwth-aachen.de/search?p=id:%22RWTH-CONV-123986%22","href":"https://publications.rwth-aachen.de/search?p=id:%22RWTH-CONV-123986%22","code":true}]}]},"links":{"outbound_url":"https://publications.rwth-aachen.de/record/62415","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Krieg, Aloys"]},{"key":"dc:creator","label":"Author","values":["Mayer, Sebastian"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:coverage","label":"Dc Coverage","values":["DE"]},{"key":"dc:date","label":"Dc Date","values":["2007"]},{"key":"dc:publisher","label":"Institution","values":["Publikationsserver der RWTH Aachen University"]},{"key":"dc:relation","label":"Dc Relation","values":["info:eu-repo/semantics/altIdentifier/urn/urn:nbn:de:hbz:82-opus-19854"]},{"key":"dc:type","label":"Dc Type","values":["info:eu-repo/semantics/doctoralThesis","info:eu-repo/semantics/publishedVersion"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["info:eu-repo/classification/ddc/510","Hilbertsche Modulform","Modulform","Hilbertsche Modulfläche","Zahlentheorie","Borcherds-Produkt","Mathematik","Hilbert modular surface","Hilbert modular form","modular form","Borcherds product","number theory"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["info:eu-repo/semantics/openAccess"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://publications.rwth-aachen.de/record/62415","https://publications.rwth-aachen.de/search?p=id:%22RWTH-CONV-123986%22"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["More than a 100 years ago David Hilbert gave his drafts of a new type of functions of several complex variables to his student Ludwig Otto Blumenthal, who made them the foundation of his Habilschrift \"Über Modulfunktionen von mehreren Veränderlichen\". Hilbert had investigated both the theory of elliptic modular functions and the theory of algebraic number fields and came up with the idea to link these areas. Elliptic modular functions are meromorphic maps from the upper half plane H into the Riemannsphere, which are invariant under the Moebiustransformations in SL(2, Z) and can be suitably extended to infinity. Hilbert replaced Z by the ring of integers of a totally real number field and obtained meromorphic modular functions on H^n, which form, similarly to modular functions, a finitely generated function field. Next to modular functions Hilbert considered also modular forms, which are holomorphic functions no longer invariant under SL(2,Z) but satisfying a certain functional equation. By taking the quotient of suitable modular forms one receives modular functions again. Hilbert had found in Blumenthal, a later professor in Aachen, the right one to work out his ideas. It took some time before further results were obtained, since on the one hand algebraic geometry and the theory of complex functions had to evolve further and on the other hand politics was directing almost all scientific efforts towards military purposes and both world wars aggravated the situation. Since that time however some progress has been made. The concrete computation of the associated rings of all Hilbert' modular forms of an algebraic number field succeeded for Q(sqrt 5), for other number fields subrings have been calculated. We extend the number of the known rings using Eisenstein series and Borcherds products, as they were formulated in Bruinier, Bundschuh, \"On Borcherds products associated with lattices of prime discriminant\", Ramanujan J., 7, 49-61 (2003) for the case of Hilbert modular forms. We compute the rings of modular forms for Q(sqrt 13) and Q(sqrt 17) for Hilbert's modular group with trivial character. We consider the well-known case Q(sqrt 5) as a benchmark for the procedure and to express the generators in Borcherds products. Those are lifts of suitable elliptic modular forms and can be rewritten as a locally uniformly convergent product of elementary factors. Their zeros including multiplicities can be computed explicitely. In addition the behavior of Hilbert modular forms under various transformations can be predicted in a way such that in many cases we can deduce obligatory zeros. There exists a map descending from the ring of Hilbert modular forms to the ring of elliptic modular forms induced by the embedding of H as diagonal in H^n. In the cases regarded in this work, Q(sqrt 5), Q(sqrt 13) and Q(sqrt 17), it suffices to find a sufficient number of generators of the ring of Hilbert modular forms. The proof that a sufficient number of generators has been determined works in each case according to the following pattern: The descent function maps a Hilbert modular form f of positive weight on an elliptic modular form. From the given generators we construct a Hilbert modular form with same weight and same image in the ring of elliptic modular forms and subtract it from f. The resulting modular form vanishes on the diagonal in H^n, so it is a multiple of the most simple of the Borcherds products which vanishes of first order on the diagonal and all equivalent points modulo Hilbert's modular group and is zero-free everywhere else. The quotient of the resulting modular form over the Borcherds product has lower weight than f and the statement follows by induction, since all modular forms of weight 0 are constant and there are no modular forms of negative weight. The work is partitioned into into several parts. We begin with the comparison of the different definitions of Hilbert modular forms. Then we present some special Hilbert modular forms, like Eisenstein series and Poincare series, and examine different elliptic modular forms wich will be needed later on. Subsequently, we introduce Borcherds products in our case and develope general characteristics of Hilbert modular forms. Finally the reduction method supplies the desired results, the generators of the rings of Hilbert modular forms for symmetric multiplier systems as well as for trivial character. An overview over related questions follows."]},{"key":"dc:source","label":"Dc Source","values":["Aachen : Publikationsserver der RWTH Aachen University 175 S. (2007). = Aachen, Techn. Hochsch., Diss., 2007"]},{"key":"dc:title","label":"Title","values":["Hilbert modular forms for the fields Q( 5), Q( 13) and Q( 17)"]}]}],"canonical_facts":{"dc:contributor":["Krieg, Aloys"],"dc:coverage":["DE"],"dc:creator":["Mayer, Sebastian"],"dc:date":["2007"],"dc:description":["More than a 100 years ago David Hilbert gave his drafts of a new type of functions of several complex variables to his student Ludwig Otto Blumenthal, who made them the foundation of his Habilschrift \"Über Modulfunktionen von mehreren Veränderlichen\". Hilbert had investigated both the theory of elliptic modular functions and the theory of algebraic number fields and came up with the idea to link these areas. Elliptic modular functions are meromorphic maps from the upper half plane H into the Riemannsphere, which are invariant under the Moebiustransformations in SL(2, Z) and can be suitably extended to infinity. Hilbert replaced Z by the ring of integers of a totally real number field and obtained meromorphic modular functions on H^n, which form, similarly to modular functions, a finitely generated function field. Next to modular functions Hilbert considered also modular forms, which are holomorphic functions no longer invariant under SL(2,Z) but satisfying a certain functional equation. By taking the quotient of suitable modular forms one receives modular functions again. Hilbert had found in Blumenthal, a later professor in Aachen, the right one to work out his ideas. It took some time before further results were obtained, since on the one hand algebraic geometry and the theory of complex functions had to evolve further and on the other hand politics was directing almost all scientific efforts towards military purposes and both world wars aggravated the situation. Since that time however some progress has been made. The concrete computation of the associated rings of all Hilbert' modular forms of an algebraic number field succeeded for Q(sqrt 5), for other number fields subrings have been calculated. We extend the number of the known rings using Eisenstein series and Borcherds products, as they were formulated in Bruinier, Bundschuh, \"On Borcherds products associated with lattices of prime discriminant\", Ramanujan J., 7, 49-61 (2003) for the case of Hilbert modular forms. We compute the rings of modular forms for Q(sqrt 13) and Q(sqrt 17) for Hilbert's modular group with trivial character. We consider the well-known case Q(sqrt 5) as a benchmark for the procedure and to express the generators in Borcherds products. Those are lifts of suitable elliptic modular forms and can be rewritten as a locally uniformly convergent product of elementary factors. Their zeros including multiplicities can be computed explicitely. In addition the behavior of Hilbert modular forms under various transformations can be predicted in a way such that in many cases we can deduce obligatory zeros. There exists a map descending from the ring of Hilbert modular forms to the ring of elliptic modular forms induced by the embedding of H as diagonal in H^n. In the cases regarded in this work, Q(sqrt 5), Q(sqrt 13) and Q(sqrt 17), it suffices to find a sufficient number of generators of the ring of Hilbert modular forms. The proof that a sufficient number of generators has been determined works in each case according to the following pattern: The descent function maps a Hilbert modular form f of positive weight on an elliptic modular form. From the given generators we construct a Hilbert modular form with same weight and same image in the ring of elliptic modular forms and subtract it from f. The resulting modular form vanishes on the diagonal in H^n, so it is a multiple of the most simple of the Borcherds products which vanishes of first order on the diagonal and all equivalent points modulo Hilbert's modular group and is zero-free everywhere else. The quotient of the resulting modular form over the Borcherds product has lower weight than f and the statement follows by induction, since all modular forms of weight 0 are constant and there are no modular forms of negative weight. The work is partitioned into into several parts. We begin with the comparison of the different definitions of Hilbert modular forms. Then we present some special Hilbert modular forms, like Eisenstein series and Poincare series, and examine different elliptic modular forms wich will be needed later on. Subsequently, we introduce Borcherds products in our case and develope general characteristics of Hilbert modular forms. Finally the reduction method supplies the desired results, the generators of the rings of Hilbert modular forms for symmetric multiplier systems as well as for trivial character. An overview over related questions follows."],"dc:identifier":["https://publications.rwth-aachen.de/record/62415","https://publications.rwth-aachen.de/search?p=id:%22RWTH-CONV-123986%22"],"dc:language":["eng"],"dc:publisher":["Publikationsserver der RWTH Aachen University"],"dc:relation":["info:eu-repo/semantics/altIdentifier/urn/urn:nbn:de:hbz:82-opus-19854"],"dc:rights":["info:eu-repo/semantics/openAccess"],"dc:source":["Aachen : Publikationsserver der RWTH Aachen University 175 S. (2007). = Aachen, Techn. Hochsch., Diss., 2007"],"dc:subject":["info:eu-repo/classification/ddc/510","Hilbertsche Modulform","Modulform","Hilbertsche Modulfläche","Zahlentheorie","Borcherds-Produkt","Mathematik","Hilbert modular surface","Hilbert modular form","modular form","Borcherds product","number theory"],"dc:title":["Hilbert modular forms for the fields Q( 5), Q( 13) and Q( 17)"],"dc:type":["info:eu-repo/semantics/doctoralThesis","info:eu-repo/semantics/publishedVersion"]},"updated_at":"2026-07-30T19:43:28Z"}