{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T23:38:52Z","timestamp":1776728332979,"version":"3.51.2"},"reference-count":10,"publisher":"American Mathematical Society (AMS)","issue":"239","license":[{"start":{"date-parts":[[2002,12,21]],"date-time":"2002-12-21T00:00:00Z","timestamp":1040428800000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    Some useful information is known about the fundamental domain for certain Hilbert modular groups. The six nonequivalent points with nontrivial isotropy in the fundamental domains under the action of the modular group for\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"bold upper Q left-parenthesis StartRoot 5 EndRoot right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"bold\">Q<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:msqrt>\n                              <mml:mn>5<\/mml:mn>\n                            <\/mml:msqrt>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathbf {Q} ( \\sqrt 5 )<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ,\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"bold upper Q left-parenthesis StartRoot 2 EndRoot right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"bold\">Q<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:msqrt>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msqrt>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathbf {Q}( \\sqrt 2 )<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"bold upper Q left-parenthesis StartRoot 3 EndRoot right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"bold\">Q<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:msqrt>\n                              <mml:mn>3<\/mml:mn>\n                            <\/mml:msqrt>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathbf {Q} ( \\sqrt 3 )<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    have been determined previously by Gundlach. In finding these points, use was made of the exact size of the isotropy groups. Here we show that the fixed points and the isotropy groups can be found without such knowledge by use of a computer scan. We consider the cases\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"bold upper Q left-parenthesis StartRoot 5 EndRoot right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"bold\">Q<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:msqrt>\n                              <mml:mn>5<\/mml:mn>\n                            <\/mml:msqrt>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathbf {Q} ( \\sqrt 5 )<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"bold upper Q left-parenthesis StartRoot 2 EndRoot right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"bold\">Q<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:msqrt>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msqrt>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathbf {Q} ( \\sqrt 2 )<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . A computer algebra system and a C compiler were essential in perfoming the computations.\n                  <\/p>","DOI":"10.1090\/s0025-5718-01-01403-x","type":"journal-article","created":{"date-parts":[[2002,9,20]],"date-time":"2002-09-20T15:46:54Z","timestamp":1032536814000},"page":"1271-1280","source":"Crossref","is-referenced-by-count":4,"title":["A computational approach to Hilbert modular group fixed points"],"prefix":"10.1090","volume":"71","author":[{"given":"Jesse","family":"Deutsch","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2001,12,21]]},"reference":[{"key":"1","series-title":"Universitext","isbn-type":"print","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4612-9950-9","volume-title":"A classical invitation to algebraic numbers and class fields","author":"Cohn, Harvey","year":"1978","ISBN":"https:\/\/id.crossref.org\/isbn\/0387903453"},{"key":"2","unstructured":"[2] J. I. Deutsch, Identities on Modular Forms in Several Variables Derivable from Hecke Transformations, Dissertation, Brown University, Providence, R.I., USA, 1986."},{"key":"3","doi-asserted-by":"crossref","unstructured":"[3] F. G\u00f6tzky, \u00dcber eine zahlentheoretische Anwendung von Modulfunktionen zweier Ver\u00e4nderlicher, Math. Ann. 100 (1928), 411\u2013437.","DOI":"10.1007\/BF01448854"},{"key":"4","doi-asserted-by":"publisher","first-page":"109","DOI":"10.1515\/crll.1965.220.109","article-title":"Die Bestimmung der Funktionen zu einigen Hilbertschen Modulgruppen","volume":"220","author":"Gundlach, Karl-Bernhard","year":"1965","journal-title":"J. Reine Angew. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0075-4102","issn-type":"print"},{"key":"5","doi-asserted-by":"publisher","first-page":"369","DOI":"10.1007\/BF02028248","article-title":"Die Fixpunkte einiger Hilbertscher Modulgruppen","volume":"157","author":"Gundlach, Karl-Bernhard","year":"1965","journal-title":"Math. Ann.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5831","issn-type":"print"},{"key":"6","unstructured":"[6] B. Haible, Private communication, 1997."},{"key":"7","series-title":"S\\'{e}rie des Conf\\'{e}rences de l'Union Math\\'{e}matique Internationale [Lecture Series of the International Mathematics Union], No. 4","volume-title":"Hilbert modular surfaces","author":"Hirzebruch, Friedrich E. P.","year":"1973"},{"key":"8","series-title":"Graduate Texts in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4684-0255-1","volume-title":"Introduction to elliptic curves and modular forms","volume":"97","author":"Koblitz, Neal","year":"1984","ISBN":"https:\/\/id.crossref.org\/isbn\/0387960295"},{"key":"9","series-title":"Graduate Texts in Mathematics, No. 7","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4684-9884-4","volume-title":"A course in arithmetic","author":"Serre, J.-P.","year":"1973"},{"key":"10","series-title":"Tata Institute of Fundamental Research Studies in Mathematics","volume-title":"Advanced analytic number theory","volume":"9","author":"Siegel, Carl Ludwig","year":"1980","edition":"2"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2002-71-239\/S0025-5718-01-01403-X\/S0025-5718-01-01403-X.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2002-71-239\/S0025-5718-01-01403-X\/S0025-5718-01-01403-X.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T23:00:48Z","timestamp":1776726048000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2002-71-239\/S0025-5718-01-01403-X\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2001,12,21]]},"references-count":10,"journal-issue":{"issue":"239","published-print":{"date-parts":[[2002,7]]}},"alternative-id":["S0025-5718-01-01403-X"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-01-01403-x","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2001,12,21]]}}}