{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T15:36:53Z","timestamp":1776785813728,"version":"3.51.2"},"reference-count":13,"publisher":"American Mathematical Society (AMS)","issue":"255","license":[{"start":{"date-parts":[[2007,3,21]],"date-time":"2007-03-21T00:00:00Z","timestamp":1174435200000},"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                    The Hilbert modular fourfold determined by the totally real quartic number field\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"k\">\n                        <mml:semantics>\n                          <mml:mi>k<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">k<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is a desingularization of a natural compactification of the quotient space\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"normal upper Gamma Subscript k Baseline minus script upper H Superscript 4\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi mathvariant=\"normal\">\n                                \u0393\n                                \n                              <\/mml:mi>\n                              <mml:mi>k<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mi class=\"MJX-variant\" mathvariant=\"normal\">\n                              \u2216\n                              \n                            <\/mml:mi>\n                            <mml:msup>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                  <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">H<\/mml:mi>\n                                <\/mml:mrow>\n                              <\/mml:mrow>\n                              <mml:mn>4<\/mml:mn>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\Gamma _k \\backslash {\\mathcal H}^4<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , where\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"normal upper Gamma Subscript k Baseline equals PSL Subscript 2 Baseline left-parenthesis script upper O Subscript k Baseline right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi mathvariant=\"normal\">\n                                \u0393\n                                \n                              <\/mml:mi>\n                              <mml:mi>k<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:msub>\n                              <mml:mstyle displaystyle=\"false\" scriptlevel=\"0\">\n                                <mml:mtext>PSL<\/mml:mtext>\n                              <\/mml:mstyle>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:msub>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                  <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">O<\/mml:mi>\n                                <\/mml:mrow>\n                              <\/mml:mrow>\n                              <mml:mi>k<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\Gamma _k=\\mbox {PSL}_2({\\mathcal O}_k)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    acts on\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"script upper H Superscript 4\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">H<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:mrow>\n                            <mml:mn>4<\/mml:mn>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">{\\mathcal H}^4<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    by fractional linear transformations via the four embeddings of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"k\">\n                        <mml:semantics>\n                          <mml:mi>k<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">k<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    into\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"bold upper R\">\n                        <mml:semantics>\n                          <mml:mi mathvariant=\"bold\">R<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\bf R<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . The arithmetic genus, equal to one plus the dimension of the space of Hilbert modular cusp forms of weight\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-parenthesis 2 comma 2 comma 2 comma 2 right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">(2,2,2,2)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , is a birational invariant useful in the classification of these varieties. In this work, we describe an algorithm allowing for the automated computation of the arithmetic genus and give sample results.\n                  <\/p>","DOI":"10.1090\/s0025-5718-06-01842-4","type":"journal-article","created":{"date-parts":[[2006,5,24]],"date-time":"2006-05-24T14:43:01Z","timestamp":1148481781000},"page":"1553-1560","source":"Crossref","is-referenced-by-count":2,"title":["Computing the arithmetic genus of Hilbert modular fourfolds"],"prefix":"10.1090","volume":"75","author":[{"given":"H.","family":"Grundman","sequence":"first","affiliation":[]},{"given":"L.","family":"Lippincott","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2006,3,21]]},"reference":[{"key":"1","unstructured":"C. Batut, K. Belabas, D. Benardi, H. Cohen, and M. Olivier. User\u2019s Guide to PARI-GP, 1998. \u27e8ftp:\/\/megrez.math.u-bordeaux.fr\/pub\/pari\u27e9."},{"key":"2","unstructured":"J. Buchmann, F. Diaz y Diaz, D. Ford, P. L\u00e9tard, M. Olivier, M. Pohst, and A. Schwarz. Tables of number fields of low degree, \u27e8ftp:\/\/megrez.math.u-bordeaux.fr\/pub\/ numberfields\/\u27e9."},{"issue":"3-4","key":"3","doi-asserted-by":"publisher","first-page":"267","DOI":"10.1006\/jsco.1996.0126","article-title":"KANT V4","volume":"24","author":"Daberkow, M.","year":"1997","journal-title":"J. Symbolic Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0747-7171","issn-type":"print"},{"key":"4","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-02638-0","volume-title":"Hilbert modular forms","author":"Freitag, Eberhard","year":"1990","ISBN":"https:\/\/id.crossref.org\/isbn\/3540505865"},{"issue":"1","key":"5","doi-asserted-by":"publisher","first-page":"47","DOI":"10.1006\/jnth.1997.2074","article-title":"Hilbert modular varieties of Galois quartic fields","volume":"63","author":"Grundman, H. G.","year":"1997","journal-title":"J. Number Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0022-314X","issn-type":"print"},{"key":"6","isbn-type":"print","doi-asserted-by":"publisher","first-page":"217","DOI":"10.1090\/fic\/041\/17","article-title":"Hilbert modular fourfolds of arithmetic genus one","author":"Grundman, H. G.","year":"2004","ISBN":"https:\/\/id.crossref.org\/isbn\/0821833537"},{"key":"7","first-page":"183","article-title":"Hilbert modular surfaces","volume":"19","author":"Hirzebruch, Friedrich E. P.","year":"1973","journal-title":"Enseign. Math. (2)","ISSN":"https:\/\/id.crossref.org\/issn\/0013-8584","issn-type":"print"},{"key":"8","series-title":"Classics in Mathematics","isbn-type":"print","volume-title":"Topological methods in algebraic geometry","author":"Hirzebruch, Friedrich","year":"1995","ISBN":"https:\/\/id.crossref.org\/isbn\/3540586636"},{"key":"9","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/BF01405200","article-title":"Hilbert modular surfaces and the classification of algebraic surfaces","volume":"23","author":"Hirzebruch, F.","year":"1974","journal-title":"Invent. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0020-9910","issn-type":"print"},{"key":"10","first-page":"43","article-title":"Classification of Hilbert modular surfaces","author":"Hirzebruch, F.","year":"1977"},{"key":"11","first-page":"87","article-title":"Berechnung von Zetafunktionen an ganzzahligen Stellen","volume":"1969","author":"Siegel, Carl Ludwig","year":"1969","journal-title":"Nachr. Akad. Wiss. G\\\"{o}ttingen Math.-Phys. Kl. II","ISSN":"https:\/\/id.crossref.org\/issn\/0065-5295","issn-type":"print"},{"key":"12","series-title":"Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-61553-5","volume-title":"Hilbert modular surfaces","volume":"16","author":"van der Geer, Gerard","year":"1988","ISBN":"https:\/\/id.crossref.org\/isbn\/3540176012"},{"issue":"1-2","key":"13","first-page":"55","article-title":"On the values at negative integers of the zeta-function of a real quadratic field","volume":"22","author":"Zagier, Don","year":"1976","journal-title":"Enseign. Math. (2)","ISSN":"https:\/\/id.crossref.org\/issn\/0013-8584","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2006-75-255\/S0025-5718-06-01842-4\/S0025-5718-06-01842-4.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2006-75-255\/S0025-5718-06-01842-4\/S0025-5718-06-01842-4.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T14:37:06Z","timestamp":1776782226000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2006-75-255\/S0025-5718-06-01842-4\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2006,3,21]]},"references-count":13,"journal-issue":{"issue":"255","published-print":{"date-parts":[[2006,7]]}},"alternative-id":["S0025-5718-06-01842-4"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-06-01842-4","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":[[2006,3,21]]}}}