{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,12]],"date-time":"2026-06-12T18:00:03Z","timestamp":1781287203927,"version":"3.54.1"},"reference-count":50,"publisher":"American Mathematical Society (AMS)","issue":"361","license":[{"start":{"date-parts":[[2026,7,31]],"date-time":"2026-07-31T00:00:00Z","timestamp":1785456000000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/501100003246","name":"Nederlandse Organisatie voor Wetenschappelijk Onderzoek","doi-asserted-by":"publisher","award":["639.031.243"],"award-info":[{"award-number":["639.031.243"]}],"id":[{"id":"10.13039\/501100003246","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    Special values of Siegel modular functions for\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper S p Subscript 2 g Baseline left-parenthesis bold upper Z right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>Sp<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>2<\/mml:mn>\n                                <mml:mi>g<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo>\n                              \u2061\n                              \n                            <\/mml:mo>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"bold\">Z<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\operatorname {Sp}_{2g}(\\mathbf {Z})<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    generate class fields of complex multiplication (CM) fields. They also yield abelian varieties with a known endomorphism ring. Smaller alternative values of modular functions that lie in the same class fields (class invariants) thus help to speed up the computation of those mathematical objects.\n                  <\/p>\n                  <p>\n                    We show that modular functions for the subgroup\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"normal upper Gamma Superscript 0 Baseline left-parenthesis upper N right-parenthesis subset-of-or-equal-to upper S p Subscript 2 g Baseline left-parenthesis bold upper Z right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msup>\n                              <mml:mi mathvariant=\"normal\">\n                                \u0393\n                                \n                              <\/mml:mi>\n                              <mml:mn>0<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>N<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>\n                              \u2286\n                              \n                            <\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>Sp<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>2<\/mml:mn>\n                                <mml:mi>g<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo>\n                              \u2061\n                              \n                            <\/mml:mo>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"bold\">Z<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\Gamma ^0 (N)\\subseteq \\operatorname {Sp}_{2g}(\\mathbf {Z})<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    yield class invariants under some splitting conditions on\u00a0\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper N\">\n                        <mml:semantics>\n                          <mml:mi>N<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">N<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , generalising results due to Schertz from classical modular functions to Siegel modular functions. We show how to obtain all Galois conjugates of a class invariant by evaluating the same modular function in CM period matrices derived from an\n                    <italic>\n                      <inline-formula content-type=\"math\/mathml\">\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper N\">\n                          <mml:semantics>\n                            <mml:mi>N<\/mml:mi>\n                            <mml:annotation encoding=\"application\/x-tex\">N<\/mml:annotation>\n                          <\/mml:semantics>\n                        <\/mml:math>\n                      <\/inline-formula>\n                      -system\n                    <\/italic>\n                    . Such a system consists of quadratic polynomials with coefficients in the real-quadratic subfield satisfying certain congruence conditions modulo\u00a0\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper N\">\n                        <mml:semantics>\n                          <mml:mi>N<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">N<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . We also examine conditions under which the minimal polynomial of a class invariant is real.\n                  <\/p>\n                  <p>Examples show that we may obtain class invariants that are much smaller than in previous constructions.<\/p>","DOI":"10.1090\/mcom\/4118","type":"journal-article","created":{"date-parts":[[2025,7,31]],"date-time":"2025-07-31T15:40:42Z","timestamp":1753976442000},"page":"2595-2636","source":"Crossref","is-referenced-by-count":0,"title":["Schertz style class invariants for higher degree CM fields"],"prefix":"10.1090","volume":"95","author":[{"given":"Andreas","family":"Enge","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Marco","family":"Streng","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[2025,7,31]]},"reference":[{"key":"1","unstructured":"B. Allombert and K. 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