{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T21:51:46Z","timestamp":1776721906759,"version":"3.51.2"},"reference-count":14,"publisher":"American Mathematical Society (AMS)","issue":"215","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    It is possible to compute\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"j left-parenthesis tau right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>j<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>\n                              \u03c4\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">j(\\tau )<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and its modular equations with no perception of its related classical group structure except at\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"normal infinity\">\n                        <mml:semantics>\n                          <mml:mi mathvariant=\"normal\">\n                            \u221e\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\infty<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . We start by taking, for\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p\">\n                        <mml:semantics>\n                          <mml:mi>p<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">p<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    prime, an unknown \u201c\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p\">\n                        <mml:semantics>\n                          <mml:mi>p<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">p<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -Newtonian\u201d polynomial equation\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"g left-parenthesis u comma v right-parenthesis equals 0\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>g<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>u<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>v<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">g(u,v)=0<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    with arbitrary coefficients (based only on Newton\u2019s polygon requirements at\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"normal infinity\">\n                        <mml:semantics>\n                          <mml:mi mathvariant=\"normal\">\n                            \u221e\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\infty<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    for\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"u equals j left-parenthesis tau right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>u<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mi>j<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>\n                              \u03c4\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">u=j(\\tau )<\/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=\"v equals j left-parenthesis p tau right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>v<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mi>j<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>p<\/mml:mi>\n                            <mml:mi>\n                              \u03c4\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">v=j(p\\tau )<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ). We then ask which choice of coefficients of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"g left-parenthesis u comma v right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>g<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>u<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>v<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">g(u,v)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    leads to some consistent Laurent series solution\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"u equals u left-parenthesis q right-parenthesis almost-equals 1 slash q\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>u<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mi>u<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>q<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>\n                              \u2248\n                              \n                            <\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo>\/<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mi>q<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">u=u(q)\\approx 1\/q<\/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=\"v equals u left-parenthesis q Superscript p Baseline right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>v<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mi>u<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>q<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>p<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">v=u(q^{p})<\/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=\"q equals exp 2 pi i tau right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>q<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mi>exp<\/mml:mi>\n                            <mml:mo>\n                              \u2061\n                              \n                            <\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mi>\n                              \u03c0\n                              \n                            <\/mml:mi>\n                            <mml:mi>i<\/mml:mi>\n                            <mml:mi>\n                              \u03c4\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">q=\\exp 2\\pi i\\tau )<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . It is conjectured that if the same Laurent series\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"u left-parenthesis q right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>u<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>q<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">u(q)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    works for\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p\">\n                        <mml:semantics>\n                          <mml:mi>p<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">p<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -Newtonian polynomials of two or more primes\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p\">\n                        <mml:semantics>\n                          <mml:mi>p<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">p<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , then there is only a bounded number of choices for the Laurent series (to within an additive constant). These choices are essentially from the set of \u201creplicable functions,\u201d which include more classical modular invariants, particularly\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"u equals j left-parenthesis tau right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>u<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mi>j<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>\n                              \u03c4\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">u=j(\\tau )<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . A demonstration for orders\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p equals 2\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>p<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">p=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=\"3\">\n                        <mml:semantics>\n                          <mml:mn>3<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">3<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is done by computation. More remarkably, if the same series\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"u left-parenthesis q right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>u<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>q<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">u(q)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    works for the\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p\">\n                        <mml:semantics>\n                          <mml:mi>p<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">p<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -Newtonian polygons of 15 special \u201cFricke-Monster\u201d values of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p\">\n                        <mml:semantics>\n                          <mml:mi>p<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">p<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , then\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-parenthesis u equals right-parenthesis j left-parenthesis tau right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>u<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mi>j<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>\n                              \u03c4\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">(u=)j(\\tau )<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is (essentially) determined uniquely. Computationally, this process stands alone, and, in a sense, modular invariants arise \u201cspontaneously.\u201d\n                  <\/p>","DOI":"10.1090\/s0025-5718-96-00726-0","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:13:45Z","timestamp":1027707225000},"page":"1295-1309","source":"Crossref","is-referenced-by-count":1,"title":["Spontaneous generation of modular invariants"],"prefix":"10.1090","volume":"65","author":[{"given":"Harvey","family":"Cohn","sequence":"first","affiliation":[]},{"given":"John","family":"McKay","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[1996]]},"reference":[{"key":"1","isbn-type":"print","doi-asserted-by":"publisher","first-page":"87","DOI":"10.1017\/CBO9780511629259.010","article-title":"Completely replicable functions","author":"Alexander, D.","year":"1992","ISBN":"https:\/\/id.crossref.org\/isbn\/0521406854"},{"key":"2","doi-asserted-by":"publisher","first-page":"134","DOI":"10.1007\/BF01359701","article-title":"Hecke operators on \u0393\u2080(\ud835\udc5a)","volume":"185","author":"Atkin, A. O. L.","year":"1970","journal-title":"Math. Ann.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5831","issn-type":"print"},{"issue":"184","key":"3","doi-asserted-by":"publisher","first-page":"787","DOI":"10.2307\/2008779","article-title":"Fricke\u2019s two-valued modular equations","volume":"51","author":"Cohn, Harvey","year":"1988","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"4","isbn-type":"print","first-page":"85","article-title":"A numerical survey of the reduction of modular curve genus by Fricke\u2019s involutions","author":"Cohn, Harvey","year":"1991","ISBN":"https:\/\/id.crossref.org\/isbn\/0387976701"},{"issue":"203","key":"5","doi-asserted-by":"publisher","first-page":"155","DOI":"10.2307\/2152943","article-title":"How branching properties determine modular equations","volume":"61","author":"Cohn, Harvey","year":"1993","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"6","doi-asserted-by":"publisher","first-page":"549","DOI":"10.2307\/1968939","article-title":"Transformationen von algebraischem Typ","volume":"40","author":"Kober, Hermann","year":"1939","journal-title":"Ann. of Math. (2)","ISSN":"https:\/\/id.crossref.org\/issn\/0003-486X","issn-type":"print"},{"issue":"3","key":"7","doi-asserted-by":"publisher","first-page":"308","DOI":"10.1112\/blms\/11.3.308","article-title":"Monstrous moonshine","volume":"11","author":"Conway, J. H.","year":"1979","journal-title":"Bull. London Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0024-6093","issn-type":"print"},{"issue":"13","key":"8","doi-asserted-by":"publisher","first-page":"5175","DOI":"10.1080\/00927879408825127","article-title":"More on replicable functions","volume":"22","author":"Ford, David","year":"1994","journal-title":"Comm. Algebra","ISSN":"https:\/\/id.crossref.org\/issn\/0092-7872","issn-type":"print"},{"key":"9","unstructured":"R. Fricke, Lehrbuch der Algebra III (Algebraische Zahlen), Vieweg, Braunschweig, 1928."},{"key":"10","doi-asserted-by":"crossref","unstructured":"R. Fricke, \u00dcber die Berechnung der Klasseninvarianten, Acta Arith. 52 (1929), 257\u2013279.","DOI":"10.1007\/BF02592687"},{"key":"11","first-page":"113","article-title":"Generalizaci\u00f3n del teorema de Laguerre","volume":"12","author":"Sispanov, Sergio","year":"1939","journal-title":"Bol. Mat.","ISSN":"https:\/\/id.crossref.org\/issn\/0120-0380","issn-type":"print"},{"issue":"1","key":"12","doi-asserted-by":"publisher","first-page":"65","DOI":"10.1017\/s1446788700013367","article-title":"On a class of non-linear functional equations connected with modular functions","volume":"22","author":"Mahler, K.","year":"1976","journal-title":"J. Austral. Math. Soc. Ser. A","ISSN":"https:\/\/id.crossref.org\/issn\/0263-6115","issn-type":"print"},{"key":"13","unstructured":"Yves Martin, On modular invariance of completely replicable functions, (preprint)."},{"issue":"1","key":"14","doi-asserted-by":"publisher","first-page":"253","DOI":"10.1080\/00927879008823911","article-title":"The \ud835\udc5e-series of monstrous moonshine and the decomposition of the head characters","volume":"18","author":"McKay, John","year":"1990","journal-title":"Comm. Algebra","ISSN":"https:\/\/id.crossref.org\/issn\/0092-7872","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/1996-65-215\/S0025-5718-96-00726-0\/S0025-5718-96-00726-0.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/1996-65-215\/S0025-5718-96-00726-0\/S0025-5718-96-00726-0.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T21:13:04Z","timestamp":1776719584000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/1996-65-215\/S0025-5718-96-00726-0\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1996]]},"references-count":14,"journal-issue":{"issue":"215","published-print":{"date-parts":[[1996,7]]}},"alternative-id":["S0025-5718-96-00726-0"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-96-00726-0","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":[[1996]]}}}