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Comp."],"abstract":"<p>\n                    In this paper we improve on existing methods to compute quadratic points on modular curves and apply them to successfully find all the quadratic points on all modular curves\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper X 0 left-parenthesis upper N right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>X<\/mml:mi>\n                              <mml:mn>0<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>N<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">X_0(N)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    of genus up to\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"8\">\n                        <mml:semantics>\n                          <mml:mn>8<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">8<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , and genus up to\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"10\">\n                        <mml:semantics>\n                          <mml:mn>10<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">10<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    with\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                    prime, for which they were previously unknown. The values of\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 consider are contained in the set\n                    <disp-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"script upper L equals StartSet 58 comma 68 comma 74 comma 76 comma 80 comma 85 comma 97 comma 98 comma 100 comma 103 comma 107 comma 109 comma 113 comma 121 comma 127 EndSet period\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">L<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mo fence=\"false\" stretchy=\"false\">{<\/mml:mo>\n                            <mml:mn>58<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>68<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>74<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>76<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>80<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>85<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>97<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>98<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>100<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>103<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>107<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>109<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>113<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>121<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>127<\/mml:mn>\n                            <mml:mo fence=\"false\" stretchy=\"false\">}<\/mml:mo>\n                            <mml:mo>.<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\begin{equation*} \\mathcal {L}=\\{58, 68, 74, 76, 80, 85, 97, 98, 100, 103, 107, 109, 113, 121, 127 \\}. \\end{equation*}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/disp-formula>\n                    We obtain that all the non-cuspidal quadratic points on\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper X 0 left-parenthesis upper N right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>X<\/mml:mi>\n                              <mml:mn>0<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>N<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">X_0(N)<\/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=\"upper N element-of script upper L\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>N<\/mml:mi>\n                            <mml:mo>\n                              \u2208\n                              \n                            <\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">L<\/mml:mi>\n                            <\/mml:mrow>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">N\\in \\mathcal {L}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    are complex multiplication (CM) points, except for one pair of Galois conjugate points on\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper X 0 left-parenthesis 103 right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>X<\/mml:mi>\n                              <mml:mn>0<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mn>103<\/mml:mn>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">X_0(103)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    defined over\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"double-struck upper Q left-parenthesis StartRoot 2885 EndRoot right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"double-struck\">Q<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:msqrt>\n                              <mml:mn>2885<\/mml:mn>\n                            <\/mml:msqrt>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathbb {Q}(\\sqrt {2885})<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . We also compute the\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"j\">\n                        <mml:semantics>\n                          <mml:mi>j<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">j<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -invariants of the elliptic curves parametrised by these points, and for the CM points determine their geometric endomorphism rings.\n                  <\/p>","DOI":"10.1090\/mcom\/3902","type":"journal-article","created":{"date-parts":[[2023,9,27]],"date-time":"2023-09-27T10:34:27Z","timestamp":1695810867000},"page":"1371-1397","source":"Crossref","is-referenced-by-count":8,"title":["Computing quadratic points on modular curves \ud835\udc4b\u2080(\ud835\udc41)"],"prefix":"10.1090","volume":"93","author":[{"given":"Nikola","family":"Ad\u017eaga","sequence":"first","affiliation":[]},{"given":"Timo","family":"Keller","sequence":"additional","affiliation":[]},{"given":"Philippe","family":"Michaud-Jacobs","sequence":"additional","affiliation":[]},{"given":"Filip","family":"Najman","sequence":"additional","affiliation":[]},{"given":"Ekin","family":"Ozman","sequence":"additional","affiliation":[]},{"given":"Borna","family":"Vukorepa","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2023,10,3]]},"reference":[{"issue":"1","key":"1","doi-asserted-by":"publisher","first-page":"15","DOI":"10.4064\/aa220110-7-3","article-title":"Quadratic Chabauty for Atkin-Lehner quotients of modular curves of prime level and genus 4, 5, 6","volume":"208","author":"Ad\u017eaga, Nikola","year":"2023","journal-title":"Acta Arith.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"},{"issue":"4","key":"2","doi-asserted-by":"publisher","first-page":"Paper No. 87, 24","DOI":"10.1007\/s40993-022-00388-9","article-title":"Rational points on hyperelliptic Atkin-Lehner quotients of modular curves and their coverings","volume":"8","author":"Ad\u017eaga, Nikola","year":"2022","journal-title":"Res. 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