{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T18:49:16Z","timestamp":1776797356834,"version":"3.51.2"},"reference-count":11,"publisher":"American Mathematical Society (AMS)","issue":"292","license":[{"start":{"date-parts":[[2015,7,17]],"date-time":"2015-07-17T00:00:00Z","timestamp":1437091200000},"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                    We extend the algorithm of Darmon\u2013Green and Darmon\u2013Pollack for computing\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                    -adic Darmon points on elliptic curves to the case of composite conductor. We also extend the algorithm of Darmon\u2013Logan for computing ATR Darmon points to treat curves of nontrivial conductor. Both cases involve an algorithmic decomposition into elementary matrices in congruence subgroups\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"normal upper Gamma 1 left-parenthesis German upper N right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi mathvariant=\"normal\">\n                                \u0393\n                                \n                              <\/mml:mi>\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi mathvariant=\"fraktur\">N<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\Gamma _1({\\mathfrak N})<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    for ideals\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"German upper N\">\n                        <mml:semantics>\n                          <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"fraktur\">N<\/mml:mi>\n                            <\/mml:mrow>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">{\\mathfrak N}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    in certain rings of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper S\">\n                        <mml:semantics>\n                          <mml:mi>S<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">S<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -integers. We use these extensions to provide additional evidence in support of the conjectures on the rationality of Darmon points.\n                  <\/p>","DOI":"10.1090\/s0025-5718-2014-02853-6","type":"journal-article","created":{"date-parts":[[2014,7,17]],"date-time":"2014-07-17T10:06:14Z","timestamp":1405591574000},"page":"875-893","source":"Crossref","is-referenced-by-count":4,"title":["Elementary matrix decomposition and the computation of Darmon points with higher conductor"],"prefix":"10.1090","volume":"84","author":[{"given":"Xavier","family":"Guitart","sequence":"first","affiliation":[]},{"given":"Marc","family":"Masdeu","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2014,7,17]]},"reference":[{"key":"1","unstructured":"[BMS67] H. Bass, J. Milnor, and J.-P. Serre. Solution of the congruence subgroup problem for \ud835\udc46\ud835\udc3f_{\ud835\udc5b}(\ud835\udc5b\u22653) and \ud835\udc46\ud835\udc5d_{2\ud835\udc5b}(\ud835\udc5b\u22652). Inst. Hautes \u00c9tudes Sci. Publ. Math., (33):59\u2013137, 1967."},{"key":"2","doi-asserted-by":"publisher","first-page":"481","DOI":"10.1080\/00927877508822057","article-title":"On the construction of division chains in algebraic number rings, with applications to \ud835\udc46\ud835\udc3f\u2082","volume":"3","author":"Cooke, George","year":"1975","journal-title":"Comm. Algebra","ISSN":"https:\/\/id.crossref.org\/issn\/0092-7872","issn-type":"print"},{"issue":"3","key":"3","doi-asserted-by":"publisher","first-page":"589","DOI":"10.2307\/3062142","article-title":"Integration on \u210b_{\ud835\udcc5}\u00d7\u210b and arithmetic applications","volume":"154","author":"Darmon, Henri","year":"2001","journal-title":"Ann. of Math. (2)","ISSN":"https:\/\/id.crossref.org\/issn\/0003-486X","issn-type":"print"},{"key":"4","series-title":"CBMS Regional Conference Series in Mathematics","isbn-type":"print","volume-title":"Rational points on modular elliptic curves","volume":"101","author":"Darmon, Henri","year":"2004","ISBN":"https:\/\/id.crossref.org\/isbn\/0821828681"},{"issue":"1","key":"5","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1080\/10586458.2002.10504467","article-title":"Elliptic curves and class fields of real quadratic fields: algorithms and evidence","volume":"11","author":"Darmon, Henri","year":"2002","journal-title":"Experiment. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1058-6458","issn-type":"print"},{"issue":"40","key":"6","doi-asserted-by":"publisher","first-page":"2153","DOI":"10.1155\/S1073792803131108","article-title":"Periods of Hilbert modular forms and rational points on elliptic curves","author":"Darmon, Henri","year":"2003","journal-title":"Int. Math. Res. Not.","ISSN":"https:\/\/id.crossref.org\/issn\/1073-7928","issn-type":"print"},{"key":"7","doi-asserted-by":"publisher","first-page":"319","DOI":"10.1007\/BF02771789","article-title":"Efficient calculation of Stark-Heegner points via overconvergent modular symbols","volume":"153","author":"Darmon, Henri","year":"2006","journal-title":"Israel J. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-2172","issn-type":"print"},{"key":"8","unstructured":"[Gaer11] J\u00e9r\u00f4me G\u00e4rtner, Points de Darmon et vari\u00e9t\u00e9s de Shimura, 2011. Ph.D. Thesis, Institut de Math\u00e9matiques de Jussieu."},{"issue":"1","key":"9","doi-asserted-by":"publisher","first-page":"85","DOI":"10.1080\/10586458.2013.738564","article-title":"Computation of ATR Darmon points on nongeometrically modular elliptic curves","volume":"22","author":"Guitart, Xavier","year":"2013","journal-title":"Exp. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1058-6458","issn-type":"print"},{"key":"10","isbn-type":"print","volume-title":"Heegner points and rigid analytic modular forms","author":"Greenberg, Matthew","year":"2006","ISBN":"https:\/\/id.crossref.org\/isbn\/9780494251584"},{"key":"11","first-page":"313","article-title":"The group \ud835\udc46\ud835\udc3f\u2082 over Dedekind rings of arithmetic type","volume":"89(131)","author":"Vaser\u0161te\u012dn, L. N.","year":"1972","journal-title":"Mat. Sb. (N.S.)","ISSN":"https:\/\/id.crossref.org\/issn\/0368-8666","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2015-84-292\/S0025-5718-2014-02853-6\/S0025-5718-2014-02853-6.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2015-84-292\/S0025-5718-2014-02853-6\/S0025-5718-2014-02853-6.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T18:16:26Z","timestamp":1776795386000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2015-84-292\/S0025-5718-2014-02853-6\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,7,17]]},"references-count":11,"journal-issue":{"issue":"292","published-print":{"date-parts":[[2015,3]]}},"alternative-id":["S0025-5718-2014-02853-6"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-2014-02853-6","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":[[2014,7,17]]}}}