{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T17:05:12Z","timestamp":1776791112282,"version":"3.51.2"},"reference-count":10,"publisher":"American Mathematical Society (AMS)","issue":"268","license":[{"start":{"date-parts":[[2010,1,29]],"date-time":"2010-01-29T00:00:00Z","timestamp":1264723200000},"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                    Let\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper 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                    be a real quadratic field, let\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                    be a prime number which is inert in\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper 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                    and let\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper K Subscript p\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>K<\/mml:mi>\n                            <mml:mi>p<\/mml:mi>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">K_p<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    be the completion of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper 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                    at\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                    . As part of a Ph.D. thesis, we constructed a certain\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 invariant\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"u element-of upper K Subscript p Superscript times\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>u<\/mml:mi>\n                            <mml:mo>\n                              \u2208\n                              \n                            <\/mml:mo>\n                            <mml:msubsup>\n                              <mml:mi>K<\/mml:mi>\n                              <mml:mi>p<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mo>\n                                  \u00d7\n                                  \n                                <\/mml:mo>\n                              <\/mml:mrow>\n                            <\/mml:msubsup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">u\\in K_p^{\\times }<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , and conjectured that\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"u\">\n                        <mml:semantics>\n                          <mml:mi>u<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">u<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is, in fact, a\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                    -unit in a suitable narrow ray class field of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper 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                    . In this paper we give numerical evidence in support of that conjecture. Our method of computation is similar to the one developed by Dasgupta and relies on partial modular symbols attached to Eisenstein series.\n                  <\/p>","DOI":"10.1090\/s0025-5718-09-02215-7","type":"journal-article","created":{"date-parts":[[2009,6,30]],"date-time":"2009-06-30T10:39:02Z","timestamp":1246358342000},"page":"2307-2345","source":"Crossref","is-referenced-by-count":1,"title":["Computation of \ud835\udc5d-units in ray class fields of real quadratic number fields"],"prefix":"10.1090","volume":"78","author":[{"given":"Hugo","family":"Chapdelaine","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2009,1,29]]},"reference":[{"key":"1","unstructured":"[Cha] H. Chapdelaine. Elliptic units in ray class fields of real quadratic number fields, version with a few corrections and supplements. available at http:\/\/www.mat. ulaval.ca\/fileadmin\/Pages_personnelles_des_profs\/hchapd\/thesis_final.pdf."},{"key":"2","unstructured":"[Cha07a] H. Chapdelaine. \ud835\udc5d-units in ray class fields of real quadratic number fields. accepted for publication in Compositio, 1:1\u201334, 2007."},{"key":"3","unstructured":"[Cha07b] H. Chapdelaine. Zeta functions twisted by additive characters, \ud835\udc5d-units and Gauss sums. International J. Number Theory, 1:1\u201340, 2007."},{"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":"3","key":"5","doi-asserted-by":"publisher","first-page":"553","DOI":"10.4153\/CJM-2007-023-0","article-title":"Computations of elliptic units for real quadratic fields","volume":"59","author":"Dasgupta, Samit","year":"2007","journal-title":"Canad. J. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0008-414X","issn-type":"print"},{"issue":"2","key":"6","doi-asserted-by":"publisher","first-page":"225","DOI":"10.1215\/00127094-2008-019","article-title":"Shintani zeta functions and Gross-Stark units for totally real fields","volume":"143","author":"Dasgupta, Samit","year":"2008","journal-title":"Duke Math. J.","ISSN":"https:\/\/id.crossref.org\/issn\/0012-7094","issn-type":"print"},{"issue":"1","key":"7","doi-asserted-by":"publisher","first-page":"301","DOI":"10.4007\/annals.2006.163.301","article-title":"Elliptic units for real quadratic fields","volume":"163","author":"Darmon, Henri","year":"2006","journal-title":"Ann. of Math. (2)","ISSN":"https:\/\/id.crossref.org\/issn\/0003-486X","issn-type":"print"},{"issue":"3","key":"8","first-page":"979","article-title":"\ud835\udc5d-adic \ud835\udc3f-series at \ud835\udc60=0","volume":"28","author":"Gross, Benedict H.","year":"1981","journal-title":"J. Fac. Sci. Univ. Tokyo Sect. IA Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0040-8980","issn-type":"print"},{"issue":"1","key":"9","first-page":"177","article-title":"On the values of abelian \ud835\udc3f-functions at \ud835\udc60=0","volume":"35","author":"Gross, Benedict H.","year":"1988","journal-title":"J. Fac. Sci. Univ. Tokyo Sect. IA Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0040-8980","issn-type":"print"},{"key":"10","series-title":"Progress in Mathematics","isbn-type":"print","volume-title":"Les conjectures de Stark sur les fonctions $L$ d'Artin en $s=0$","volume":"47","author":"Tate, John","year":"1984","ISBN":"https:\/\/id.crossref.org\/isbn\/0817631887"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2009-78-268\/S0025-5718-09-02215-7\/S0025-5718-09-02215-7.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2009-78-268\/S0025-5718-09-02215-7\/S0025-5718-09-02215-7.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T16:13:07Z","timestamp":1776787987000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2009-78-268\/S0025-5718-09-02215-7\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,1,29]]},"references-count":10,"journal-issue":{"issue":"268","published-print":{"date-parts":[[2009,10]]}},"alternative-id":["S0025-5718-09-02215-7"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-09-02215-7","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":[[2009,1,29]]}}}