{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T17:40:33Z","timestamp":1776793233690,"version":"3.51.2"},"reference-count":7,"publisher":"American Mathematical Society (AMS)","issue":"274","license":[{"start":{"date-parts":[[2011,9,2]],"date-time":"2011-09-02T00:00:00Z","timestamp":1314921600000},"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 an imaginary quadratic field with class number one and let\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"German p subset-of script upper O Subscript upper K\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"fraktur\">p<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mo>\n                              \u2282\n                              \n                            <\/mml:mo>\n                            <mml:msub>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">O<\/mml:mi>\n                              <\/mml:mrow>\n                              <mml:mi>K<\/mml:mi>\n                            <\/mml:msub>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathfrak {p} \\subset \\mathcal {O}_K<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    be a degree one prime ideal of norm\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                    not dividing\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"6 d Subscript upper K\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mn>6<\/mml:mn>\n                            <mml:msub>\n                              <mml:mi>d<\/mml:mi>\n                              <mml:mi>K<\/mml:mi>\n                            <\/mml:msub>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">6d_K<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . In this paper we generalize an algorithm of Schoof to compute the class numbers of ray class fields\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper K Subscript German p\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>K<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi mathvariant=\"fraktur\">p<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:mrow>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">K_{\\mathfrak {p}}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    heuristically. We achieve this by using elliptic units analytically constructed by Stark and the Galois action on them given by Shimura\u2019s reciprocity law. We have discovered a very interesting phenomenon where\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                    divides the class number of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper K Subscript German p\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>K<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi mathvariant=\"fraktur\">p<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:mrow>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">K_{\\mathfrak {p}}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . This is a counterexample to the elliptic analogue of Vandiver\u2019s conjecture.\n                  <\/p>","DOI":"10.1090\/s0025-5718-2010-02413-5","type":"journal-article","created":{"date-parts":[[2010,12,29]],"date-time":"2010-12-29T14:39:42Z","timestamp":1293633582000},"page":"1099-1122","source":"Crossref","is-referenced-by-count":3,"title":["Class numbers of ray class fields of imaginary quadratic fields"],"prefix":"10.1090","volume":"80","author":[{"given":"Omer","family":"Kucuksakalli","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2010,9,2]]},"reference":[{"issue":"1","key":"1","first-page":"1","article-title":"Bernoulli numbers, Hurwitz numbers, \ud835\udc5d-adic \ud835\udc3f-functions and Kummer\u2019s criterion","volume":"101","author":"Lozano-Robledo, \u00c1lvaro","year":"2007","journal-title":"RACSAM. Rev. R. Acad. Cienc. Exactas F\\'{\\i}s. Nat. Ser. 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