{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T20:49:24Z","timestamp":1776804564033,"version":"3.51.2"},"reference-count":24,"publisher":"American Mathematical Society (AMS)","issue":"315","license":[{"start":{"date-parts":[[2019,4,12]],"date-time":"2019-04-12T00:00:00Z","timestamp":1555027200000},"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 give a systematic method of providing numerical evidence for higher order Stark-type conjectures such as (in chronological order) Stark\u2019s conjecture 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\">\n                        <mml:semantics>\n                          <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                            <mml:mi mathvariant=\"double-struck\">Q<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathbb {Q}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , Rubin\u2019s conjecture, Popescu\u2019s conjecture, and a conjecture due to Burns that constitutes a generalization of Brumer\u2019s classical conjecture on annihilation of class groups. Our approach is general and could be used for any abelian extension of number fields, independent of the signature and type of places (finite or infinite) that split completely in the extension.\n                  <\/p>\n                  <p>\n                    We then employ our techniques in the situation where\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                    is a totally real, abelian, ramified cubic extension of a real quadratic field. We numerically verify the conjectures listed above for all fields\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                    of this type with absolute discriminant less than\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"10 Superscript 12\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mn>10<\/mml:mn>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mn>12<\/mml:mn>\n                            <\/mml:mrow>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">10^{12}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , for a total of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"19197\">\n                        <mml:semantics>\n                          <mml:mn>19197<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">19197<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    examples. The places that split completely in these extensions are always taken to be the two real archimedean places of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"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 we are in a situation where all the\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                    -truncated\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper L\">\n                        <mml:semantics>\n                          <mml:mi>L<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">L<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -functions have order of vanishing at least two.\n                  <\/p>","DOI":"10.1090\/mcom\/3337","type":"journal-article","created":{"date-parts":[[2017,11,15]],"date-time":"2017-11-15T11:16:24Z","timestamp":1510744584000},"page":"389-420","source":"Crossref","is-referenced-by-count":2,"title":["Numerical evidence for higher order Stark-type conjectures"],"prefix":"10.1090","volume":"88","author":[{"given":"Kevin","family":"McGown","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jonathan","family":"Sands","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Daniel","family":"Valli\u00e8res","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2018,4,12]]},"reference":[{"key":"1","doi-asserted-by":"publisher","first-page":"153","DOI":"10.1515\/crll.1932.167.153","article-title":"\u00dcber Einheiten relativ galoisscher Zahlk\u00f6rper","volume":"167","author":"Artin, E.","year":"1932","journal-title":"J. Reine Angew. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0075-4102","issn-type":"print"},{"key":"2","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1090\/chel\/366","volume-title":"Class field theory","author":"Artin, Emil","year":"2009","ISBN":"https:\/\/id.crossref.org\/isbn\/9780821844267"},{"key":"3","first-page":"181","article-title":"Higher regulators and values of \ud835\udc3f-functions","author":"Be\u012dlinson, A. A.","year":"1984"},{"key":"4","isbn-type":"print","first-page":"333","article-title":"\ud835\udc3f-functions and Tamagawa numbers of motives","author":"Bloch, Spencer","year":"1990","ISBN":"https:\/\/id.crossref.org\/isbn\/0817634274"},{"issue":"3","key":"5","doi-asserted-by":"publisher","first-page":"451","DOI":"10.1007\/s00222-007-0052-3","article-title":"Congruences between derivatives of abelian \ud835\udc3f-functions at \ud835\udc60=0","volume":"169","author":"Burns, David","year":"2007","journal-title":"Invent. 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