{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,12]],"date-time":"2026-06-12T03:29:32Z","timestamp":1781234972388,"version":"3.54.1"},"reference-count":34,"publisher":"American Mathematical Society (AMS)","issue":"242","license":[{"start":{"date-parts":[[2003,6,25]],"date-time":"2003-06-25T00:00:00Z","timestamp":1056499200000},"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 derive a mass formula for\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"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                    -dimensional unimodular lattices having any prescribed root system. We use Katsurada\u2019s formula for the Fourier coefficients of Siegel Eisenstein series to compute these masses for all root systems of even unimodular 32-dimensional lattices and odd unimodular lattices of dimension\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"n less-than-or-equal-to 30\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>\n                              \u2264\n                              \n                            <\/mml:mo>\n                            <mml:mn>30<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">n\\leq 30<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . In particular, we find the mass of even unimodular 32-dimensional lattices with no roots, and the mass of odd unimodular lattices with no roots in dimension\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"n less-than-or-equal-to 30\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>\n                              \u2264\n                              \n                            <\/mml:mo>\n                            <mml:mn>30<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">n\\leq 30<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , verifying Bacher and Venkov\u2019s enumerations in dimensions 27 and 28. We also compute better lower bounds on the number of inequivalent unimodular lattices in dimensions 26 to 30 than those afforded by the Minkowski-Siegel mass constants.\n                  <\/p>","DOI":"10.1090\/s0025-5718-02-01455-2","type":"journal-article","created":{"date-parts":[[2003,2,10]],"date-time":"2003-02-10T11:10:52Z","timestamp":1044875452000},"page":"839-863","source":"Crossref","is-referenced-by-count":31,"title":["A mass formula for unimodular lattices with no roots"],"prefix":"10.1090","volume":"72","author":[{"given":"Oliver","family":"King","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[2002,6,25]]},"reference":[{"issue":"2","key":"1","doi-asserted-by":"publisher","first-page":"91","DOI":"10.1155\/S1073792894000115","article-title":"Unimodular lattices without nontrivial automorphisms","author":"Bacher, Roland","year":"1994","journal-title":"Internat. Math. Res. Notices","ISSN":"https:\/\/id.crossref.org\/issn\/1073-7928","issn-type":"print"},{"key":"2","unstructured":"R. Bacher and B. B. Venkov, R\u00e9seaux entiers unimodulaires sans racines en dimension 27 et 28, R\u00e9seaux euclidiens, designs sph\u00e9riques et formes modulaires, 212\u2013267, Monogr. Enseign. Math., 37, Enseignement Math., Geneva, 2001."},{"issue":"429","key":"3","doi-asserted-by":"publisher","first-page":"iv+70","DOI":"10.1090\/memo\/0429","article-title":"Positive definite unimodular lattices with trivial automorphism groups","volume":"85","author":"Bannai, Etsuko","year":"1990","journal-title":"Mem. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-9266","issn-type":"print"},{"key":"4","unstructured":"R. E. Borcherds, The Leech lattice and other lattices, Ph.D. Dissertation, University of Cambridge, 1984. Available at arXiv:math.NT\/9911195 Much of this material also appears in [5]."},{"issue":"3","key":"5","doi-asserted-by":"publisher","first-page":"525","DOI":"10.1215\/S0012-7094-00-10536-4","article-title":"Classification of positive definite lattices","volume":"105","author":"Borcherds, Richard E.","year":"2000","journal-title":"Duke Math. J.","ISSN":"https:\/\/id.crossref.org\/issn\/0012-7094","issn-type":"print"},{"key":"6","first-page":"656","article-title":"The Weierstrass condition for multiple integral variation problems","volume":"5","author":"Graves, Lawrence M.","year":"1939","journal-title":"Duke Math. J.","ISSN":"https:\/\/id.crossref.org\/issn\/0012-7094","issn-type":"print"},{"issue":"1857","key":"7","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1098\/rspa.1988.0107","article-title":"Low-dimensional lattices. IV. The mass formula","volume":"419","author":"Conway, J. H.","year":"1988","journal-title":"Proc. Roy. Soc. London Ser. 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