{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,7]],"date-time":"2026-08-07T16:09:59Z","timestamp":1786118999854,"version":"build-2736575974"},"reference-count":23,"publisher":"American Mathematical Society (AMS)","issue":"362","license":[{"start":{"date-parts":[[2026,8,8]],"date-time":"2026-08-08T00:00:00Z","timestamp":1786147200000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/501100017700","name":"Henan Provincial Science and Technology Research Project","doi-asserted-by":"publisher","award":["225200810032"],"award-info":[{"award-number":["225200810032"]}],"id":[{"id":"10.13039\/501100017700","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100017700","name":"Henan Provincial Science and Technology Research Project","doi-asserted-by":"publisher","award":["12271405"],"award-info":[{"award-number":["12271405"]}],"id":[{"id":"10.13039\/501100017700","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["225200810032"],"award-info":[{"award-number":["225200810032"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12271405"],"award-info":[{"award-number":["12271405"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    Classifications and representations are two main topics in the theory of quadratic forms. In this paper, we consider these topics of ternary quadratic forms. For a given squarefree integer\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper 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                    , first we give the classification of positive definite ternary quadratic forms of level\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"4 upper N\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mn>4<\/mml:mn>\n                            <mml:mi>N<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">4N<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    explicitly. Second, we give explicit formulas of the weighted sum of representations over each class in every genus of ternary quadratic forms of level\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"4 upper N\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mn>4<\/mml:mn>\n                            <mml:mi>N<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">4N<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , which are involved with modified Hurwitz class number. In the proof of the main results, we use the relations among ternary quadratic forms, quaternion algebras, and Jacobi forms. As a corollary, we get the formula for the class number of positive ternary quadratic forms of level\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"4 upper N\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mn>4<\/mml:mn>\n                            <mml:mi>N<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">4N<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . As applications, we derive an explicit base of Eisenstein series space of modular forms of weight\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"3 slash 2\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mn>3<\/mml:mn>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo>\/<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">3\/2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and level\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"4 upper N\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mn>4<\/mml:mn>\n                            <mml:mi>N<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">4N<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , and give new proofs of some interesting identities involving representation number of ternary quadratic forms.\n                  <\/p>","DOI":"10.1090\/mcom\/4130","type":"journal-article","created":{"date-parts":[[2025,8,8]],"date-time":"2025-08-08T19:44:16Z","timestamp":1754682256000},"page":"3041-3079","source":"Crossref","is-referenced-by-count":0,"title":["The classification and representations of positive definite ternary quadratic forms of level 4\ud835\udc41"],"prefix":"10.1090","volume":"95","author":[{"given":"Yifan","family":"Luo","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Haigang","family":"Zhou","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[2025,8,8]]},"reference":[{"issue":"1","key":"1","doi-asserted-by":"publisher","first-page":"258","DOI":"10.1016\/j.jnt.2011.09.001","article-title":"On representation of an integer as the sum of three squares and ternary quadratic forms with the discriminants \ud835\udc5d\u00b2, 16\ud835\udc5d\u00b2","volume":"132","author":"Berkovich, Alexander","year":"2012","journal-title":"J. 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