{"id":"https://openalex.org/W4392764279","doi":"https://doi.org/10.1090/mcom/3961","title":"CM points, class numbers, and the Mahler measures of \ud835\udc65\u00b3+\ud835\udc66\u00b3+1-\ud835\udc58\ud835\udc65\ud835\udc66","display_name":"CM points, class numbers, and the Mahler measures of \ud835\udc65\u00b3+\ud835\udc66\u00b3+1-\ud835\udc58\ud835\udc65\ud835\udc66","publication_year":2024,"publication_date":"2024-03-09","ids":{"openalex":"https://openalex.org/W4392764279","doi":"https://doi.org/10.1090/mcom/3961"},"language":"lv","primary_location":{"id":"doi:10.1090/mcom/3961","is_oa":false,"landing_page_url":"http://dx.doi.org/10.1090/mcom/3961","pdf_url":null,"source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5004745198","display_name":"Zhengyu Tao","orcid":"https://orcid.org/0000-0002-2884-6732"},"institutions":[{"id":"https://openalex.org/I881766915","display_name":"Nanjing University","ror":"https://ror.org/01rxvg760","country_code":"CN","type":"education","lineage":["https://openalex.org/I881766915"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Zhengyu Tao","raw_affiliation_strings":["Department of Mathematics, Nanjing University, Nanjing 210093, People\u00e2\u0080\u0099s Republic of China","Department of Mathematics, Nanjing University, Nanjing 210093, People\u2019s Republic of China","Department of Mathematics, Nanjing University, Nanjing 210093, People's Republic of China"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Nanjing University, Nanjing 210093, People\u00e2\u0080\u0099s Republic of China","institution_ids":["https://openalex.org/I881766915"]},{"raw_affiliation_string":"Department of Mathematics, Nanjing University, Nanjing 210093, People\u2019s Republic of China","institution_ids":["https://openalex.org/I881766915"]},{"raw_affiliation_string":"Department of Mathematics, Nanjing University, Nanjing 210093, People's Republic of China","institution_ids":["https://openalex.org/I881766915"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5084859356","display_name":"Xuejun Guo","orcid":"https://orcid.org/0000-0002-0162-8532"},"institutions":[{"id":"https://openalex.org/I881766915","display_name":"Nanjing University","ror":"https://ror.org/01rxvg760","country_code":"CN","type":"education","lineage":["https://openalex.org/I881766915"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Xuejun Guo","raw_affiliation_strings":["Department of Mathematics, Nanjing University, Nanjing 210093, People\u00e2\u0080\u0099s Republic of China","Department of Mathematics, Nanjing University, Nanjing 210093, People\u2019s Republic of China","Department of Mathematics, Nanjing University, Nanjing 210093, People's Republic of China"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Nanjing University, Nanjing 210093, People\u00e2\u0080\u0099s Republic of China","institution_ids":["https://openalex.org/I881766915"]},{"raw_affiliation_string":"Department of Mathematics, Nanjing University, Nanjing 210093, People\u2019s Republic of China","institution_ids":["https://openalex.org/I881766915"]},{"raw_affiliation_string":"Department of Mathematics, Nanjing University, Nanjing 210093, People's Republic of China","institution_ids":["https://openalex.org/I881766915"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5004745198"],"corresponding_institution_ids":["https://openalex.org/I881766915"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.06968513,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10061","display_name":"Algebraic Geometry and Number Theory","score":0.9994999766349792,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10061","display_name":"Algebraic Geometry and Number Theory","score":0.9994999766349792,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11166","display_name":"Analytic Number Theory Research","score":0.9993000030517578,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11151","display_name":"Advanced Topology and Set Theory","score":0.9987999796867371,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.9078945517539978},{"id":"https://openalex.org/keywords/class","display_name":"Class (philosophy)","score":0.6674398183822632},{"id":"https://openalex.org/keywords/arithmetic","display_name":"Arithmetic","score":0.33527135848999023},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.32874178886413574},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.07080608606338501},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.04409012198448181}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.9078945517539978},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.6674398183822632},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.33527135848999023},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.32874178886413574},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.07080608606338501},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.04409012198448181}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1090/mcom/3961","is_oa":false,"landing_page_url":"http://dx.doi.org/10.1090/mcom/3961","pdf_url":null,"source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":27,"referenced_works":["https://openalex.org/W1586485728","https://openalex.org/W1983635056","https://openalex.org/W1998754688","https://openalex.org/W2016094420","https://openalex.org/W2029988180","https://openalex.org/W2038383731","https://openalex.org/W2108965900","https://openalex.org/W2110093674","https://openalex.org/W2112102145","https://openalex.org/W2491759990","https://openalex.org/W2793292115","https://openalex.org/W2900405975","https://openalex.org/W2963841585","https://openalex.org/W3026017058","https://openalex.org/W3036381431","https://openalex.org/W3047588414","https://openalex.org/W3100383786","https://openalex.org/W3130884874","https://openalex.org/W3183733065","https://openalex.org/W4232274958","https://openalex.org/W4240694056","https://openalex.org/W4243943870","https://openalex.org/W4316829925","https://openalex.org/W4376863352","https://openalex.org/W4385606257","https://openalex.org/W6631811706","https://openalex.org/W6676131352"],"related_works":["https://openalex.org/W4391375266","https://openalex.org/W1979597421","https://openalex.org/W2007980826","https://openalex.org/W2061531152","https://openalex.org/W3002753104","https://openalex.org/W2077600819","https://openalex.org/W2142036596","https://openalex.org/W2072657027","https://openalex.org/W2600246793","https://openalex.org/W4238204885"],"abstract_inverted_index":{"We":[0],"study":[1],"the":[2,6,75,80,124,191,309,350],"Mahler":[3,125,193,310],"measures":[4,126,311],"of":[5,127,175,183,188,312,349,364],"polynomial":[7],"family":[8],"<inline-formula":[9,95,128,162,195,235,285,313,351,370],"content-type=\"math/mathml\">":[10,96,129,163,196,236,286,314,352,371],"<mml:math":[11,97,130,164,197,237,287,315,353,372],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[12,98,131,165,198,238,288,316,354,373],"alttext=\"upper":[13,132,166,317,355],"Q":[14,133,318,376],"Subscript":[15,134,319],"k":[16,32,135,204,320],"Baseline":[17,136,321],"left-parenthesis":[18,137,203,322,377],"x":[19,24,33,138,323],"comma":[20,139,324],"y":[21,27,140,325],"right-parenthesis":[22],"equals":[23,240],"cubed":[25,28],"plus":[26,29],"1":[30],"minus":[31],"y\">":[34],"<mml:semantics>":[35,101,142,168,206,251,292,327,357,382],"<mml:mrow>":[36,102,143,207,252,256,293,328,383],"<mml:msub>":[37,144,329],"<mml:mi>Q</mml:mi>":[38,145,330],"<mml:mi>k</mml:mi>":[39,63,146,220,253,331],"</mml:msub>":[40,147,332],"<mml:mo":[41,46,148,153,212,218,221,333,338,389,394],"stretchy=\"false\">(</mml:mo>":[42,149,219,334,390],"<mml:mi>x</mml:mi>":[43,50,64,150,335],"<mml:mo>,</mml:mo>":[44,151,336],"<mml:mi>y</mml:mi>":[45,55,65,152,337],"stretchy=\"false\">)</mml:mo>":[47,154,222,339,395],"<mml:mo>=</mml:mo>":[48,254],"<mml:msup>":[49,54],"<mml:mn>3</mml:mn>":[51,56,106,263,266,392],"</mml:msup>":[52,57],"<mml:mo>+</mml:mo>":[53,58],"<mml:mn>1</mml:mn>":[59],"<mml:mo>\u2212<!--":[60],"\u2212":[61],"--></mml:mo>":[62,105,215,260,297],"</mml:mrow>":[66,107,155,217,223,265,268,299,340,388,396],"<mml:annotation":[67,108,156,170,224,269,300,341,359,397],"encoding=\"application/x-tex\">Q_k(x,y)":[68],"=":[69],"x^3+y^3+1-kxy</mml:annotation>":[70],"</mml:semantics>":[71,111,158,172,227,274,303,343,361,401],"</mml:math>":[72,112,159,173,228,275,304,344,362,402],"</inline-formula>":[73,160,229,305,345],"using":[74],"method":[76],"previously":[77],"developed":[78],"by":[79,231],"authors.":[81],"An":[82],"algorithm":[83],"is":[84],"implemented":[85],"to":[86,118,161],"search":[87],"for":[88],"complex":[89],"multiplication":[90],"points":[91,117],"with":[92,307],"class":[93],"numbers":[94],"alttext=\"less-than-or-slanted-equals":[99],"3\">":[100],"<mml:mo>\u2a7d<!--":[103],"\u2a7d":[104],"encoding=\"application/x-tex\">\\leqslant":[109],"3</mml:annotation>":[110],"</inline-formula>,":[113,276],"we":[114,277],"employ":[115],"these":[116],"derive":[119],"interesting":[120],"formulas":[121],"that":[122,282],"link":[123],"right-parenthesis\">":[141,205,326,381],"encoding=\"application/x-tex\">Q_k(x,y)</mml:annotation>":[157,342],"L\">":[167,356],"<mml:mi>L</mml:mi>":[169,358],"encoding=\"application/x-tex\">L</mml:annotation>":[171,360],"</inline-formula>-values":[174],"modular":[176],"forms.":[177],"As":[178],"by-products,":[179],"some":[180,347],"conjectural":[181],"identities":[182],"Samart":[184,232],"are":[185],"confirmed,":[186],"one":[187],"them":[189],"involves":[190],"modified":[192],"measure":[194],"alttext=\"ModifyingAbove":[199],"n":[200],"With":[201],"tilde":[202],"<mml:mrow":[208,384],"class=\"MJX-TeXAtom-ORD\">":[209,385],"<mml:mover>":[210],"<mml:mi>n</mml:mi>":[211],"stretchy=\"false\">~<!--":[213],"~":[214],"</mml:mover>":[216],"encoding=\"application/x-tex\">\\tilde":[225],"{n}(k)</mml:annotation>":[226],"introduced":[230],"recently.":[233],"For":[234],"alttext=\"k":[239],"NestedRootIndex":[241],"3":[242,248,379],"NestedStartRoot":[243],"729":[244],"plus-or-minus":[245],"405":[246],"StartRoot":[247,378],"EndRoot":[249,380],"NestedEndRoot\">":[250],"<mml:mroot>":[255],"<mml:mn>729</mml:mn>":[257],"<mml:mo>\u00b1<!--":[258],"\u00b1":[259],"<mml:mn>405</mml:mn>":[261],"<mml:msqrt>":[262,391],"</mml:msqrt>":[264,393],"</mml:mroot>":[267],"encoding=\"application/x-tex\">k=\\sqrt":[270],"[3]{729\\pm":[271],"405\\sqrt":[272],"{3}}</mml:annotation>":[273],"also":[278],"prove":[279],"an":[280],"equality":[281],"expresses":[283],"a":[284],"alttext=\"2":[289],"times":[290],"2\">":[291],"<mml:mn>2</mml:mn>":[294,298],"<mml:mo>\u00d7<!--":[295],"\u00d7":[296],"encoding=\"application/x-tex\">2\\times":[301],"2</mml:annotation>":[302],"determinant":[306],"entries":[308],"as":[346],"multiple":[348],"</inline-formula>-value":[363],"two":[365],"isogenous":[366],"elliptic":[367],"curves":[368],"over":[369],"alttext=\"double-struck":[374],"upper":[375],"<mml:mi":[386],"mathvariant=\"double-struck\">Q</mml:mi>":[387],"encoding=\"application/x-tex\">\\mathbb":[398],"{Q}(\\sqrt":[399],"{3})</mml:annotation>":[400],"</inline-formula>.":[403]},"counts_by_year":[],"updated_date":"2026-01-05T23:09:59.299951","created_date":"2025-10-10T00:00:00"}
