Alex Best, L. Betts, Theresa Kumpitsch, Martin Lüdtke, Angus McAndrew, Lie Qian, Elie Studnia, Yujie Xu
{"title":"Refined Selmer equations for the thrice-punctured line in depth two","authors":"Alex Best, L. Betts, Theresa Kumpitsch, Martin Lüdtke, Angus McAndrew, Lie Qian, Elie Studnia, Yujie Xu","doi":"10.1090/mcom/3898","DOIUrl":null,"url":null,"abstract":"Kim gave a new proof of Siegel’s Theorem that there are only finitely many <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper S\"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding=\"application/x-tex\">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-integral points on <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper P Subscript double-struck upper Z Superscript 1 Baseline minus StartSet 0 comma 1 comma normal infinity EndSet\"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">P</mml:mi> </mml:mrow> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">Z</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msubsup> <mml:mo class=\"MJX-variant\">∖<!-- ∖ --></mml:mo> <mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi> <mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\mathbb {P}^1_\\mathbb {Z}\\setminus \\{0,1,\\infty \\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. One advantage of Kim’s method is that it in principle allows one to actually find these points, but the calculations grow vastly more complicated as the size of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper S\"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding=\"application/x-tex\">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> increases. In this paper, we implement a refinement of Kim’s method to explicitly compute various examples where <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper S\"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding=\"application/x-tex\">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has size <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"2\"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding=\"application/x-tex\">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which has been introduced by Betts and Dogra. In so doing, we exhibit new examples of a natural generalization of a conjecture of Kim.","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"50 1","pages":"0"},"PeriodicalIF":2.2000,"publicationDate":"2023-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematics of Computation","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1090/mcom/3898","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 1
Abstract
Kim gave a new proof of Siegel’s Theorem that there are only finitely many SS-integral points on PZ1∖{0,1,∞}\mathbb {P}^1_\mathbb {Z}\setminus \{0,1,\infty \}. One advantage of Kim’s method is that it in principle allows one to actually find these points, but the calculations grow vastly more complicated as the size of SS increases. In this paper, we implement a refinement of Kim’s method to explicitly compute various examples where SS has size 22 which has been introduced by Betts and Dogra. In so doing, we exhibit new examples of a natural generalization of a conjecture of Kim.
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