{"title":"Colored multizeta values in positive characteristic","authors":"Ryotaro Harada","doi":"10.1515/forum-2023-0226","DOIUrl":"https://doi.org/10.1515/forum-2023-0226","url":null,"abstract":"In 2004, Thakur introduced a positive characteristic analogue of multizeta values. Later, in 2017, he mentioned the two colored variants which are positive characteristic analogues of colored multizeta values in his survey of multizeta values in positive characteristic. In this paper, we study one of those two variants. We establish their fundamental properties, that include their non-vanishing, sum-shuffle relations, 𝑡-motivic interpretation and linear independence. For the linear independence results, we prove that there are no nontrivial <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mover accent=\"true\"> <m:mi>k</m:mi> <m:mo>̄</m:mo> </m:mover> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0226_ineq_0001.png\" /> <jats:tex-math>overline{k}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-linear relations among the colored multizeta values with different weights.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"54 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140585665","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A p-adic analog of Hasse--Davenport product relation involving ϵ-factors","authors":"Dani Szpruch","doi":"10.1515/forum-2023-0347","DOIUrl":"https://doi.org/10.1515/forum-2023-0347","url":null,"abstract":"In this paper we prove some generalizations of the classical Hasse–Davenport product relation for certain arithmetic factors defined on a <jats:italic>p</jats:italic>-adic field <jats:italic>F</jats:italic>, among them one finds the ϵ-factors appearing in Tate’s thesis. We then show that these generalizations are equivalent to some representation theoretic identities relating the determinant of ramified local coefficients matrices defined for coverings of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msub> <m:mi>SL</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>F</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0347_eq_0339.png\" /> <jats:tex-math>{mathrm{SL}_{2}(F)}</jats:tex-math> </jats:alternatives> </jats:inline-formula> to Plancherel measures and γ-factors.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"17 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140300120","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Some q-supercongruences from a q-analogue of Watson's 3 F 2 summation","authors":"Victor J. W. Guo","doi":"10.1515/forum-2023-0475","DOIUrl":"https://doi.org/10.1515/forum-2023-0475","url":null,"abstract":"We give some <jats:italic>q</jats:italic>-supercongruences from a <jats:italic>q</jats:italic>-analogue of Watson’s <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mmultiscripts> <m:mi>F</m:mi> <m:mn>2</m:mn> <m:none /> <m:mprescripts /> <m:mn>3</m:mn> <m:none /> </m:mmultiscripts> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0475_eq_0206.png\" /> <jats:tex-math>{{}_{3}F_{2}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> summation and the method of “creative microscoping”, introduced by the author and Zudilin. These <jats:italic>q</jats:italic>-supercongruences may be considered as further generalizations of the (A.2) supercongruence of Van Hamme modulo <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>p</m:mi> <m:mn>3</m:mn> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0475_eq_0181.png\" /> <jats:tex-math>{p^{3}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> or <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>p</m:mi> <m:mn>2</m:mn> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0475_eq_0180.png\" /> <jats:tex-math>{p^{2}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> for any odd prime <jats:italic>p</jats:italic>. Meanwhile, we confirm a supercongruence conjecture of Wang and Yue through establishing its <jats:italic>q</jats:italic>-analogue.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"233 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140300124","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Balesh Kumar, Murugesan Manickam, Karam Deo Shankhadhar
{"title":"Simultaneous nonvanishing of central L-values with large level","authors":"Balesh Kumar, Murugesan Manickam, Karam Deo Shankhadhar","doi":"10.1515/forum-2024-0014","DOIUrl":"https://doi.org/10.1515/forum-2024-0014","url":null,"abstract":"For a given normalized newform <jats:italic>f</jats:italic> of large prime level, we establish a lower bound with respect to the level for the number of normalized newforms <jats:italic>g</jats:italic> of the same weight and level as of <jats:italic>f</jats:italic> such that the central <jats:italic>L</jats:italic>-values of <jats:italic>f</jats:italic> and <jats:italic>g</jats:italic> both twisted by a quadratic character do not vanish.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"21 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140300118","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Arithmetic progression in a finite field with prescribed norms","authors":"Kaustav Chatterjee, Hariom Sharma, Aastha Shukla, Shailesh Kumar Tiwari","doi":"10.1515/forum-2024-0026","DOIUrl":"https://doi.org/10.1515/forum-2024-0026","url":null,"abstract":"Given a prime power <jats:italic>q</jats:italic> and a positive integer <jats:italic>n</jats:italic>, let <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>𝔽</m:mi> <m:msup> <m:mi>q</m:mi> <m:mi>n</m:mi> </m:msup> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2024-0026_eq_0435.png\" /> <jats:tex-math>{mathbb{F}_{q^{n}}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> represent a finite extension of degree <jats:italic>n</jats:italic> of the finite field <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>𝔽</m:mi> <m:mi>q</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2024-0026_eq_0722.png\" /> <jats:tex-math>{{mathbb{F}_{q}}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. In this article, we investigate the existence of <jats:italic>m</jats:italic> elements in arithmetic progression, where every element is primitive and at least one is normal with prescribed norms. Moreover, for <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>n</m:mi> <m:mo>≥</m:mo> <m:mn>6</m:mn> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2024-0026_eq_0646.png\" /> <jats:tex-math>{ngeq 6}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>q</m:mi> <m:mo>=</m:mo> <m:msup> <m:mn>3</m:mn> <m:mi>k</m:mi> </m:msup> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2024-0026_eq_0678.png\" /> <jats:tex-math>{q=3^{k}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>m</m:mi> <m:mo>=</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2024-0026_eq_0621.png\" /> <jats:tex-math>{m=2}</jats:tex-math> </jats:alternatives> </jats:inline-formula> we establish that there are only 10 possible exceptions.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"21 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140299673","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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