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 k̄overline{k}-linear relations among the colored multizeta values with different weights.
2004 年，Thakur 提出了多奇塔值的正特征类似物。之后，在 2017 年，他在关于正特征多奇塔值的研究中提到了两个彩色变体，它们是彩色多奇塔值的正特征类似物。在本文中，我们将研究这两种变体中的一种。我们建立了它们的基本性质，包括它们的非凡性、和-洗牌关系、𝑡-动机解释和线性独立性。关于线性独立性结果，我们证明了在彩色多线{k}之间不存在非私密的 k overline{k} -线性关系。 -线性关系。
{"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}
Let 𝜅 be an inaccessible cardinal, 𝔘 a universal algebra, and <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mo>∼</m:mo> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2022-0375_ineq_0001.png" /> <jats:tex-math>sim</jats:tex-math> </jats:alternatives> </jats:inline-formula> the equivalence relation on <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi mathvariant="fraktur">U</m:mi> <m:mi>κ</m:mi> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2022-0375_ineq_0002.png" /> <jats:tex-math>mathfrak{U}^{kappa}</jats:tex-math> </jats:alternatives> </jats:inline-formula> of eventual equality. From mild assumptions on 𝜅, we give general constructions of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="script">E</m:mi> <m:mo>∈</m:mo> <m:mi>End</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msup> <m:mi mathvariant="fraktur">U</m:mi> <m:mi>κ</m:mi> </m:msup> <m:mo rspace="0em">/</m:mo> <m:mo lspace="0em" rspace="0em">∼</m:mo> <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-2022-0375_ineq_0003.png" /> <jats:tex-math>mathcal{E}inoperatorname{End}(mathfrak{U}^{kappa}/{sim})</jats:tex-math> </jats:alternatives> </jats:inline-formula> satisfying <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi mathvariant="script">E</m:mi> <m:mo lspace="0.222em" rspace="0.222em">∘</m:mo> <m:mi mathvariant="script">E</m:mi> </m:mrow> <m:mo>=</m:mo> <m:mi mathvariant="script">E</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2022-0375_ineq_0004.png" /> <jats:tex-math>mathcal{E}circmathcal{E}=mathcal{E}</jats:tex-math> </jats:alternatives> </jats:inline-formula> which do not descend from <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="normal">Δ</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mi>End</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msup> <m:mi mathvariant="fraktur">U</m:mi> <m:mi>κ</m:mi> </m:msup> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2022-0375_ineq_0005.png" /> <jats:tex-math>Deltainoperatorname{End}(mathfrak{U}^{kappa})</jats:tex-math> </jats:alternatives> </jats:inline-formula> having small strong supports. As an application, there exists an <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="script">E</m:mi> <m:mo>∈</m:mo> <m:mi>End</m:mi>
让 𝜅 是一个不可访问的红心，𝔘 是一个普遍代数，∼ sim 是 U κ mathfrak{U}^{kappa} 上最终相等的等价关系。根据对𝜅、我们给出了 E∈ End ( U κ / ∼ ) 的一般构造 mathcal{E}inoperatorname{End}(mathfrak{U}^{kappa}/{sim}) 满足 E ∘ E = E mathcal{E}circmathcal{E}=mathcal{E} 它不会从具有小强支持的 Δ∈ End ( U κ ) Deltainoperatorname{End}(mathfrak{U}^{kappa}) 下降。作为应用，存在一个 E∈ End ( Z κ / ∼ ) （ （mathcal{E}inoperatorname{End}(mathbb{Z}^{kappa}}/{sim})），它不是来自一个 Δ∈ End ( Z κ ) （ （Deltainoperatorname{End}(mathbb{Z}^{kappa})）。
{"title":"On projections of the tails of a power","authors":"Samuel M. Corson, Saharon Shelah","doi":"10.1515/forum-2022-0375","DOIUrl":"https://doi.org/10.1515/forum-2022-0375","url":null,"abstract":"Let 𝜅 be an inaccessible cardinal, 𝔘 a universal algebra, and <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mo>∼</m:mo> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2022-0375_ineq_0001.png\" /> <jats:tex-math>sim</jats:tex-math> </jats:alternatives> </jats:inline-formula> the equivalence relation on <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi mathvariant=\"fraktur\">U</m:mi> <m:mi>κ</m:mi> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2022-0375_ineq_0002.png\" /> <jats:tex-math>mathfrak{U}^{kappa}</jats:tex-math> </jats:alternatives> </jats:inline-formula> of eventual equality. From mild assumptions on 𝜅, we give general constructions of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi mathvariant=\"script\">E</m:mi> <m:mo>∈</m:mo> <m:mi>End</m:mi> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:msup> <m:mi mathvariant=\"fraktur\">U</m:mi> <m:mi>κ</m:mi> </m:msup> <m:mo rspace=\"0em\">/</m:mo> <m:mo lspace=\"0em\" rspace=\"0em\">∼</m:mo> <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-2022-0375_ineq_0003.png\" /> <jats:tex-math>mathcal{E}inoperatorname{End}(mathfrak{U}^{kappa}/{sim})</jats:tex-math> </jats:alternatives> </jats:inline-formula> satisfying <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mrow> <m:mi mathvariant=\"script\">E</m:mi> <m:mo lspace=\"0.222em\" rspace=\"0.222em\">∘</m:mo> <m:mi mathvariant=\"script\">E</m:mi> </m:mrow> <m:mo>=</m:mo> <m:mi mathvariant=\"script\">E</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2022-0375_ineq_0004.png\" /> <jats:tex-math>mathcal{E}circmathcal{E}=mathcal{E}</jats:tex-math> </jats:alternatives> </jats:inline-formula> which do not descend from <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi mathvariant=\"normal\">Δ</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mi>End</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:msup> <m:mi mathvariant=\"fraktur\">U</m:mi> <m:mi>κ</m:mi> </m:msup> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2022-0375_ineq_0005.png\" /> <jats:tex-math>Deltainoperatorname{End}(mathfrak{U}^{kappa})</jats:tex-math> </jats:alternatives> </jats:inline-formula> having small strong supports. As an application, there exists an <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi mathvariant=\"script\">E</m:mi> <m:mo>∈</m:mo> <m:mi>End</m:mi> ","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"45 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140585372","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}
Given a proper holomorphic surjective morphism f:X→Y{f:Xrightarrow Y} between compact Kähler manifolds, and a Nakano semipositive holomorphic vector bundle E on X, we prove Kollár-type vanishing theorems on cohomologies with coefficients in Rqf∗(ωX(E))⊗F{R^{q}f_{ast}(omega_{X}(E))otimes F}, where F is a k-positive vector bundle on Y. The main inputs in the proof are the deep results on the Nakano semipositivity of the higher direct images due to Berndtsson and Mourougane–Takayama, and an L2{L^{2}}-Dolbeault resolution of the higher direct image sheaf Rqf∗(ωX(E)){R^{q}f_{ast}(omega_{X}(E))}, which is of interest in itself.
给定紧凑 Kähler 流形之间的适当全态投射态 f : X → Y {f:Xrightarrow Y} 和 X 上的中野半正全态向量束 E，我们证明了 R q f ∗ ( ω X ( E ) ) 中系数的同调上的 Kollár 型消失定理。 ⊗ F {R^{q}f_{ast}(omega_{X}(E))otimes F} 。 证明的主要输入是 Berndtsson 和 Mourougane-Takayama 关于高直达像的中野半实在性的深入结果，以及一个 L 2 {L^{2}} -Dolbeault 解析。 高直映像 Sheaf R q f ∗ ( ω X ( E ) ) 的 L 2 {L^{2}} -Dolbeault 解析。 {R^{q}f_{ast}(omega_{X}(E))} {R^{q}f_{ast}(omega_{X}(E))} ，这本身就很有趣。
{"title":"A Kollár-type vanishing theorem for k-positive vector bundles","authors":"Chen Zhao","doi":"10.1515/forum-2023-0332","DOIUrl":"https://doi.org/10.1515/forum-2023-0332","url":null,"abstract":"Given a proper holomorphic surjective morphism <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>f</m:mi> <m:mo>:</m:mo> <m:mrow> <m:mi>X</m:mi> <m:mo>→</m:mo> <m:mi>Y</m:mi> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0332_eq_0235.png\" /> <jats:tex-math>{f:Xrightarrow Y}</jats:tex-math> </jats:alternatives> </jats:inline-formula> between compact Kähler manifolds, and a Nakano semipositive holomorphic vector bundle <jats:italic>E</jats:italic> on <jats:italic>X</jats:italic>, we prove Kollár-type vanishing theorems on cohomologies with coefficients in <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mrow> <m:msup> <m:mi>R</m:mi> <m:mi>q</m:mi> </m:msup> <m:mo></m:mo> <m:msub> <m:mi>f</m:mi> <m:mo>∗</m:mo> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:msub> <m:mi>ω</m:mi> <m:mi>X</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>E</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mo>⊗</m:mo> <m:mi>F</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0332_eq_0130.png\" /> <jats:tex-math>{R^{q}f_{ast}(omega_{X}(E))otimes F}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, where <jats:italic>F</jats:italic> is a <jats:italic>k</jats:italic>-positive vector bundle on <jats:italic>Y</jats:italic>. The main inputs in the proof are the deep results on the Nakano semipositivity of the higher direct images due to Berndtsson and Mourougane–Takayama, and an <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>L</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-2023-0332_eq_0114.png\" /> <jats:tex-math>{L^{2}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-Dolbeault resolution of the higher direct image sheaf <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msup> <m:mi>R</m:mi> <m:mi>q</m:mi> </m:msup> <m:mo></m:mo> <m:msub> <m:mi>f</m:mi> <m:mo>∗</m:mo> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:msub> <m:mi>ω</m:mi> <m:mi>X</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>E</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <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-2023-0332_eq_0132.png\" /> <jats:tex-math>{R^{q}f_{ast}(omega_{X}(E))}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, which is of interest in itself.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"18 71 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140300115","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}
In this paper we prove some generalizations of the classical Hasse–Davenport product relation for certain arithmetic factors defined on a p-adic field F, 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 SL2(F){mathrm{SL}_{2}(F)} to Plancherel measures and γ-factors.
在本文中，我们证明了经典的哈塞-达文波特乘积关系对于定义在 p-adic 场 F 上的某些算术因子的一些泛化，其中包括塔特论文中出现的 ϵ 因子。然后，我们证明这些广义等价于一些表示论的同义词，这些同义词涉及为 SL 2 ( F ) {mathrm{SL}_{2}(F)} 的覆盖而定义的斜线化局部系数矩阵的行列式与 Plancherel 度量和 γ 因子。
{"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}
We give some q-supercongruences from a q-analogue of Watson’s F23{{}_{3}F_{2}} summation and the method of “creative microscoping”, introduced by the author and Zudilin. These q-supercongruences may be considered as further generalizations of the (A.2) supercongruence of Van Hamme modulo p3{p^{3}} or p2{p^{2}} for any odd prime p. Meanwhile, we confirm a supercongruence conjecture of Wang and Yue through establishing its q-analogue.
我们从沃森的 F 2 3 {{}_{3}F_{2}} 求和的 q-analogue 以及作者和祖迪林提出的 "创造性微观 "方法中给出了一些 q-supercongruences 。这些 q 超共形可以看作是凡-哈姆（Van Hamme）对任意奇素数 p 的 p 3 {p^{3}} 或 p 2 {p^{2}} 模的 (A.2) 超共形的进一步推广。
{"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
For a given normalized newform f of large prime level, we establish a lower bound with respect to the level for the number of normalized newforms g of the same weight and level as of f such that the central L-values of f and g both twisted by a quadratic character do not vanish.
对于给定的大素数级的规范化新形 f，我们为与 f 具有相同权重和级数的规范化新形 g 的数量建立了一个与级数相关的下限，使得 f 和 g 的中心 L 值都不会因二次方特征的扭曲而消失。
{"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}
Given a prime power q and a positive integer n, let 𝔽qn{mathbb{F}_{q^{n}}} represent a finite extension of degree n of the finite field 𝔽q{{mathbb{F}_{q}}}. In this article, we investigate the existence of m elements in arithmetic progression, where every element is primitive and at least one is normal with prescribed norms. Moreover, for n≥6{ngeq 6}, q=3k{q=3^{k}}, m=2{m=2} we establish that there are only 10 possible exceptions.
给定一个质幂 q 和一个正整数 n，让 𝔽 q n {{mathbb{F}_{q^{n}}} 表示有限域 𝔽 q {{mathbb{F}_{q}}} 的 n 阶有限扩展。本文将研究算术级数中是否存在 m 个元素，其中每个元素都是基元，且至少有一个元素是具有规定规范的正则元素。此外，对于 n ≥ 6 {ngeq 6} , q = 3 k {q=3^{k}} , m = 2 {m=2 , m = 2 {m=2}，我们可以确定只有 10 个可能的例外。
{"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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