Let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0027763023000399_inline1.png" /> <jats:tex-math> $Sigma $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> be a nonempty subset of the set of prime numbers which is either equal to the entire set of prime numbers or of cardinality one. In the present paper, we continue our study of the pro-<jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0027763023000399_inline2.png" /> <jats:tex-math> $Sigma $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> fundamental groups of hyperbolic curves and their associated configuration spaces over algebraically closed fields in which the primes of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0027763023000399_inline3.png" /> <jats:tex-math> $Sigma $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> are invertible. The present paper focuses on the topic of <jats:italic>comparison</jats:italic> between the theory developed in earlier papers concerning <jats:italic>pro-</jats:italic><jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0027763023000399_inline4.png" /> <jats:tex-math> $Sigma $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> fundamental groups and various <jats:italic>discrete</jats:italic> versions of this theory. We begin by developing a theory concerning certain combinatorial analogues of the <jats:italic>section conjecture</jats:italic> and <jats:italic>Grothendieck conjecture</jats:italic>. This portion of the theory is <jats:italic>purely combinatorial</jats:italic> and essentially follows from a result concerning the <jats:italic>existence of fixed points</jats:italic> of actions of finite groups on finite graphs (satisfying certain conditions). We then examine various applications of this purely combinatorial theory to <jats:italic>scheme theory</jats:italic>. Next, we verify various results in the theory of discrete fundamental groups of hyperbolic topological surfaces to the effect that various properties of <jats:italic>(discrete) subgroups</jats:italic> of such groups hold if and only if analogous properties hold for the closures of these subgroups in the <jats:italic>profinite completions</jats:italic> of the discrete fundamental groups under consideration. These results make possible a fairly <jats:italic>straightforward translation</jats:italic>, into <jats:italic>discrete versions</jats:italic>, of pro-<jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0027763023000399_inline5.png" /> <jats:tex-math> $Sigma $ </jats:tex-math> </jats:alternatives> </jats:inl
{"title":"TOPICS SURROUNDING THE COMBINATORIAL ANABELIAN GEOMETRY OF HYPERBOLIC CURVES IV: DISCRETENESS AND SECTIONS","authors":"YUICHIRO HOSHI, SHINICHI MOCHIZUKI","doi":"10.1017/nmj.2023.39","DOIUrl":"https://doi.org/10.1017/nmj.2023.39","url":null,"abstract":"Let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0027763023000399_inline1.png\" /> <jats:tex-math> $Sigma $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> be a nonempty subset of the set of prime numbers which is either equal to the entire set of prime numbers or of cardinality one. In the present paper, we continue our study of the pro-<jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0027763023000399_inline2.png\" /> <jats:tex-math> $Sigma $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> fundamental groups of hyperbolic curves and their associated configuration spaces over algebraically closed fields in which the primes of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0027763023000399_inline3.png\" /> <jats:tex-math> $Sigma $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> are invertible. The present paper focuses on the topic of <jats:italic>comparison</jats:italic> between the theory developed in earlier papers concerning <jats:italic>pro-</jats:italic><jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0027763023000399_inline4.png\" /> <jats:tex-math> $Sigma $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> fundamental groups and various <jats:italic>discrete</jats:italic> versions of this theory. We begin by developing a theory concerning certain combinatorial analogues of the <jats:italic>section conjecture</jats:italic> and <jats:italic>Grothendieck conjecture</jats:italic>. This portion of the theory is <jats:italic>purely combinatorial</jats:italic> and essentially follows from a result concerning the <jats:italic>existence of fixed points</jats:italic> of actions of finite groups on finite graphs (satisfying certain conditions). We then examine various applications of this purely combinatorial theory to <jats:italic>scheme theory</jats:italic>. Next, we verify various results in the theory of discrete fundamental groups of hyperbolic topological surfaces to the effect that various properties of <jats:italic>(discrete) subgroups</jats:italic> of such groups hold if and only if analogous properties hold for the closures of these subgroups in the <jats:italic>profinite completions</jats:italic> of the discrete fundamental groups under consideration. These results make possible a fairly <jats:italic>straightforward translation</jats:italic>, into <jats:italic>discrete versions</jats:italic>, of pro-<jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0027763023000399_inline5.png\" /> <jats:tex-math> $Sigma $ </jats:tex-math> </jats:alternatives> </jats:inl","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":"35 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139499121","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We extend the notion of y-variables (coefficients) in cluster algebras to cluster scattering diagrams (CSDs). Accordingly, we extend the dilogarithm identity associated with a period in a cluster pattern to the one associated with a loop in a CSD. We show that these identities are constructed from and reduced to trivial ones by applying the pentagon identity possibly infinitely many times.
我们将簇代数中 y 变量（系数）的概念扩展到簇散射图（CSD）。相应地，我们将与簇模式中的周期相关的稀对数特性扩展为与 CSD 中的环相关的稀对数特性。我们证明，通过应用可能无限次的五边形特征，这些特征可以构造并简化为微不足道的特征。
{"title":"DILOGARITHM IDENTITIES IN CLUSTER SCATTERING DIAGRAMS","authors":"TOMOKI NAKANISHI","doi":"10.1017/nmj.2023.15","DOIUrl":"https://doi.org/10.1017/nmj.2023.15","url":null,"abstract":"<p>We extend the notion of <span>y</span>-variables (coefficients) in cluster algebras to cluster scattering diagrams (CSDs). Accordingly, we extend the dilogarithm identity associated with a period in a cluster pattern to the one associated with a loop in a CSD. We show that these identities are constructed from and reduced to trivial ones by applying the pentagon identity possibly infinitely many times.</p>","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":"29 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138826458","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We investigate sections of the arithmetic fundamental group $pi _1(X)$ where X is either a smooth affinoid p-adic curve, or a formal germ of a p-adic curve, and prove that they can be lifted (unconditionally) to sections of cuspidally abelian Galois groups. As a consequence, if X admits a compactification Y, and the exact sequence of $pi _1(X)$splits, then $text {index} (Y)=1$. We also exhibit a necessary and sufficient condition for a section of $pi _1(X)$ to arise from a rational point of Y. One of the key ingredients in our investigation is the fact, we prove in this paper in case X is affinoid, that the Picard group of X is finite.
我们研究了算术基本群 $pi _1(X)$的截面，其中 X 是光滑的affinoid p-adic曲线，或者是 p-adic曲线的形式胚芽，并证明它们可以（无条件地）提升到簕杜鹃无边际伽罗瓦群的截面。因此，如果 X 允许一个紧凑化 Y，并且 $pi _1(X)$ 的精确序列分裂，那么 $text {index} (Y)=1$ 。我们还展示了 $pi _1(X)$ 的一个部分从 Y 的一个有理点产生的必要条件和充分条件。我们研究的一个关键因素是，我们在本文中证明了在 X 是affinoid的情况下，X 的 Picard 群是有限的。
{"title":"LOCAL SECTIONS OF ARITHMETIC FUNDAMENTAL GROUPS OF p-ADIC CURVES","authors":"MOHAMED SAÏDI","doi":"10.1017/nmj.2023.33","DOIUrl":"https://doi.org/10.1017/nmj.2023.33","url":null,"abstract":"<p>We investigate <span>sections</span> of the arithmetic fundamental group <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231219130139179-0193:S0027763023000338:S0027763023000338_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$pi _1(X)$</span></span></img></span></span> where <span>X</span> is either a <span>smooth affinoid p-adic curve</span>, or a <span>formal germ of a p-adic curve</span>, and prove that they can be lifted (unconditionally) to sections of cuspidally abelian Galois groups. As a consequence, if <span>X</span> admits a compactification <span>Y</span>, and the exact sequence of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231219130139179-0193:S0027763023000338:S0027763023000338_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$pi _1(X)$</span></span></img></span></span> <span>splits</span>, then <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231219130139179-0193:S0027763023000338:S0027763023000338_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$text {index} (Y)=1$</span></span></img></span></span>. We also exhibit a necessary and sufficient condition for a section of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231219130139179-0193:S0027763023000338:S0027763023000338_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$pi _1(X)$</span></span></img></span></span> to arise from a <span>rational point</span> of <span>Y</span>. One of the key ingredients in our investigation is the fact, we prove in this paper in case <span>X</span> is affinoid, that the Picard group of <span>X</span> is <span>finite</span>.</p>","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":"38 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138821352","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
<p>A quiver representation assigns a vector space to each vertex, and a linear map to each arrow of a quiver. When one considers the category <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231212132346299-0928:S0027763023000375:S0027763023000375_inline2.png"><span data-mathjax-type="texmath"><span>$mathrm {Vect}(mathbb {F}_1)$</span></span></img></span></span> of vector spaces “over <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231212132346299-0928:S0027763023000375:S0027763023000375_inline3.png"><span data-mathjax-type="texmath"><span>$mathbb {F}_1$</span></span></img></span></span>” (the field with one element), one obtains <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231212132346299-0928:S0027763023000375:S0027763023000375_inline4.png"><span data-mathjax-type="texmath"><span>$mathbb {F}_1$</span></span></img></span></span>-representations of a quiver. In this paper, we study representations of a quiver over the field with one element in connection to coefficient quivers. To be precise, we prove that the category <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231212132346299-0928:S0027763023000375:S0027763023000375_inline5.png"><span data-mathjax-type="texmath"><span>$mathrm {Rep}(Q,mathbb {F}_1)$</span></span></img></span></span> is equivalent to the (suitably defined) category of coefficient quivers over <span>Q</span>. This provides a conceptual way to see Euler characteristics of a class of quiver Grassmannians as the number of “<span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231212132346299-0928:S0027763023000375:S0027763023000375_inline6.png"><span data-mathjax-type="texmath"><span>$mathbb {F}_1$</span></span></img></span></span>-rational points” of quiver Grassmannians. We generalize techniques originally developed for string and band modules to compute the Euler characteristics of quiver Grassmannians associated with <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231212132346299-0928:S0027763023000375:S0027763023000375_inline7.png"><span data-mathjax-type="texmath"><span>$mathbb {F}_1$</span></span></img></span></span>-representations. These techniques apply to a large class of <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231212132346299-0928:S0027763023000375:S0027763023000375_inline8.png"><span data-mathjax-type="texmath"><span>$mathbb {F}_1$</span></span></img></span></span>-representations, which we call the <span><span><img
{"title":"COEFFICIENT QUIVERS, -REPRESENTATIONS, AND EULER CHARACTERISTICS OF QUIVER GRASSMANNIANS","authors":"JAIUNG JUN, ALEXANDER SISTKO","doi":"10.1017/nmj.2023.37","DOIUrl":"https://doi.org/10.1017/nmj.2023.37","url":null,"abstract":"<p>A quiver representation assigns a vector space to each vertex, and a linear map to each arrow of a quiver. When one considers the category <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231212132346299-0928:S0027763023000375:S0027763023000375_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$mathrm {Vect}(mathbb {F}_1)$</span></span></img></span></span> of vector spaces “over <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231212132346299-0928:S0027763023000375:S0027763023000375_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$mathbb {F}_1$</span></span></img></span></span>” (the field with one element), one obtains <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231212132346299-0928:S0027763023000375:S0027763023000375_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$mathbb {F}_1$</span></span></img></span></span>-representations of a quiver. In this paper, we study representations of a quiver over the field with one element in connection to coefficient quivers. To be precise, we prove that the category <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231212132346299-0928:S0027763023000375:S0027763023000375_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$mathrm {Rep}(Q,mathbb {F}_1)$</span></span></img></span></span> is equivalent to the (suitably defined) category of coefficient quivers over <span>Q</span>. This provides a conceptual way to see Euler characteristics of a class of quiver Grassmannians as the number of “<span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231212132346299-0928:S0027763023000375:S0027763023000375_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$mathbb {F}_1$</span></span></img></span></span>-rational points” of quiver Grassmannians. We generalize techniques originally developed for string and band modules to compute the Euler characteristics of quiver Grassmannians associated with <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231212132346299-0928:S0027763023000375:S0027763023000375_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$mathbb {F}_1$</span></span></img></span></span>-representations. These techniques apply to a large class of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231212132346299-0928:S0027763023000375:S0027763023000375_inline8.png\"><span data-mathjax-type=\"texmath\"><span>$mathbb {F}_1$</span></span></img></span></span>-representations, which we call the <span><span><img","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":"99 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138581377","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For homogeneous polynomials $G_1,ldots ,G_k$ over a finite field, their Dwork complex is defined by Adolphson and Sperber, based on Dwork’s theory. In this article, we will construct an explicit cochain map from the Dwork complex of $G_1,ldots ,G_k$ to the Monsky–Washnitzer complex associated with some affine bundle over the complement $mathbb {P}^nsetminus X_G$ of the common zero $X_G$ of $G_1,ldots ,G_k$ , which computes the rigid cohomology of $mathbb {P}^nsetminus X_G$ . We verify that this cochain map realizes the rigid cohomology of $mathbb {P}^nsetminus X_G$ as a direct summand of the Dwork cohomology of $G_1,ldots ,G_k$ . We also verify that the comparison map is compatible with the Frobenius and the Dwork operator defined on both complexes, respectively. Consequently, we extend Katz’s comparison results in [19] for projective hypersurface complements to arbitrary projective complements.
对于有限域上的齐次多项式$G_1，ldots,G_k$，它们的Dwork复形由Adolphson和Sperber根据Dwork理论定义。在本文中，我们将构造一个显式的协链映射，从$G_1，ldots,G_k$的Dwork复形到$G_1，ldots,G_k$的公零$X_G$的补$mathbb {P} n set- X_G$上与某个仿射束相关联的Monsky-Washnitzer复形，计算$mathbb {P} n set- X_G$的刚性上同调。我们证明了这个协链映射实现了$mathbb {P}^n set- X_G$的刚性上同调作为$G_1，ldots,G_k$的Dwork上同调的直接和。我们还验证了比较映射分别与两个复合体上定义的Frobenius算子和Dwork算子兼容。因此，我们将Katz在[19]中关于射影超曲面补的比较结果推广到任意射影补。
{"title":"ON A COMPARISON BETWEEN DWORK AND RIGID COHOMOLOGIES OF PROJECTIVE COMPLEMENTS","authors":"JUNYEONG PARK","doi":"10.1017/nmj.2023.32","DOIUrl":"https://doi.org/10.1017/nmj.2023.32","url":null,"abstract":"For homogeneous polynomials <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0027763023000326_inline1.png\" /> <jats:tex-math> $G_1,ldots ,G_k$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> over a finite field, their Dwork complex is defined by Adolphson and Sperber, based on Dwork’s theory. In this article, we will construct an explicit cochain map from the Dwork complex of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0027763023000326_inline2.png\" /> <jats:tex-math> $G_1,ldots ,G_k$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> to the Monsky–Washnitzer complex associated with some affine bundle over the complement <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0027763023000326_inline3.png\" /> <jats:tex-math> $mathbb {P}^nsetminus X_G$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> of the common zero <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0027763023000326_inline4.png\" /> <jats:tex-math> $X_G$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0027763023000326_inline5.png\" /> <jats:tex-math> $G_1,ldots ,G_k$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, which computes the rigid cohomology of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0027763023000326_inline6.png\" /> <jats:tex-math> $mathbb {P}^nsetminus X_G$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. We verify that this cochain map realizes the rigid cohomology of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0027763023000326_inline7.png\" /> <jats:tex-math> $mathbb {P}^nsetminus X_G$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> as a direct summand of the Dwork cohomology of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0027763023000326_inline8.png\" /> <jats:tex-math> $G_1,ldots ,G_k$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. We also verify that the comparison map is compatible with the Frobenius and the Dwork operator defined on both complexes, respectively. Consequently, we extend Katz’s comparison results in [19] for projective hypersurface complements to arbitrary projective complements.","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":"220 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138543526","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
There are several ways to convert a closure or interior operation to a different operation that has particular desirable properties. In this paper, we axiomatize three ways to do so, drawing on disparate examples from the literature, including tight closure, basically full closure, and various versions of integral closure. In doing so, we explore several such desirable properties, including hereditary, residual, and cofunctorial, and see how they interact with other properties such as the finitistic property.
{"title":"HOW TO EXTEND CLOSURE AND INTERIOR OPERATIONS TO MORE MODULES","authors":"NEIL EPSTEIN, REBECCA R. G., JANET VASSILEV","doi":"10.1017/nmj.2023.36","DOIUrl":"https://doi.org/10.1017/nmj.2023.36","url":null,"abstract":"There are several ways to convert a closure or interior operation to a different operation that has particular desirable properties. In this paper, we axiomatize three ways to do so, drawing on disparate examples from the literature, including tight closure, basically full closure, and various versions of integral closure. In doing so, we explore several such desirable properties, including <jats:italic>hereditary</jats:italic>, <jats:italic>residual</jats:italic>, and <jats:italic>cofunctorial</jats:italic>, and see how they interact with other properties such as the <jats:italic>finitistic</jats:italic> property.","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":"377 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138505276","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The problem of classifying elliptic curves over $mathbb Q$ with a given discriminant has received much attention. The analogous problem for genus $2$ curves has only been tackled when the absolute discriminant is a power of $2$ . In this article, we classify genus $2$ curves C defined over ${mathbb Q}$ with at least two rational Weierstrass points and whose absolute discriminant is an odd prime. In fact, we show that such a curve C must be isomorphic to a specialization of one of finitely many $1$ -parameter families of genus $2$ curves. In particular, we provide genus $2$ analogues to Neumann–Setzer families of elliptic curves over the rationals.
{"title":"GENUS CURVES WITH BAD REDUCTION AT ONE ODD PRIME","authors":"ANDRZEJ DĄBROWSKI, MOHAMMAD SADEK","doi":"10.1017/nmj.2023.35","DOIUrl":"https://doi.org/10.1017/nmj.2023.35","url":null,"abstract":"The problem of classifying elliptic curves over <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0027763023000351_inline2.png\" /> <jats:tex-math> $mathbb Q$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> with a given discriminant has received much attention. The analogous problem for genus <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0027763023000351_inline3.png\" /> <jats:tex-math> $2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> curves has only been tackled when the absolute discriminant is a power of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0027763023000351_inline4.png\" /> <jats:tex-math> $2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. In this article, we classify genus <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0027763023000351_inline5.png\" /> <jats:tex-math> $2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> curves <jats:italic>C</jats:italic> defined over <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0027763023000351_inline6.png\" /> <jats:tex-math> ${mathbb Q}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> with at least two rational Weierstrass points and whose absolute discriminant is an odd prime. In fact, we show that such a curve <jats:italic>C</jats:italic> must be isomorphic to a specialization of one of finitely many <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0027763023000351_inline7.png\" /> <jats:tex-math> $1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-parameter families of genus <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0027763023000351_inline8.png\" /> <jats:tex-math> $2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> curves. In particular, we provide genus <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0027763023000351_inline9.png\" /> <jats:tex-math> $2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> analogues to Neumann–Setzer families of elliptic curves over the rationals.","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":"13 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138539100","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ action button.
{"title":"NMJ volume 252 Cover and Back matter","authors":"","doi":"10.1017/nmj.2023.31","DOIUrl":"https://doi.org/10.1017/nmj.2023.31","url":null,"abstract":"An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ action button.","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":"82 11","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135863973","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
$pi _1(X)$ where X is either a smooth affinoid p-adic curve, or a formal germ of a p-adic curve, and prove that they can be lifted (unconditionally) to sections of cuspidally abelian Galois groups. As a consequence, if X admits a compactification Y, and the exact sequence of 
$pi _1(X)$ splits, then 
$text {index} (Y)=1$. We also exhibit a necessary and sufficient condition for a section of 
$pi _1(X)$ to arise from a rational point of Y. One of the key ingredients in our investigation is the fact, we prove in this paper in case X is affinoid, that the Picard group of X is finite.
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