Pub Date : 2022-01-01DOI: 10.4310/cjm.2022.v10.n2.a1
Yong Hu, Haoran Wang
{"title":"On the $operatorname{mod}p$ cohomology for $mathrm{GL}_2$: the non-semisimple case","authors":"Yong Hu, Haoran Wang","doi":"10.4310/cjm.2022.v10.n2.a1","DOIUrl":"https://doi.org/10.4310/cjm.2022.v10.n2.a1","url":null,"abstract":"","PeriodicalId":48573,"journal":{"name":"Cambridge Journal of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70404577","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}
Pub Date : 2021-10-29DOI: 10.4310/cjm.2021.v9.n4.a3
Jianbin Guo, Changshou Lin, Yifan Yang
Let H = H∪Q∪{∞}, where H is the complex upper half-plane, and Q(z) be a meromorphic modular form of weight 4 on SL(2,Z) such that the differential equation L : y(z) = Q(z)y(z) is Fuchsian on H. In this paper, we consider the problem when L is apparent on H, i.e., the ratio of any two nonzero solutions of L is single-valued and meromorphic on H. Such a modular differential equation is closely related to the existence of a conformal metric ds = eu|dz|2 on H with curvature 1/2 that is invariant under z 7→ γ · z for all γ ∈ SL(2,Z). Let ±κ∞ be the local exponents of L at ∞. In the case κ∞ ∈ 1 2 Z, we obtain the following results: (a) a complete characterization of Q(z) such that L is apparent on H with only one singularity (up to SL(2,Z)-equivalence) at i = √ −1 or ρ = (1 + √ 3i)/2, and (b) a complete characterization of Q(z) such that L is apparent on H with singularities only at i and ρ. We provide two proofs of the results, one using Riemann’s existence theorem and the other using Eremenko’s theorem on the existence of conformal metric on the sphere. In the case κ∞ / ∈ 1 2 Z, we let r∞ ∈ (0, 1/2) be defined by r∞ ≡ ±κ∞ mod 1. Assume that r∞ / ∈ {1/12, 5/12}. A special case of an earlier result of Eremenko and Tarasov says that 1/12 < r∞ < 5/12 is the necessary and sufficient condition for the existence of the invariant metric. The threshold case r∞ ∈ {1/12, 5/12} is more delicate. We show that in the threshold case, an invariant metric exists if and only if L has two linearly independent solutions whose squares are meromorphic modular forms of weight −2 with a pair of conjugate characters on SL(2,Z). In the non-existence case, our example shows that the monodromy data of L are related to periods of the elliptic curve y = x − 1728.
{"title":"Metrics with Positive constant curvature and modular differential equations","authors":"Jianbin Guo, Changshou Lin, Yifan Yang","doi":"10.4310/cjm.2021.v9.n4.a3","DOIUrl":"https://doi.org/10.4310/cjm.2021.v9.n4.a3","url":null,"abstract":"Let H = H∪Q∪{∞}, where H is the complex upper half-plane, and Q(z) be a meromorphic modular form of weight 4 on SL(2,Z) such that the differential equation L : y(z) = Q(z)y(z) is Fuchsian on H. In this paper, we consider the problem when L is apparent on H, i.e., the ratio of any two nonzero solutions of L is single-valued and meromorphic on H. Such a modular differential equation is closely related to the existence of a conformal metric ds = eu|dz|2 on H with curvature 1/2 that is invariant under z 7→ γ · z for all γ ∈ SL(2,Z). Let ±κ∞ be the local exponents of L at ∞. In the case κ∞ ∈ 1 2 Z, we obtain the following results: (a) a complete characterization of Q(z) such that L is apparent on H with only one singularity (up to SL(2,Z)-equivalence) at i = √ −1 or ρ = (1 + √ 3i)/2, and (b) a complete characterization of Q(z) such that L is apparent on H with singularities only at i and ρ. We provide two proofs of the results, one using Riemann’s existence theorem and the other using Eremenko’s theorem on the existence of conformal metric on the sphere. In the case κ∞ / ∈ 1 2 Z, we let r∞ ∈ (0, 1/2) be defined by r∞ ≡ ±κ∞ mod 1. Assume that r∞ / ∈ {1/12, 5/12}. A special case of an earlier result of Eremenko and Tarasov says that 1/12 < r∞ < 5/12 is the necessary and sufficient condition for the existence of the invariant metric. The threshold case r∞ ∈ {1/12, 5/12} is more delicate. We show that in the threshold case, an invariant metric exists if and only if L has two linearly independent solutions whose squares are meromorphic modular forms of weight −2 with a pair of conjugate characters on SL(2,Z). In the non-existence case, our example shows that the monodromy data of L are related to periods of the elliptic curve y = x − 1728.","PeriodicalId":48573,"journal":{"name":"Cambridge Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2021-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46066949","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}
Pub Date : 2021-06-28DOI: 10.4310/cjm.2023.v11.n2.a3
B. Bhatt, P. Scholze
Let $K$ be a complete discretely valued field of mixed characteristic $(0,p)$ with perfect residue field. We prove that the category of prismatic $F$-crystals on $mathcal O_K$ is equivalent to the category of lattices in crystalline $G_K$-representations.
{"title":"Prismatic $F$-crystals and crystalline Galois representations","authors":"B. Bhatt, P. Scholze","doi":"10.4310/cjm.2023.v11.n2.a3","DOIUrl":"https://doi.org/10.4310/cjm.2023.v11.n2.a3","url":null,"abstract":"Let $K$ be a complete discretely valued field of mixed characteristic $(0,p)$ with perfect residue field. We prove that the category of prismatic $F$-crystals on $mathcal O_K$ is equivalent to the category of lattices in crystalline $G_K$-representations.","PeriodicalId":48573,"journal":{"name":"Cambridge Journal of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2021-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41511790","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}
Pub Date : 2021-05-28DOI: 10.4310/cjm.2023.v11.n3.a1
K. Choi, Robert Haslhofer, Or Hershkovits
In this paper, we classify all noncollapsed singularity models for the mean curvature flow of 3-dimensional hypersurfaces in $mathbb{R}^4$ or more generally in $4$-manifolds. Specifically, we prove that every noncollapsed translating hypersurface in $mathbb{R}^4$ is either $mathbb{R}times$2d-bowl, or a 3d round bowl, or belongs to the one-parameter family of 3d oval bowls constructed by Hoffman-Ilmanen-Martin-White.
{"title":"Classification of noncollapsed translators in $mathbb{R}^4$","authors":"K. Choi, Robert Haslhofer, Or Hershkovits","doi":"10.4310/cjm.2023.v11.n3.a1","DOIUrl":"https://doi.org/10.4310/cjm.2023.v11.n3.a1","url":null,"abstract":"In this paper, we classify all noncollapsed singularity models for the mean curvature flow of 3-dimensional hypersurfaces in $mathbb{R}^4$ or more generally in $4$-manifolds. Specifically, we prove that every noncollapsed translating hypersurface in $mathbb{R}^4$ is either $mathbb{R}times$2d-bowl, or a 3d round bowl, or belongs to the one-parameter family of 3d oval bowls constructed by Hoffman-Ilmanen-Martin-White.","PeriodicalId":48573,"journal":{"name":"Cambridge Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2021-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45681462","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}
Pub Date : 2021-03-29DOI: 10.4310/cjm.2022.v10.n1.a1
Z. An, Lan-Hsuan Huang
We prove the existence and local uniqueness of asymptotically flat, static vacuum metrics with arbitrarily prescribed Bartnik boundary data that are close to the induced boundary data on any star-shaped hypersurface or a general family of perturbed hypersurfaces in the Euclidean space. It confirms the existence part of the Bartnik static extension conjecture for large classes of boundary data, and the static vacuum metric obtained is geometrically unique in a neighborhood of the Euclidean metric.
{"title":"Existence of static vacuum extensions with prescribed Bartnik boundary data","authors":"Z. An, Lan-Hsuan Huang","doi":"10.4310/cjm.2022.v10.n1.a1","DOIUrl":"https://doi.org/10.4310/cjm.2022.v10.n1.a1","url":null,"abstract":"We prove the existence and local uniqueness of asymptotically flat, static vacuum metrics with arbitrarily prescribed Bartnik boundary data that are close to the induced boundary data on any star-shaped hypersurface or a general family of perturbed hypersurfaces in the Euclidean space. It confirms the existence part of the Bartnik static extension conjecture for large classes of boundary data, and the static vacuum metric obtained is geometrically unique in a neighborhood of the Euclidean metric.","PeriodicalId":48573,"journal":{"name":"Cambridge Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2021-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44738640","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}
Pub Date : 2021-03-03DOI: 10.4310/CJM.2023.v11.n3.a2
Simion Filip, Valentino Tosatti
We construct canonical positive currents and heights on the boundary of the ample cone of a K3 surface. These are equivariant for the automorphism group and fit together into a continuous family, defined over an enlarged boundary of the ample cone. Along the way, we construct preferred representatives for certain height functions and currents on elliptically fibered surfaces.
{"title":"Canonical currents and heights for K3 surfaces","authors":"Simion Filip, Valentino Tosatti","doi":"10.4310/CJM.2023.v11.n3.a2","DOIUrl":"https://doi.org/10.4310/CJM.2023.v11.n3.a2","url":null,"abstract":"We construct canonical positive currents and heights on the boundary of the ample cone of a K3 surface. These are equivariant for the automorphism group and fit together into a continuous family, defined over an enlarged boundary of the ample cone. Along the way, we construct preferred representatives for certain height functions and currents on elliptically fibered surfaces.","PeriodicalId":48573,"journal":{"name":"Cambridge Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2021-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43971633","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}
Pub Date : 2021-01-17DOI: 10.4310/cjm.2022.v10.n3.a1
P. Colmez, Gabriel Dospinescu, Wiesława Nizioł
Cohomology of affinoids does not behave well; often, this can be remedied by making affinoids overconvergent. In this paper, we focus on dimension 1 and compute, using analogs of pants decompositions of Riemann surfaces, various cohomologies of affinoids. To give a meaning to these decompositions we modify slightly the notion of $p$-adic formal scheme, which gives rise to the adoc (an interpolation between adic and ad hoc) geometry. It turns out that cohomology of affinoids (in dimension 1) is not that pathological. From this we deduce a computation of cohomologies of curves without boundary (like the Drinfeld half-plane and its coverings). In particular, we obtain a description of their $p$-adic pro-'etale cohomology in terms of de the Rham complex and the Hyodo-Kato cohomology, the later having properties similar to the ones of $ell$-adic pro-'etale cohomology, for $ellneq p$.
仿射的上同性表现不佳;通常，这可以通过使仿射过度收敛来补救。在本文中，我们将重点放在维1上，并使用类似于黎曼曲面的分解，计算各种仿射的上同调。为了给这些分解赋予意义，我们稍微修改了$p$-adic形式方案的概念，这就产生了adoc(在adic和ad hoc之间的插值)几何。事实证明，仿射的上同调(在1维)并不是那么病态。由此我们推导出无边界曲线(如德林菲尔德半平面及其覆盖)的上同调的计算。特别地，我们用Rham复形和Hyodo-Kato上同调来描述它们的$p$-adic上同调，后者对于$ well neq p$具有类似于$ well $-adic上同调的性质。
{"title":"Cohomologie des courbes analytiques $p$-adiques","authors":"P. Colmez, Gabriel Dospinescu, Wiesława Nizioł","doi":"10.4310/cjm.2022.v10.n3.a1","DOIUrl":"https://doi.org/10.4310/cjm.2022.v10.n3.a1","url":null,"abstract":"Cohomology of affinoids does not behave well; often, this can be remedied by making affinoids overconvergent. In this paper, we focus on dimension 1 and compute, using analogs of pants decompositions of Riemann surfaces, various cohomologies of affinoids. To give a meaning to these decompositions we modify slightly the notion of $p$-adic formal scheme, which gives rise to the adoc (an interpolation between adic and ad hoc) geometry. It turns out that cohomology of affinoids (in dimension 1) is not that pathological. From this we deduce a computation of cohomologies of curves without boundary (like the Drinfeld half-plane and its coverings). In particular, we obtain a description of their $p$-adic pro-'etale cohomology in terms of de the Rham complex and the Hyodo-Kato cohomology, the later having properties similar to the ones of $ell$-adic pro-'etale cohomology, for $ellneq p$.","PeriodicalId":48573,"journal":{"name":"Cambridge Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2021-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47704490","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}
Pub Date : 2021-01-01DOI: 10.4310/cjm.2021.v9.n2.a1
Shuang Miao, S. Shahshahani, Sijue Wu
{"title":"Well-posedness of free boundary hard phase fluids in Minkowski background and their Newtonian limit","authors":"Shuang Miao, S. Shahshahani, Sijue Wu","doi":"10.4310/cjm.2021.v9.n2.a1","DOIUrl":"https://doi.org/10.4310/cjm.2021.v9.n2.a1","url":null,"abstract":"","PeriodicalId":48573,"journal":{"name":"Cambridge Journal of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70404515","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}
Pub Date : 2021-01-01DOI: 10.4310/cjm.2021.v9.n3.a2
N. M. Katz, P. Tiep
We construct hypergeometric sheaves whose geometric monodromy groups are the finite symplectic groups Sp2n(q) for any odd n ≥ 3, for q any power of an odd prime p. We construct other hypergeometric sheaves whose geometric monodromy groups are the finite unitary groups GUn(q), for any even n ≥ 2, for q any power of any prime p. Suitable Kummer pullbacks of these sheaves yield local systems on A, whose geometric monodromy groups are Sp2n(q), respectively SUn(q), in their total Weil representation of degree q, and whose trace functions are simple-to-remember one-parameter families of two-variable exponential sums. The main novelty of this paper is two-fold. First, it treats unitary groups GUn(q) with n even via hypergeometric sheaves for the first time. Second, in both the symplectic and the unitary cases, it uses a maximal torus which is a product of two sub-tori to furnish a generator of local monodromy at 0.
{"title":"Hypergeometric sheaves and finite symplectic and unitary groups","authors":"N. M. Katz, P. Tiep","doi":"10.4310/cjm.2021.v9.n3.a2","DOIUrl":"https://doi.org/10.4310/cjm.2021.v9.n3.a2","url":null,"abstract":"We construct hypergeometric sheaves whose geometric monodromy groups are the finite symplectic groups Sp2n(q) for any odd n ≥ 3, for q any power of an odd prime p. We construct other hypergeometric sheaves whose geometric monodromy groups are the finite unitary groups GUn(q), for any even n ≥ 2, for q any power of any prime p. Suitable Kummer pullbacks of these sheaves yield local systems on A, whose geometric monodromy groups are Sp2n(q), respectively SUn(q), in their total Weil representation of degree q, and whose trace functions are simple-to-remember one-parameter families of two-variable exponential sums. The main novelty of this paper is two-fold. First, it treats unitary groups GUn(q) with n even via hypergeometric sheaves for the first time. Second, in both the symplectic and the unitary cases, it uses a maximal torus which is a product of two sub-tori to furnish a generator of local monodromy at 0.","PeriodicalId":48573,"journal":{"name":"Cambridge Journal of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70404570","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}
Pub Date : 2020-12-24DOI: 10.4310/cjm.2021.v9.n4.a2
Tristan Ozuch
We find new obstructions to the desingularization of compact Einstein orbifolds by smooth Einstein metrics. These new obstructions, specific to the compact situation, raise the question of whether a compact Einstein 4-orbifold which is limit of Einstein metrics bubbling out Eguchi-Hanson metrics has to be Kähler. We then test these obstructions to discuss if it is possible to produce a Ricci-flat but not Kähler metric by the most promising desingularization configuration proposed by Page in 1981. We identify 84 obstructions which, once compared to the 57 degrees of freedom, indicate that almost all flat orbifold metrics on T/Z2 should not be limit of Ricci-flat metrics with generic holonomy while bubbling out Eguchi-Hanson metrics. Perhaps surprisingly, in the most symmetric situation, we also identify a 14-dimensional family of desingularizations satisfying all of our 84 obstructions.
{"title":"Higher order obstructions to the desingularization of Einstein metrics","authors":"Tristan Ozuch","doi":"10.4310/cjm.2021.v9.n4.a2","DOIUrl":"https://doi.org/10.4310/cjm.2021.v9.n4.a2","url":null,"abstract":"We find new obstructions to the desingularization of compact Einstein orbifolds by smooth Einstein metrics. These new obstructions, specific to the compact situation, raise the question of whether a compact Einstein 4-orbifold which is limit of Einstein metrics bubbling out Eguchi-Hanson metrics has to be Kähler. We then test these obstructions to discuss if it is possible to produce a Ricci-flat but not Kähler metric by the most promising desingularization configuration proposed by Page in 1981. We identify 84 obstructions which, once compared to the 57 degrees of freedom, indicate that almost all flat orbifold metrics on T/Z2 should not be limit of Ricci-flat metrics with generic holonomy while bubbling out Eguchi-Hanson metrics. Perhaps surprisingly, in the most symmetric situation, we also identify a 14-dimensional family of desingularizations satisfying all of our 84 obstructions.","PeriodicalId":48573,"journal":{"name":"Cambridge Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2020-12-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46606552","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：