Pub Date : 2020-10-13DOI: 10.2140/tunis.2022.4.249
G. Henniart, Marie-France Vign'eras
We investigate the irreducible cuspidal $C$-representations of a reductive $p$-adic group $G$ over a field $C$ of characteristic different from $p$. When $C$ is algebraically closed, for many groups $G$, a list of cuspidal $C$-types $(J,lambda)$ has been produced satisfying exhaustion, sometimes for a restricted kind of cuspidal representations, and often unicity. We verify that those lists verify Aut($C$)-stability and we produce similar lists when $C$ is no longer assumed algebraically closed. Our other main results concern supercuspidality. This notion makes sense for the representations $lambda$ in the cuspidal $C$-types $(J,lambda)$ as above, which involve finite reductive groups. We check that an irreducible cuspidal representation of $G$ induced from $lambda$ is supercuspidal if and only $lambda$ is supercuspidal.
{"title":"Representations of a reductive p-adic group in\u0000characteristic distinct from p","authors":"G. Henniart, Marie-France Vign'eras","doi":"10.2140/tunis.2022.4.249","DOIUrl":"https://doi.org/10.2140/tunis.2022.4.249","url":null,"abstract":"We investigate the irreducible cuspidal $C$-representations of a reductive $p$-adic group $G$ over a field $C$ of characteristic different from $p$. When $C$ is algebraically closed, for many groups $G$, a list of cuspidal $C$-types $(J,lambda)$ has been produced satisfying exhaustion, sometimes for a restricted kind of cuspidal representations, and often unicity. We verify that those lists verify Aut($C$)-stability and we produce similar lists when $C$ is no longer assumed algebraically closed. Our other main results concern supercuspidality. This notion makes sense for the representations $lambda$ in the cuspidal $C$-types $(J,lambda)$ as above, which involve finite reductive groups. We check that an irreducible cuspidal representation of $G$ induced from $lambda$ is supercuspidal if and only $lambda$ is supercuspidal.","PeriodicalId":36030,"journal":{"name":"Tunisian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48210429","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We introduce twisted differential calculus of negative level and prove a descent theorem: Frobenius pullback provides an equivalence between finitely presented modules endowed with a topologically quasi-nilpotent twisted connection of level minus one and those of level zero. We explain how this is related to the existence of a Cartier operator on prismatic crystals. For the sake of readability, we limit ourselves to the case of dimension one.
{"title":"Twisted differential operators of negative level and prismatic crystals","authors":"Michel Gros, B. L. Stum, Adolfo Quir'os","doi":"10.2140/tunis.2022.4.19","DOIUrl":"https://doi.org/10.2140/tunis.2022.4.19","url":null,"abstract":"We introduce twisted differential calculus of negative level and prove a descent theorem: Frobenius pullback provides an equivalence between finitely presented modules endowed with a topologically quasi-nilpotent twisted connection of level minus one and those of level zero. We explain how this is related to the existence of a Cartier operator on prismatic crystals. For the sake of readability, we limit ourselves to the case of dimension one.","PeriodicalId":36030,"journal":{"name":"Tunisian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42763459","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-06-19DOI: 10.2140/tunis.2021.3.801
E. L. Page, M. Peign'e, C. Pham
Let (Z n) n≥0 with Z n = (Z n (i, j)) 1≤i,j≤p be a p multi-type critical branching process in random environment, and let M n be the expectation of Z n given a fixed environment. We prove theorems on convergence in distribution of sequences of branching processes Zn |Mn| /|Z n | > 0 and ln Zn √ n /|Z n | > 0. These theorems extend similar results for single-type critical branching process in random environment.
让Z Z (n) n≥0和n = (Z (i, j) 1≤i, j≤p be a p multi-type连接branching的过程中随机环境,期待》,让M n是Z a n赐予固定环境。我们在distribution of sequences of证明theorems on集的branching processes Zn | Mn | n - Z | | > 0和ln Zn√n - Z | | > 0。这些版本也保留了类似于在随机环境中克隆过程的特性。
{"title":"Central limit theorem for a critical multitype branching process in random environments","authors":"E. L. Page, M. Peign'e, C. Pham","doi":"10.2140/tunis.2021.3.801","DOIUrl":"https://doi.org/10.2140/tunis.2021.3.801","url":null,"abstract":"Let (Z n) n≥0 with Z n = (Z n (i, j)) 1≤i,j≤p be a p multi-type critical branching process in random environment, and let M n be the expectation of Z n given a fixed environment. We prove theorems on convergence in distribution of sequences of branching processes Zn |Mn| /|Z n | > 0 and ln Zn √ n /|Z n | > 0. These theorems extend similar results for single-type critical branching process in random environment.","PeriodicalId":36030,"journal":{"name":"Tunisian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46582637","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove a strengthened sector lemma for irreducible, finite-dimensional, locally finite, essential, cocompact CAT(0) cube complexes under the additional hypothesis that the complex is emph{hyperplane-essential}; we prove that every quarterspace contains a halfspace. In aid of this, we present simplified proofs of known results about loxodromic isometries of the contact graph, avoiding the use of disc diagrams. This paper has an expository element; in particular, we collect results about cube complexes proved by combining Ramsey's theorem and Dilworth's theorem. We illustrate the use of these tricks with a discussion of the Tits alternative for cubical groups, and ask some questions about ``quantifying'' statements related to rank-rigidity and the Tits alternative.
{"title":"Large facing tuples and a strengthened sector lemma","authors":"M. Hagen","doi":"10.2140/tunis.2022.4.55","DOIUrl":"https://doi.org/10.2140/tunis.2022.4.55","url":null,"abstract":"We prove a strengthened sector lemma for irreducible, finite-dimensional, locally finite, essential, cocompact CAT(0) cube complexes under the additional hypothesis that the complex is emph{hyperplane-essential}; we prove that every quarterspace contains a halfspace. In aid of this, we present simplified proofs of known results about loxodromic isometries of the contact graph, avoiding the use of disc diagrams. \u0000This paper has an expository element; in particular, we collect results about cube complexes proved by combining Ramsey's theorem and Dilworth's theorem. We illustrate the use of these tricks with a discussion of the Tits alternative for cubical groups, and ask some questions about ``quantifying'' statements related to rank-rigidity and the Tits alternative.","PeriodicalId":36030,"journal":{"name":"Tunisian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46419276","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-01-01DOI: 10.2140/TUNIS.2020.2.217
M. Igarashi
{"title":"Duality relations among multiple series with three parameters","authors":"M. Igarashi","doi":"10.2140/TUNIS.2020.2.217","DOIUrl":"https://doi.org/10.2140/TUNIS.2020.2.217","url":null,"abstract":"","PeriodicalId":36030,"journal":{"name":"Tunisian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/TUNIS.2020.2.217","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45562989","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-01-01DOI: 10.2140/tunis.2020.2.535
D. Bakry, L. Mbarki
We provide a simplified proof of the hypergroup property for the Jacobi polynomials on the unit interval, and investigate various generalizations of this property for the family of Dirichlet laws on the simplex.
{"title":"The Markov sequence problem for the Jacobi polynomials and on the simplex","authors":"D. Bakry, L. Mbarki","doi":"10.2140/tunis.2020.2.535","DOIUrl":"https://doi.org/10.2140/tunis.2020.2.535","url":null,"abstract":"We provide a simplified proof of the hypergroup property for the Jacobi polynomials on the unit interval, and investigate various generalizations of this property for the family of Dirichlet laws on the simplex.","PeriodicalId":36030,"journal":{"name":"Tunisian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/tunis.2020.2.535","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48544899","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-01-01DOI: 10.2140/TUNIS.2020.2.197
M. Dimassi
In the semiclassical regime (i.e., (cid:15) (cid:38) 0), we study the effect of a slowly varying potential V ((cid:15) t , (cid:15) z ) on the magnetic Schrödinger operator P = D 2 x + ( D z + µ x ) 2 on a strip [− a , a ] × (cid:82) z . The potential V ( t , z ) is assumed to be smooth. We derive the semiclassical dynamics and we describe the asymptotic structure of the spectrum and the resonances of the operator P + V ((cid:15) t , (cid:15) z ) for (cid:15) small enough. All our results depend on the eigenvalues corresponding to D 2 x + (µ x + k ) 2 on L 2 ( [− a , a ] ) with Dirichlet boundary condition. x ≤ a } . The Fourier transfor-mation with respect to z reduces the spectral problem of P to an analysis of the ( k depending) eigenvalues E 0 ( k ), E 1 ( k ), . . . of the Sturm-Liouville operator on the interval [− a , a ] with Dirichlet boundary condition at − a and a .
在半经典区(即(cid:15) (cid:38) 0)中，我们研究了慢变电位V ((cid:15) t， (cid:15) z)对条带[- a, a] × (cid:82) z上的磁Schrödinger算符P = d2 x + (D z +µx) 2的影响。假设势能V (t, z)是光滑的。我们导出了半经典动力学，并描述了谱的渐近结构和算子P + V ((cid:15) t， (cid:15) z)在(cid:15)足够小时的共振。我们所有的结果都依赖于d2 +(µx + k) 2在l2([−a, a])上对应的特征值，并具有Dirichlet边界条件。X≤a}。关于z的傅里叶变换将P的频谱问题简化为(取决于k的)特征值e0 (k)， e1 (k)，…的分析。在- a和- a处具有Dirichlet边界条件的区间[- a, a]上Sturm-Liouville算子的性质。
{"title":"Semiclassical approximation of the magnetic Schrödinger operator on a strip : dynamics and spectrum","authors":"M. Dimassi","doi":"10.2140/TUNIS.2020.2.197","DOIUrl":"https://doi.org/10.2140/TUNIS.2020.2.197","url":null,"abstract":"In the semiclassical regime (i.e., (cid:15) (cid:38) 0), we study the effect of a slowly varying potential V ((cid:15) t , (cid:15) z ) on the magnetic Schrödinger operator P = D 2 x + ( D z + µ x ) 2 on a strip [− a , a ] × (cid:82) z . The potential V ( t , z ) is assumed to be smooth. We derive the semiclassical dynamics and we describe the asymptotic structure of the spectrum and the resonances of the operator P + V ((cid:15) t , (cid:15) z ) for (cid:15) small enough. All our results depend on the eigenvalues corresponding to D 2 x + (µ x + k ) 2 on L 2 ( [− a , a ] ) with Dirichlet boundary condition. x ≤ a } . The Fourier transfor-mation with respect to z reduces the spectral problem of P to an analysis of the ( k depending) eigenvalues E 0 ( k ), E 1 ( k ), . . . of the Sturm-Liouville operator on the interval [− a , a ] with Dirichlet boundary condition at − a and a .","PeriodicalId":36030,"journal":{"name":"Tunisian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/TUNIS.2020.2.197","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48961629","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-12-12DOI: 10.2140/tunis.2021.3.689
B. Conrad, M. Temkin
In this paper we study two types of descent in the category of Berkovich analytic spaces: flat descent and descent with respect to an extension of the ground field. Quite surprisingly, the deepest results in this direction seem to be of the second type, including the descent of properties of being a good analytic space and being a morphism without boundary.
{"title":"Descent for nonarchimedean analytic spaces","authors":"B. Conrad, M. Temkin","doi":"10.2140/tunis.2021.3.689","DOIUrl":"https://doi.org/10.2140/tunis.2021.3.689","url":null,"abstract":"In this paper we study two types of descent in the category of Berkovich analytic spaces: flat descent and descent with respect to an extension of the ground field. Quite surprisingly, the deepest results in this direction seem to be of the second type, including the descent of properties of being a good analytic space and being a morphism without boundary.","PeriodicalId":36030,"journal":{"name":"Tunisian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2019-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48148418","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-12-09DOI: 10.2140/tunis.2022.4.587
Jonathan Wise
We describe a 3-step filtration on all logarithmic abelian varieties with constant degeneration. The obstruction to descending this filtration, as a variegated extension, from logarithmic geometry to algebraic geometry is encoded in a bilinear pairing valued in the characteristic monoid of the base. This pairing is realized as the monodromy pairing in p-adic, l-adic, and Betti cohomolgies, and recovers the Picard-Lefschetz transformation in the case of Jacobians. The Hodge realization of the filtration is the monodromy weight filtration on the limit mixed Hodge structure.
{"title":"The monodromy pairing for logarithmic\u00001-motifs","authors":"Jonathan Wise","doi":"10.2140/tunis.2022.4.587","DOIUrl":"https://doi.org/10.2140/tunis.2022.4.587","url":null,"abstract":"We describe a 3-step filtration on all logarithmic abelian varieties with constant degeneration. The obstruction to descending this filtration, as a variegated extension, from logarithmic geometry to algebraic geometry is encoded in a bilinear pairing valued in the characteristic monoid of the base. This pairing is realized as the monodromy pairing in p-adic, l-adic, and Betti cohomolgies, and recovers the Picard-Lefschetz transformation in the case of Jacobians. The Hodge realization of the filtration is the monodromy weight filtration on the limit mixed Hodge structure.","PeriodicalId":36030,"journal":{"name":"Tunisian Journal of Mathematics","volume":"20 5","pages":""},"PeriodicalIF":0.9,"publicationDate":"2019-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41266667","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-11-05DOI: 10.2140/tunis.2021.3.657
M. Harris
The Ichino-Ikeda conjecture, and its generalization to unitary groups by N. Harris, has given explicit formulas for central critical values of a large class of Rankin-Selberg tensor products. Although the conjecture is not proved in full generality, there has been considerable progress, especially for $L$-values of the form $L(1/2,BC(pi) times BC(pi'))$, where $pi$ and $pi'$ are cohomological automorphic representations of unitary groups $U(V)$ and $U(V')$, respectively. Here $V$ and $V'$ are hermitian spaces over a CM field, $V$ of dimension $n$, $V'$ of codimension $1$ in $V$, and $BC$ denotes the twisted base change to $GL(n) times GL(n-1)$. This paper contains the first steps toward generalizing the construction of my paper with Tilouine on triple product $L$-functions to this situation. We assume $pi$ is a holomorphic representation and $pi'$ varies in an ordinary Hida family (of antiholomorphic forms). The construction of the measure attached to $pi$ uses recent work of Eischen, Fintzen, Mantovan, and Varma.
{"title":"Square root p-adic L-functions, I : Construction\u0000of a one-variable measure","authors":"M. Harris","doi":"10.2140/tunis.2021.3.657","DOIUrl":"https://doi.org/10.2140/tunis.2021.3.657","url":null,"abstract":"The Ichino-Ikeda conjecture, and its generalization to unitary groups by N. Harris, has given explicit formulas for central critical values of a large class of Rankin-Selberg tensor products. Although the conjecture is not proved in full generality, there has been considerable progress, especially for $L$-values of the form $L(1/2,BC(pi) times BC(pi'))$, where $pi$ and $pi'$ are cohomological automorphic representations of unitary groups $U(V)$ and $U(V')$, respectively. Here $V$ and $V'$ are hermitian spaces over a CM field, $V$ of dimension $n$, $V'$ of codimension $1$ in $V$, and $BC$ denotes the twisted base change to $GL(n) times GL(n-1)$. \u0000This paper contains the first steps toward generalizing the construction of my paper with Tilouine on triple product $L$-functions to this situation. We assume $pi$ is a holomorphic representation and $pi'$ varies in an ordinary Hida family (of antiholomorphic forms). The construction of the measure attached to $pi$ uses recent work of Eischen, Fintzen, Mantovan, and Varma.","PeriodicalId":36030,"journal":{"name":"Tunisian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2019-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44143485","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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