Algebra & Number Theory最新文献

{"title":"Degree growth for tame automorphisms of an affine quadric threefold","authors":"Nguyen-Bac Dang","doi":"10.2140/ant.2024.18.1","DOIUrl":"https://doi.org/10.2140/ant.2024.18.1","url":null,"abstract":"<p>We consider the degree sequences of the tame automorphisms preserving an affine quadric threefold. Using some valuative estimates derived from the work of Shestakov and Umirbaev and the action of this group on a <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi> CAT</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--><mo stretchy=\"false\">(</mo><mn>0</mn><mo stretchy=\"false\">)</mo></math>, Gromov-hyperbolic square complex constructed by Bisi, Furter and Lamy, we prove that the dynamical degrees of tame elements avoid any value strictly between 1 and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mfrac><mrow><mn>4</mn></mrow>\u0000<mrow><mn>3</mn></mrow></mfrac></math>. As an application, these methods allow us to characterize when the growth exponent of the degree of a random product of finitely many tame automorphisms is positive. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"19 6","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138293925","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the variation of Frobenius eigenvalues in a skew-abelian Iwasawa tower","authors":"Asvin G.","doi":"10.2140/ant.2023.17.2151","DOIUrl":"https://doi.org/10.2140/ant.2023.17.2151","url":null,"abstract":"<p>We study towers of varieties over a finite field such as <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msup>\u0000<mo>=</mo>\u0000<mi>f</mi><mo stretchy=\"false\">(</mo><msup><mrow><mi>x</mi></mrow><mrow><msup><mrow><mi>ℓ</mi></mrow><mrow><mi>n</mi></mrow></msup>\u0000</mrow></msup><mo stretchy=\"false\">)</mo></math> and prove that the characteristic polynomials of the Frobenius on the étale cohomology show a surprising <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>ℓ</mi></math>-adic convergence. We prove this by proving a more general statement about the convergence of certain invariants related to a skew-abelian cohomology group. The key ingredient is a generalization of Fermat’s little theorem to matrices. Along the way, we will prove that many natural sequences of polynomials <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mo stretchy=\"false\">(</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">)</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></msub>\u0000<mo>∈</mo> <msub><mrow><mi>ℤ</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><msup><mrow><mo stretchy=\"false\">[</mo><mi>x</mi><mo stretchy=\"false\">]</mo></mrow><mrow><mi>ℕ</mi></mrow></msup></math> converge <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>ℓ</mi></math>-adically and give explicit rates of convergence. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"11 20","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71493546","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Limit multiplicity for unitary groups and the stable trace formula","authors":"Mathilde Gerbelli-Gauthier","doi":"10.2140/ant.2023.17.2181","DOIUrl":"https://doi.org/10.2140/ant.2023.17.2181","url":null,"abstract":"<p>We give upper bounds on limit multiplicities of certain nontempered representations of unitary groups <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>U</mi><mo stretchy=\"false\">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy=\"false\">)</mo></math>, conditionally on the endoscopic classification of representations. Our result applies to some cohomological representations, and we give applications to the growth of cohomology of cocompact arithmetic subgroups of unitary groups. The representations considered are transfers of products of characters and discrete series on endoscopic groups, and the bounds are obtained using Arthur’s stabilization of the trace formula and the classification established by Mok, and Kaletha, Minguez, Shin and White. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"11 22","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71493545","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The de Rham–Fargues–Fontaine cohomology","authors":"Arthur-César Le Bras, Alberto Vezzani","doi":"10.2140/ant.2023.17.2097","DOIUrl":"https://doi.org/10.2140/ant.2023.17.2097","url":null,"abstract":"<p>We show how to attach to any rigid analytic variety <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>V</mi> </math> over a perfectoid space <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>P</mi></math> a rigid analytic motive over the Fargues–Fontaine curve <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi mathvariant=\"bold-script\">𝒳</mi><mo stretchy=\"false\">(</mo><mi>P</mi><mo stretchy=\"false\">)</mo></math> functorially in <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>V</mi> </math> and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>P</mi></math>. We combine this construction with the overconvergent relative de Rham cohomology to produce a complex of solid quasicoherent sheaves over <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi mathvariant=\"bold-script\">𝒳</mi><mo stretchy=\"false\">(</mo><mi>P</mi><mo stretchy=\"false\">)</mo></math>, and we show that its cohomology groups are vector bundles if <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>V</mi> </math> is smooth and proper over <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>P</mi></math> or if <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>V</mi> </math> is quasicompact and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>P</mi></math> is a perfectoid field, thus proving and generalizing a conjecture of Scholze. The main ingredients of the proofs are explicit <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi mathvariant=\"double-struck\">𝔹</mi></mrow><mrow><mn>1</mn></mrow></msup></math>-homotopies, the motivic proper base change and the formalism of solid quasicoherent sheaves. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"54 43","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71514518","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A number theoretic characterization of E-smooth and (FRS) morphisms : estimates on the number of ℤ∕pkℤ-points","authors":"Raf Cluckers, Itay Glazer, Yotam I. Hendel","doi":"10.2140/ant.2023.17.2229","DOIUrl":"https://doi.org/10.2140/ant.2023.17.2229","url":null,"abstract":"<p>We provide uniform estimates on the number of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>ℤ</mi><mo>∕</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>k</mi></mrow></msup><mi>ℤ</mi></math>-points lying on fibers of flat morphisms between smooth varieties whose fibers have rational singularities, termed (FRS) morphisms. For each individual fiber, the estimates were known by work of Avni and Aizenbud, but we render them uniform over all fibers. The proof technique for individual fibers is based on Hironaka’s resolution of singularities and Denef’s formula, but breaks down in the uniform case. Instead, we use recent results from the theory of motivic integration. Our estimates are moreover equivalent to the (FRS) property, just like in the absolute case by Avni and Aizenbud. In addition, we define new classes of morphisms, called <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>E</mi></math>-smooth morphisms (<math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>E</mi>\u0000<mo>∈</mo>\u0000<mi>ℕ</mi></math>), which refine the (FRS) property, and use the methods we developed to provide uniform number-theoretic estimates as above for their fibers. Similar estimates are given for fibers of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>𝜀</mi></math>-jet flat morphisms, improving previous results by the last two authors. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"11 21","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71493544","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"GKM-theory for torus actions on cyclic quiver Grassmannians","authors":"Martina Lanini, Alexander Pütz","doi":"10.2140/ant.2023.17.2055","DOIUrl":"https://doi.org/10.2140/ant.2023.17.2055","url":null,"abstract":"<p>We define and investigate algebraic torus actions on quiver Grassmannians for nilpotent representations of the equioriented cycle. Examples of such varieties are type <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>A</mi></math> flag varieties, their linear degenerations and finite-dimensional approximations of both the affine flag variety and affine Grassmannian for <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi> GL</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--></mrow><mrow><mi>n</mi></mrow></msub></math>. We show that these quiver Grassmannians equipped with our specific torus action are GKM-varieties and that their moment graph admits a combinatorial description in terms of the coefficient quiver of the underlying quiver representations. By adapting to our setting results by Gonzales, we are able to prove that moment graph techniques can be applied to construct module bases for the equivariant cohomology of the quiver Grassmannians listed above. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"11 19","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71493547","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The behavior of essential dimension under specialization, II","authors":"Zinovy Reichstein, Federico Scavia","doi":"10.2140/ant.2023.17.1925","DOIUrl":"https://doi.org/10.2140/ant.2023.17.1925","url":null,"abstract":"<p>Let <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>G</mi></math> be a linear algebraic group over a field. We show that, under mild assumptions, in a family of primitive generically free <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>G</mi></math>-varieties over a base variety <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>B</mi></math>, the essential dimension of the geometric fibers may drop on a countable union of Zariski closed subsets of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>B</mi></math> and stays constant away from this countable union. We give several applications of this result. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"11 24","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71493542","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Fitting ideals of class groups for CM abelian extensions","authors":"Mahiro Atsuta, Takenori Kataoka","doi":"10.2140/ant.2023.17.1901","DOIUrl":"https://doi.org/10.2140/ant.2023.17.1901","url":null,"abstract":"<p>Let <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>K</mi></math> be a finite abelian CM-extension of a totally real field <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>k</mi></math> and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>T</mi></math> a suitable finite set of finite primes of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>k</mi></math>. We determine the Fitting ideal of the minus component of the <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>T</mi></math>-ray class group of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>K</mi></math>, except for the <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>2</mn></math>-component, assuming the validity of the equivariant Tamagawa number conjecture. As an application, we give a necessary and sufficient condition for the Stickelberger element to lie in that Fitting ideal. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"11 23","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71493543","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Bézoutians and the 𝔸1-degree","authors":"Thomas Brazelton, Stephen McKean, Sabrina Pauli","doi":"10.2140/ant.2023.17.1985","DOIUrl":"https://doi.org/10.2140/ant.2023.17.1985","url":null,"abstract":"<p>We prove that both the local and global <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi mathvariant=\"double-struck\">𝔸</mi></mrow><mrow><mn>1</mn></mrow></msup></math>-degree of an endomorphism of affine space can be computed in terms of the multivariate Bézoutian. In particular, we show that the Bézoutian bilinear form, the Scheja–Storch form, and the <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi mathvariant=\"double-struck\">𝔸</mi></mrow><mrow><mn>1</mn></mrow></msup></math>-degree for complete intersections are isomorphic. Our global theorem generalizes Cazanave’s theorem in the univariate case, and our local theorem generalizes Kass–Wickelgren’s theorem on EKL forms and the local degree. This result provides an algebraic formula for local and global degrees in motivic homotopy theory. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"54 47","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71514516","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On self-correspondences on curves","authors":"Joël Bellaïche","doi":"10.2140/ant.2023.17.1867","DOIUrl":"https://doi.org/10.2140/ant.2023.17.1867","url":null,"abstract":"<p>We study the algebraic dynamics of self-correspondences on a curve. A self-correspondence on a (proper and smooth) curve <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>C</mi></math> over an algebraically closed field is the data of another curve <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>D</mi></math> and two nonconstant separable morphisms <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>π</mi></mrow><mrow><mn>1</mn></mrow></msub></math> and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></msub></math> from <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>D</mi></math> to <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>C</mi></math>. A subset <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>S</mi></math> of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>C</mi></math> is <span>complete</span> if <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msubsup><mrow><mi>π</mi></mrow><mrow><mn>1</mn></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mo stretchy=\"false\">(</mo><mi>S</mi><mo stretchy=\"false\">)</mo>\u0000<mo>=</mo> <msubsup><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mo stretchy=\"false\">(</mo><mi>S</mi><mo stretchy=\"false\">)</mo></math>. We show that self-correspondences are divided into two classes: those that have only finitely many finite complete sets, and those for which <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>C</mi></math> is a union of finite complete sets. The latter ones are called <span>finitary</span>, and happen only when <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi> deg</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--><msub><mrow><mi>π</mi></mrow><mrow><mn>1</mn></mrow></msub>\u0000<mo>=</mo><mi> deg</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--><msub><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></msub></math> and have a trivial dynamics. For a nonfinitary self-correspondence in characteristic zero, we give a sharp bound for the number of étale finite complete sets. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"54 45","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71514517","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}