Algebra & Number Theory最新文献

{"title":"Differentially large fields","authors":"Omar León Sánchez, Marcus Tressl","doi":"10.2140/ant.2024.18.249","DOIUrl":"https://doi.org/10.2140/ant.2024.18.249","url":null,"abstract":"<p>We introduce the notion of <span>differential largeness </span>for fields equipped with several commuting derivations (as an analogue to largeness of fields). We lay out the foundations of this new class of “tame” differential fields. We state several characterizations and exhibit plenty of examples and applications. Our results strongly indicate that differentially large fields will play a key role in differential field arithmetic. For instance, we characterize differential largeness in terms of being existentially closed in their power series field (furnished with natural derivations), we give explicit constructions of differentially large fields in terms of iterated powers series, we prove that the class of differentially large fields is elementary, and we show that differential largeness is preserved under algebraic extensions, therefore showing that their algebraic closure is differentially closed. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"236 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139695775","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":"Sur les espaces homogènes de Borovoi–Kunyavskii","authors":"Mạnh Linh Nguyễn","doi":"10.2140/ant.2024.18.349","DOIUrl":"https://doi.org/10.2140/ant.2024.18.349","url":null,"abstract":"<p>Nous établissons le principe de Hasse et l’approximation faible pour certains espaces homogènes de <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi> SL</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--></mrow><mrow><mi>m</mi></mrow></msub></math> à stabilisateur géométrique nilpotent de classe 2, construits par Borovoi et Kunyavskii. Ces espaces homogènes vérifient donc une conjecture de Colliot-Thélène concernant l’obstruction de Brauer–Manin pour les variétés géométriquement rationnellement connexes. </p><p> We establish the Hasse principle and the weak approximation property for certain homogeneous spaces of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi> SL</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--></mrow><mrow><mi>m</mi></mrow></msub></math> whose geometric stabilizer is of nilpotency class 2, which were constructed by Borovoi and Kunyavskii. These homogeneous spaces verify thus a conjecture of Colliot-Thélène on the Brauer–Manin obstruction for geometrically rationally connected varieties. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"4 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139695827","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":"Projective orbifolds of Nikulin type","authors":"Chiara Camere, Alice Garbagnati, Grzegorz Kapustka, Michał Kapustka","doi":"10.2140/ant.2024.18.165","DOIUrl":"https://doi.org/10.2140/ant.2024.18.165","url":null,"abstract":"<p>We study projective irreducible symplectic orbifolds of dimension four that are deformations of partial resolutions of quotients of hyperkähler manifolds of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>K</mi><msup><mrow><mn>3</mn></mrow><mrow><mo stretchy=\"false\">[</mo><mn>2</mn><mo stretchy=\"false\">]</mo></mrow></msup></math>-type by symplectic involutions; we call them orbifolds of Nikulin type. We first classify those projective orbifolds that are really quotients, by describing all families of projective fourfolds of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>K</mi><msup><mrow><mn>3</mn></mrow><mrow><mo stretchy=\"false\">[</mo><mn>2</mn><mo stretchy=\"false\">]</mo></mrow></msup></math>-type with a symplectic involution and the relation with their quotients, and then study their deformations. We compute the Riemann–Roch formula for Weil divisors on orbifolds of Nikulin type and using this we describe the first known locally complete family of singular irreducible symplectic varieties as double covers of special complete intersections <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mo stretchy=\"false\">(</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo stretchy=\"false\">)</mo></math> in <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>ℙ</mi></mrow><mrow><mn>6</mn></mrow></msup></math>. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"20 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138293892","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":"Semisimple algebras and PI-invariants of finite dimensional algebras","authors":"Eli Aljadeff, Yakov Karasik","doi":"10.2140/ant.2024.18.133","DOIUrl":"https://doi.org/10.2140/ant.2024.18.133","url":null,"abstract":"&lt;p&gt;Let &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;Γ&lt;/mi&gt;&lt;/math&gt; be the &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;mspace width=\"-0.17em\"&gt;&lt;/mspace&gt;&lt;/math&gt;-ideal of identities of an affine PI-algebra over an algebraically closed field &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/math&gt; of characteristic zero. Consider the family &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi mathvariant=\"bold-script\"&gt;ℳ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;Γ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt; of finite dimensional algebras &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi mathvariant=\"normal\"&gt;Σ&lt;/mi&gt;&lt;/math&gt; with &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt; Id&lt;/mi&gt;&lt;mo&gt; ⁡&lt;!--FUNCTION APPLICATION--&gt; &lt;/mo&gt;&lt;!--nolimits--&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;mi mathvariant=\"normal\"&gt;Σ&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;\u0000&lt;mo&gt;=&lt;/mo&gt;\u0000&lt;mi&gt;Γ&lt;/mi&gt;&lt;/math&gt;. By Kemer’s theory &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi mathvariant=\"bold-script\"&gt;ℳ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;Γ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt; is not empty. We show there exists &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;A&lt;/mi&gt;\u0000&lt;mo&gt;∈&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi mathvariant=\"bold-script\"&gt;ℳ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;Γ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt; with Wedderburn–Malcev decomposition &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mi&gt;≅&lt;/mi&gt;&lt;mo&gt; ⁡&lt;!--FUNCTION APPLICATION--&gt;&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;\u0000&lt;mo&gt;⊕&lt;/mo&gt; &lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;J&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;, where &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;J&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt; is the Jacobson’s radical and &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt; is a semisimple supplement with the property that if &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mi&gt;≅&lt;/mi&gt;&lt;mo&gt; ⁡&lt;!--FUNCTION APPLICATION--&gt;&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;\u0000&lt;mo&gt;⊕&lt;/mo&gt; &lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;J&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;\u0000&lt;mo&gt;∈&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi mathvariant=\"bold-script\"&gt;ℳ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;Γ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt; then &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt; is a direct summand of &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;. In particular &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt; is unique minimal, thus an invariant of &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;Γ&lt;/mi&gt;&lt;/math&gt;","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"19 26","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138293893","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 weighted one-level density of families of L-functions","authors":"Alessandro Fazzari","doi":"10.2140/ant.2024.18.87","DOIUrl":"https://doi.org/10.2140/ant.2024.18.87","url":null,"abstract":"<p>This paper is devoted to a weighted version of the one-level density of the nontrivial zeros of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>L</mi></math>-functions, tilted by a power of the <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>L</mi></math>-function evaluated at the central point. Assuming the Riemann hypothesis and the ratio conjecture, for some specific families of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>L</mi></math>-functions, we prove that the same structure suggested by the density conjecture also holds in this weighted investigation, if the exponent of the weight is small enough. Moreover, we speculate about the general case, conjecturing explicit formulae for the weighted kernels. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"19 25","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138293894","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":"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}