Algebra & Number Theory最新文献

{"title":"On Tamagawa numbers of CM tori","authors":"Pei-Xin Liang, Yasuhiro Oki, Hsin-Yi Yang, Chia-Fu Yu","doi":"10.2140/ant.2024.18.583","DOIUrl":"https://doi.org/10.2140/ant.2024.18.583","url":null,"abstract":"<p>We investigate the problem of computing Tamagawa numbers of CM tori. This problem arises naturally from the problem of counting polarized abelian varieties with commutative endomorphism algebras over finite fields, and polarized CM abelian varieties and components of unitary Shimura varieties in the works of Achter, Altug, Garcia and Gordon and of Guo, Sheu and Yu, respectively. We make a systematic study on Galois cohomology groups in a more general setting and compute the Tamagawa numbers of CM tori associated to various Galois CM fields. Furthermore, we show that every (positive or negative) power of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>2</mn></math> is the Tamagawa number of a CM tori, proving the analogous conjecture of Ono for CM tori. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"34 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139898804","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":"p-groups, p-rank, and semistable reduction of coverings of curves","authors":"Yu Yang","doi":"10.2140/ant.2024.18.281","DOIUrl":"https://doi.org/10.2140/ant.2024.18.281","url":null,"abstract":"<p>We prove various explicit formulas concerning <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>-rank of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>-coverings of pointed semistable curves over discrete valuation rings. In particular, we obtain a full generalization of Raynaud’s formula for <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>-rank of fibers over <span>nonmarked smooth </span>closed points in the case of <span>arbitrary </span>closed points. As an application, for abelian <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>-coverings, we give an affirmative answer to an open problem concerning boundedness of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>-rank asked by Saïdi more than twenty years ago. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"5 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139695844","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":"Decidability via the tilting correspondence","authors":"Konstantinos Kartas","doi":"10.2140/ant.2024.18.209","DOIUrl":"https://doi.org/10.2140/ant.2024.18.209","url":null,"abstract":"<p>We prove a relative decidability result for perfectoid fields. This applies to show that the fields <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>ℚ</mi></mrow><mrow><mi>p</mi></mrow></msub><mo stretchy=\"false\">(</mo><msup><mrow><mi>p</mi></mrow><mrow><mn>1</mn><mo>∕</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>∞</mi></mrow></msup>\u0000</mrow></msup><mo stretchy=\"false\">)</mo></math> and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>ℚ</mi></mrow><mrow><mi>p</mi></mrow></msub><mo stretchy=\"false\">(</mo><msub><mrow><mi>ζ</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>∞</mi></mrow></msup></mrow></msub><mo stretchy=\"false\">)</mo></math> are (existentially) decidable relative to the perfect hull of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi mathvariant=\"double-struck\">𝔽</mi></mrow><mrow><mi>p</mi></mrow></msub><mo stretchy=\"false\">(</mo><mo stretchy=\"false\">(</mo><mi>t</mi><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">)</mo></math> and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msubsup><mrow><mi>ℚ</mi></mrow><mrow><mi>p</mi></mrow><mrow><mi>a</mi><mi>b</mi></mrow></msubsup></math> is (existentially) decidable relative to the perfect hull of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mover accent=\"false\"><mrow><mi mathvariant=\"double-struck\">𝔽</mi></mrow><mo accent=\"true\">¯</mo></mover></mrow><mrow><mi>p</mi></mrow></msub><mo stretchy=\"false\">(</mo><mo stretchy=\"false\">(</mo><mi>t</mi><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">)</mo></math>. We also prove some unconditional decidability results in mixed characteristic via reduction to characteristic <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"35 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139695696","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":"Partial sums of typical multiplicative functions over short moving intervals","authors":"Mayank Pandey, Victor Y. Wang, Max Wenqiang Xu","doi":"10.2140/ant.2024.18.389","DOIUrl":"https://doi.org/10.2140/ant.2024.18.389","url":null,"abstract":"<p>We prove that the <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>k</mi></math>-th positive integer moment of partial sums of Steinhaus random multiplicative functions over the interval <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mo stretchy=\"false\">(</mo><mi>x</mi><mo>,</mo><mi>x</mi>\u0000<mo>+</mo>\u0000<mi>H</mi><mo stretchy=\"false\">]</mo></math> matches the corresponding Gaussian moment, as long as <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>H</mi>\u0000<mo>≪</mo>\u0000<mi>x</mi><mo>∕</mo><msup><mrow><mo stretchy=\"false\">(</mo><mi>log</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--><mi>x</mi><mo stretchy=\"false\">)</mo></mrow><mrow><mn>2</mn><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>2</mn><mo>+</mo><mi>o</mi><mo stretchy=\"false\">(</mo><mn>1</mn><mo stretchy=\"false\">)</mo>\u0000</mrow></msup></math> and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>H</mi></math> tends to infinity with <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>x</mi></math>. We show that properly normalized partial sums of typical multiplicative functions arising from realizations of random multiplicative functions have Gaussian limiting distribution in short moving intervals <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mo stretchy=\"false\">(</mo><mi>x</mi><mo>,</mo><mi>x</mi>\u0000<mo>+</mo>\u0000<mi>H</mi><mo stretchy=\"false\">]</mo></math> with <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>H</mi>\u0000<mo>≪</mo>\u0000<mi>X</mi><mo>∕</mo><msup><mrow><mo stretchy=\"false\">(</mo><mi>log</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--><mi>X</mi><mo stretchy=\"false\">)</mo></mrow><mrow><mi>W</mi><mo stretchy=\"false\">(</mo><mi>X</mi><mo stretchy=\"false\">)</mo></mrow></msup></math> tending to infinity with <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>X</mi></math>, where <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>x</mi></math> is uniformly chosen from <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mo stretchy=\"false\">{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi>…</mi><mo> ⁡<!--FUNCTION APPLICATION--></mo><mo>,</mo><mi>X</mi><mo stretchy=\"false\">}</mo></math>, and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>W</mi><mo stretchy=\"false\">(</mo><mi>X</mi><mo stretchy=\"false\">)</mo></math> tends to infinity with <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>X</mi></math> arbitrarily slowly. This makes some initial progress on a recent question of Harper. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"13 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139695819","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 deterministic algorithm for Harder–Narasimhan filtrations for representations of acyclic quivers","authors":"Chi-Yu Cheng","doi":"10.2140/ant.2024.18.319","DOIUrl":"https://doi.org/10.2140/ant.2024.18.319","url":null,"abstract":"<p>Let <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>M</mi></math> be a representation of an acyclic quiver <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>Q</mi></math> over an infinite field <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>k</mi></math>. We establish a deterministic algorithm for computing the Harder–Narasimhan filtration of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>M</mi></math>. The algorithm is polynomial in the dimensions of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>M</mi></math>, the weights that induce the Harder–Narasimhan filtration of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>M</mi></math>, and the number of paths in <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>Q</mi></math>. As a direct application, we also show that when <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>k</mi></math> is algebraically closed and when <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>M</mi></math> is unstable, the same algorithm produces Kempf’s maximally destabilizing one parameter subgroups for <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>M</mi></math>. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"2 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139695805","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":"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}