Algebra & Number Theory最新文献

{"title":"Prismatic G-displays and descent theory","authors":"Kazuhiro Ito","doi":"10.2140/ant.2025.19.1685","DOIUrl":"https://doi.org/10.2140/ant.2025.19.1685","url":null,"abstract":"<p>For a smooth affine group scheme <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>G</mi></math> over the ring of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>-adic integers <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>ℤ</mi></mrow><mrow><mi>p</mi></mrow></msub></math> and a cocharacter <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>μ</mi></math> of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>G</mi></math>, we study <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>G</mi></math>-<math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>μ</mi></math>-displays over the prismatic site of Bhatt and Scholze. In particular, we obtain several descent results for them. If <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>G</mi>\u0000<mo>=</mo><msub><mrow><mi> GL</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--></mrow><mrow><mi>n</mi></mrow></msub></math>, then our <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>G</mi></math>-<math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>μ</mi></math>-displays can be thought of as Breuil–Kisin modules with some additional conditions. The relation between our <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>G</mi></math>-<math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>μ</mi></math>-displays and prismatic <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>F</mi></math>-gauges introduced by Drinfeld and Bhatt–Lurie is also discussed. </p><p> In fact, our main results are formulated and proved for smooth affine group schemes over the ring of integers <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi mathvariant=\"bold-script\">𝒪</mi></mrow><mrow><mi>E</mi></mrow></msub></math> of any finite extension <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>E</mi></math> of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>ℚ</mi></mrow><mrow><mi>p</mi></mrow></msub></math> by using <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi mathvariant=\"bold-script\">𝒪</mi></mrow><mrow><mi>E</mi></mrow></msub></math>-prisms, which are <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi mathvariant=\"bold-script\">𝒪</mi></mrow><mrow><mi>E</mi></mrow></msub></math>-analogues of prisms. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"1 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144677858","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":"Rigidity of modular morphisms via Fujita decomposition","authors":"Giulio Codogni, Víctor González Alonso, Sara Torelli","doi":"10.2140/ant.2025.19.1671","DOIUrl":"https://doi.org/10.2140/ant.2025.19.1671","url":null,"abstract":"<p>We prove that the Torelli, Prym and spin-Torelli morphisms, as well as covering maps between moduli stacks of smooth projective curves, cannot be deformed. The proofs use properties of the Fujita decomposition of the Hodge bundle of families of curves. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"67 2 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144678176","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":"Arithmetic Siegel–Weil formula on 𝒳0(N)","authors":"Baiqing Zhu","doi":"10.2140/ant.2025.19.1771","DOIUrl":"https://doi.org/10.2140/ant.2025.19.1771","url":null,"abstract":"<p>We establish the arithmetic Siegel–Weil formula on the modular curve <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi mathvariant=\"bold-script\">𝒳</mi></mrow><mrow><mn>0</mn></mrow></msub><mo stretchy=\"false\">(</mo><mi>N</mi><mo stretchy=\"false\">)</mo></math> for arbitrary level <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>N</mi></math>, i.e., we relate the arithmetic degrees of special cycles on <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi mathvariant=\"bold-script\">𝒳</mi></mrow><mrow><mn>0</mn></mrow></msub><mo stretchy=\"false\">(</mo><mi>N</mi><mo stretchy=\"false\">)</mo></math> to the derivatives of Fourier coefficients of a genus-2 Eisenstein series. We prove this formula by a precise identity between the local arithmetic intersection numbers on the Rapoport–Zink space associated to <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi mathvariant=\"bold-script\">𝒳</mi></mrow><mrow><mn>0</mn></mrow></msub><mo stretchy=\"false\">(</mo><mi>N</mi><mo stretchy=\"false\">)</mo></math> and the derivatives of local representation densities of quadratic forms. When <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>N</mi></math> is odd and square-free, this gives a different proof of the main results in work of Sankaran, Shi and Yang. This local identity is proved by relating it to an identity in one dimension higher, but at hyperspecial level. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"1 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144677861","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":"Metaplectic cusp forms and the large sieve","authors":"Alexander Dunn","doi":"10.2140/ant.2025.19.1823","DOIUrl":"https://doi.org/10.2140/ant.2025.19.1823","url":null,"abstract":"<p>We prove a power saving upper bound for the sum of Fourier coefficients <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>ρ</mi></mrow><mrow><mi>f</mi></mrow></msub><mo stretchy=\"false\">(</mo><mo>⋅</mo><mo stretchy=\"false\">)</mo></math> of a fixed cubic metaplectic cusp form <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>f</mi></math> over primes. Our result is the cubic analogue of a celebrated 1990 theorem of Duke and Iwaniec, and the cuspidal analogue of a theorem due to the author and Radziwiłł for the bias in cubic Gauss sums. </p><p> The proof has two main inputs, both of independent interest. Firstly, we prove a new large sieve estimate for a bilinear form whose kernel function is <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>ρ</mi></mrow><mrow><mi>f</mi></mrow></msub><mo stretchy=\"false\">(</mo><mo>⋅</mo><mo stretchy=\"false\">)</mo></math>. The proof of the bilinear estimate uses a number field version of circle method due to Browning and Vishe, Voronoi summation, and Gauss–Ramanujan sums. Secondly, we use Voronoi summation and the cubic large sieve of Heath-Brown to prove an estimate for a linear form involving <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>ρ</mi></mrow><mrow><mi>f</mi></mrow></msub><mo stretchy=\"false\">(</mo><mo>⋅</mo><mo stretchy=\"false\">)</mo></math>. Our linear estimate overcomes a bottleneck occurring at level of distribution <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mfrac><mrow><mn>2</mn></mrow>\u0000<mrow><mn>3</mn></mrow></mfrac></math>. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"17 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144677862","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 core of monomial ideals","authors":"Louiza Fouli, Jonathan Montaño, Claudia Polini, Bernd Ulrich","doi":"10.2140/ant.2025.19.1463","DOIUrl":"https://doi.org/10.2140/ant.2025.19.1463","url":null,"abstract":"<p>The core of an ideal is defined as the intersection of all of its reductions. We provide an explicit description for the core of a monomial ideal <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>I</mi></math> satisfying certain residual conditions, showing that <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi> core</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--><mo stretchy=\"false\">(</mo><mi>I</mi><mo stretchy=\"false\">)</mo></math> coincides with the largest monomial ideal contained in a general reduction of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>I</mi></math>. We prove that the class of lex-segment ideals satisfies these residual conditions and study the core of lex-segment ideals generated in one degree. For monomial ideals that do not necessarily satisfy the residual conditions and that are generated in one degree, we conjecture an explicit formula for the core, and make progress towards this conjecture. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"10 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144278610","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":"Rational points of rigid-analytic sets : a Pila–Wilkie-type theorem","authors":"Gal Binyamini, Fumiharu Kato","doi":"10.2140/ant.2025.19.1581","DOIUrl":"https://doi.org/10.2140/ant.2025.19.1581","url":null,"abstract":"<p>We establish a rigid-analytic analog of the Pila–Wilkie counting theorem, giving subpolynomial upper bounds for the number of rational points in the transcendental part of a <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>ℚ</mi></mrow><mrow><mi>p</mi></mrow></msub></math>-analytic set and the number of rational functions in a <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi mathvariant=\"double-struck\">𝔽</mi></mrow><mrow><mi>q</mi></mrow></msub><mo stretchy=\"false\">(</mo><mo stretchy=\"false\">(</mo><mi>t</mi><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">)</mo></math>-analytic set. For <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>ℤ</mi><mo stretchy=\"false\">(</mo><mo stretchy=\"false\">(</mo><mi>t</mi><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">)</mo></math>-analytic sets, we prove such bounds uniformly for the specialization to every nonarchimedean local field. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"33 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144278613","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":"Weyl sums with multiplicative coefficients and joint equidistribution","authors":"Matteo Bordignon, Cynthia Bortolotto, Bryce Kerr","doi":"10.2140/ant.2025.19.1549","DOIUrl":"https://doi.org/10.2140/ant.2025.19.1549","url":null,"abstract":"<p>We generalise a result of Montgomery and Vaughan regarding exponential sums with multiplicative coefficients to the setting of Weyl sums. As applications, we establish a joint equidistribution result for roots of polynomial congruences and polynomial values which generalises a result of Hooley. We also obtain some new results for mixed character sums. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"221 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144278611","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":"Extending the unconditional support in an Iwaniec–Luo–Sarnak family","authors":"Lucile Devin, Daniel Fiorilli, Anders Södergren","doi":"10.2140/ant.2025.19.1621","DOIUrl":"https://doi.org/10.2140/ant.2025.19.1621","url":null,"abstract":"<p>We study the harmonically weighted one-level density of low-lying zeros of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>L</mi><!--mstyle--><mtext> -</mtext><!--/mstyle--></math>functions in the family of holomorphic newforms of fixed even weight <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>k</mi></math> and prime level <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>N</mi></math> tending to infinity. For this family, Iwaniec, Luo and Sarnak proved that the Katz–Sarnak prediction for the one-level density holds unconditionally when the support of the Fourier transform of the implied test function is contained in <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mo stretchy=\"false\">(</mo><mo>−</mo><mn>3</mn><mspace width=\"-0.17em\"></mspace><mo>∕</mo><mspace width=\"-0.17em\"></mspace><mn>2</mn><mo>,</mo><mn>3</mn><mspace width=\"-0.17em\"></mspace><mo>∕</mo><mspace width=\"-0.17em\"></mspace><mn>2</mn><mo stretchy=\"false\">)</mo></math>. This result was improved by Ricotta–Royer, who increased the admissible support for <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>k</mi>\u0000<mo>≥</mo> <mn>4</mn></math> in a way that is asymptotically as good as the best known GRH result. We extend the admissible support for all <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>k</mi>\u0000<mo>≥</mo> <mn>2</mn></math> to <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mo stretchy=\"false\">(</mo><mo>−</mo><msub><mrow><mi>Θ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>,</mo><msub><mrow><mi>Θ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy=\"false\">)</mo></math>, where <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>Θ</mi></mrow><mrow><mn>2</mn></mrow></msub>\u0000<mo>=</mo> <mn>1</mn><mo>.</mo><mn>8</mn><mn>6</mn><mn>6</mn><mi>…</mi><mo> ⁡<!--FUNCTION APPLICATION--></mo></math> and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>Θ</mi></mrow><mrow><mi>k</mi></mrow></msub></math> tends monotonically to <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>2</mn></math> asymptotically five times faster than what was previously known. The main novelty in our analysis is the use of zero-density estimates for Dirichlet <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>L</mi></math>-functions. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"22 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144278615","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 maximum gonality of a curve over a finite field","authors":"Xander Faber, Jon Grantham, Everett W. Howe","doi":"10.2140/ant.2025.19.1637","DOIUrl":"https://doi.org/10.2140/ant.2025.19.1637","url":null,"abstract":"<p>The gonality of a smooth geometrically connected curve over a field <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>k</mi></math> is the smallest degree of a nonconstant <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>k</mi></math>-morphism from the curve to the projective line. In general, the gonality of a curve of genus <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>g</mi>\u0000<mo>≥</mo> <mn>2</mn></math> is at most <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>2</mn><mi>g</mi>\u0000<mo>−</mo> <mn>2</mn></math>. Over finite fields, a result of F. K. Schmidt from the 1930s can be used to prove that the gonality is at most <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>g</mi>\u0000<mo>+</mo> <mn>1</mn></math>. Via a mixture of geometry and computation, we improve this bound: for a curve of genus <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>g</mi>\u0000<mo>≥</mo> <mn>5</mn></math> over a finite field, the gonality is at most <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>g</mi></math>. For genus <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>g</mi>\u0000<mo>=</mo> <mn>3</mn></math> and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>g</mi>\u0000<mo>=</mo> <mn>4</mn></math>, the same result holds with exactly <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>2</mn><mn>1</mn><mn>7</mn></math> exceptions: there are two curves of genus <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>4</mn></math> and gonality <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>5</mn></math>, and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>2</mn><mn>1</mn><mn>5</mn></math> curves of genus <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>3</mn></math> and gonality <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>4</mn></math>. The genus-<math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>4</mn></math> examples were found in other papers, and we reproduce their equations here; in supplementary material, we provide equations for the genus-<math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>3</mn></math> examples. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"7 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144278653","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":"Solvable and nonsolvable finite groups of the same order type","authors":"Paweł Piwek","doi":"10.2140/ant.2025.19.1663","DOIUrl":"https://doi.org/10.2140/ant.2025.19.1663","url":null,"abstract":"<p>We construct two groups of size <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mn>2</mn></mrow><mrow><mn>3</mn><mn>6</mn><mn>5</mn></mrow></msup>\u0000<mo>⋅</mo> <msup><mrow><mn>3</mn></mrow><mrow><mn>1</mn><mn>0</mn><mn>5</mn></mrow></msup>\u0000<mo>⋅</mo> <msup><mrow><mn>7</mn></mrow><mrow><mn>1</mn><mn>0</mn><mn>4</mn></mrow></msup></math>: a solvable group <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>G</mi></math> and a nonsolvable group <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>H</mi></math> such that for every integer <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>n</mi></math> the groups have the same number of elements of order <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>n</mi></math>. This answers a question posed in 1987 by John G. Thompson.</p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"609 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144278654","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}