Algebra & Number Theory最新文献

{"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}

{"title":"Pullback formulas for arithmetic cycles on orthogonal Shimura varieties","authors":"Benjamin Howard","doi":"10.2140/ant.2025.19.1495","DOIUrl":"https://doi.org/10.2140/ant.2025.19.1495","url":null,"abstract":"<p>On an orthogonal Shimura variety, one has a collection of arithmetic special cycles in the Gillet–Soulé arithmetic Chow group. We describe how these cycles behave under pullback to an embedded orthogonal Shimura variety of lower dimension. The bulk of the paper is devoted to cases in which the special cycles intersect the embedded Shimura variety improperly, which requires that we analyze logarithmic expansions of Green currents on the deformation to the normal bundle of the embedding. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"224 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144279032","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":"Ideals in enveloping algebras of affine Kac–Moody algebras","authors":"Rekha Biswal, Susan J. Sierra","doi":"10.2140/ant.2025.19.1199","DOIUrl":"https://doi.org/10.2140/ant.2025.19.1199","url":null,"abstract":"<p>Let <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>L</mi></math> be an affine Kac–Moody algebra, with central element <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>c</mi></math>, and let <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>λ</mi>\u0000<mo>∈</mo>\u0000<mi>ℂ</mi></math>. We study two-sided ideals in the central quotient <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>U</mi></mrow><mrow><mi>λ</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>L</mi><mo stretchy=\"false\">)</mo>\u0000<mo>:</mo><mo>=</mo>\u0000<mi>U</mi><mo stretchy=\"false\">(</mo><mi>L</mi><mo stretchy=\"false\">)</mo><mo>∕</mo><mo stretchy=\"false\">(</mo><mi>c</mi>\u0000<mo>−</mo>\u0000<mi>λ</mi><mo stretchy=\"false\">)</mo></math> of the universal enveloping algebra of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>L</mi></math> and prove: </p><ol>\u0000<li>\u0000<p>If <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>λ</mi><mo>≠</mo><mn>0</mn></math> then <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>U</mi></mrow><mrow><mi>λ</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>L</mi><mo stretchy=\"false\">)</mo></math> is simple. </p></li>\u0000<li>\u0000<p>The algebra <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>U</mi></mrow><mrow><mn>0</mn></mrow></msub><mo stretchy=\"false\">(</mo><mi>L</mi><mo stretchy=\"false\">)</mo></math> has <span>just-infinite growth</span>, in the sense that any proper quotient has polynomial growth.</p></li></ol>\u0000<p> As an immediate corollary, we show that the annihilator of any nontrivial integrable highest-weight representation of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>L</mi></math> is centrally generated, extending a result of Chari for Verma modules. </p><p> We also show that universal enveloping algebras of loop algebras and current algebras of finite-dimensional simple Lie algebras have just-infinite growth, and prove similar results to the two results above for quotients of symmetric algebras of these Lie algebras by Poisson ideals. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"25 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144066644","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":"Geometry-of-numbers methods in the cusp","authors":"Arul Shankar, Artane Siad, Ashvin A. Swaminathan, Ila Varma","doi":"10.2140/ant.2025.19.1099","DOIUrl":"https://doi.org/10.2140/ant.2025.19.1099","url":null,"abstract":"<p>We develop new methods for counting integral orbits having bounded invariants that lie inside the cusps of fundamental domains for coregular representations. We illustrate these methods for a representation of cardinal interest in number theory, namely that of the split orthogonal group acting on the space of quadratic forms. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"57 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144066643","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":"Semistable representations as limits of crystalline representations","authors":"Anand Chitrao, Eknath Ghate, Seidai Yasuda","doi":"10.2140/ant.2025.19.1049","DOIUrl":"https://doi.org/10.2140/ant.2025.19.1049","url":null,"abstract":"&lt;p&gt;We construct an explicit sequence &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;V&lt;/mi&gt; &lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&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;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt; of crystalline representations of exceptional weights converging to a given irreducible two-dimensional semistable representation &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;V&lt;/mi&gt; &lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi mathvariant=\"bold-script\"&gt;ℒ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt; of &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt; Gal&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;msub&gt;&lt;mrow&gt;&lt;mover accent=\"false\"&gt;&lt;mrow&gt;&lt;mi&gt;ℚ&lt;/mi&gt; &lt;/mrow&gt;&lt;mo accent=\"true\"&gt;¯&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;/&lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ℚ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;/math&gt;. The convergence takes place in the blow-up space of two-dimensional trianguline representations studied by Colmez and Chenevier. The process of blow-up is described in detail in the rigid-analytic setting and may be of independent interest. Also, we recover a formula of Stevens expressing the &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi mathvariant=\"bold-script\"&gt;ℒ&lt;/mi&gt;&lt;/math&gt;-invariant as a logarithmic derivative. &lt;/p&gt;&lt;p&gt; Our result can be used to compute the reduction of &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;V&lt;/mi&gt; &lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi mathvariant=\"bold-script\"&gt;ℒ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt; in terms of the reductions of the &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;V&lt;/mi&gt; &lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&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;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;. For instance, using the zig-zag conjecture we recover (resp. extend) the work of Breuil and Mézard and Guerberoff and Park computing the reductions of the &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;V&lt;/mi&gt; &lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi mathvariant=\"bold-script\"&gt;ℒ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt; for weights &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/math&gt; at most &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;p&lt;/mi&gt;\u0000&lt;mo&gt;−&lt;/mo&gt; &lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt; (resp. &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;p&lt;/mi&gt;\u0000&lt;mo&gt;+&lt;/mo&gt; &lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/math&gt; at least on the inertia subgroup. In the cases where zig-zag is known, we are further able to obtain some new information about the reductions for small odd weights. &lt;/p&gt;&lt;p&gt; In the cases where zig-zag is known, we are further able to obtain some new information about the reduct","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"6 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144066642","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}