Algebra & Number Theory最新文献

{"title":"Breuil–Mézard conjectures for central division algebras","authors":"Andrea Dotto","doi":"10.2140/ant.2025.19.213","DOIUrl":"https://doi.org/10.2140/ant.2025.19.213","url":null,"abstract":"<p>We formulate an analogue of the Breuil–Mézard conjecture for the group of units of a central division algebra over a <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>-adic local field, and we prove that it follows from the conjecture for <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi> GL</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--></mrow><mrow><mi>n</mi></mrow></msub></math>. To do so we construct a transfer of inertial types and Serre weights between the maximal compact subgroups of these two groups, in terms of Deligne–Lusztig theory, and we prove its compatibility with mod <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math> reduction, via the inertial Jacquet–Langlands correspondence and certain explicit character formulas. We also prove analogous statements for <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>ℓ</mi></math>-adic coefficients. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"10 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143071530","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":"Canonical integral models for Shimura varieties of toral type","authors":"Patrick Daniels","doi":"10.2140/ant.2025.19.247","DOIUrl":"https://doi.org/10.2140/ant.2025.19.247","url":null,"abstract":"<p>We prove the Pappas–Rapoport conjecture on the existence of canonical integral models of Shimura varieties with parahoric level structure in the case where the Shimura variety is defined by a torus. As an important ingredient, we show, using the Bhatt–Scholze theory of prismatic <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>F</mi></math>-crystals, that there is a fully faithful functor from <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi mathvariant=\"bold-script\">𝒢</mi></math>-valued crystalline representations of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi> Gal</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--><mo stretchy=\"false\">(</mo><mover accent=\"false\"><mrow><mi>K</mi></mrow><mo accent=\"true\">¯</mo></mover><mo>∕</mo><mi>K</mi><mo stretchy=\"false\">)</mo></math> to <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi mathvariant=\"bold-script\">𝒢</mi></math>-shtukas over <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi> Spd</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--><mo stretchy=\"false\">(</mo><msub><mrow><mi mathvariant=\"bold-script\">𝒪</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\"><mi mathvariant=\"bold-script\">𝒢</mi></math> is a parahoric group scheme over <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 <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi mathvariant=\"bold-script\">𝒪</mi></mrow><mrow><mi>K</mi></mrow></msub></math> is the ring of integers in a <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>-adic field <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>K</mi></math>. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"50 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143071531","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":"Index of coregularity zero log Calabi–Yau pairs","authors":"Stefano Filipazzi, Mirko Mauri, Joaquín Moraga","doi":"10.2140/ant.2025.19.383","DOIUrl":"https://doi.org/10.2140/ant.2025.19.383","url":null,"abstract":"<p>We study the index of log Calabi–Yau pairs <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mo stretchy=\"false\">(</mo><mi>X</mi><mo>,</mo><mi>B</mi><mo stretchy=\"false\">)</mo></math> of coregularity 0. We show that <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>2</mn><mi>λ</mi><mo stretchy=\"false\">(</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>X</mi></mrow></msub>\u0000<mo>+</mo>\u0000<mi>B</mi><mo stretchy=\"false\">)</mo>\u0000<mo>∼</mo> <mn>0</mn></math>, where <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>λ</mi></math> is the Weil index of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mo stretchy=\"false\">(</mo><mi>X</mi><mo>,</mo><mi>B</mi><mo stretchy=\"false\">)</mo></math>. This is in contrast to the case of klt Calabi–Yau varieties, where the index can grow doubly exponentially with the dimension. Our sharp bound on the index extends to the context of generalized log Calabi–Yau pairs, semi-log canonical pairs, and isolated log canonical singularities of coregularity 0. As a consequence, we show that the index of a variety appearing in the Gross–Siebert program or in the Kontsevich–Soibelman program is at most <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>2</mn></math>. Finally, we discuss applications to Calabi–Yau varieties endowed with a finite group action, including holomorphic symplectic varieties endowed with a purely nonsymplectic automorphism. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"39 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143071526","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 reduced arc spaces of toric varieties","authors":"Ilya Dumanski, Evgeny Feigin, Ievgen Makedonskyi, Igor Makhlin","doi":"10.2140/ant.2025.19.313","DOIUrl":"https://doi.org/10.2140/ant.2025.19.313","url":null,"abstract":"<p>An arc space of an affine cone over a projective toric variety is known to be nonreduced in general. It was demonstrated recently that the reduced scheme structure of arc spaces is very meaningful from algebro-geometric, representation-theoretic and combinatorial points of view. In this paper we develop a general machinery for the description of the reduced arc spaces of affine cones over toric varieties. We apply our techniques to a number of classical cases and explore some connections with representation theory of current algebras. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"157 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143072092","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":"Divisibility of character values of the symmetric group by prime powers","authors":"Sarah Peluse, Kannan Soundararajan","doi":"10.2140/ant.2025.19.365","DOIUrl":"https://doi.org/10.2140/ant.2025.19.365","url":null,"abstract":"<p>Let <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>k</mi></math> be a positive integer. We show that, as <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>n</mi></math> goes to infinity, almost every entry of the character table of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math> is divisible by <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>k</mi></math>. This proves a conjecture of Miller. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"22 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143072091","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 geometric Breuil–Mézard conjecture for two-dimensional potentially Barsotti–Tate Galois representations","authors":"Ana Caraiani, Matthew Emerton, Toby Gee, David Savitt","doi":"10.2140/ant.2025.19.287","DOIUrl":"https://doi.org/10.2140/ant.2025.19.287","url":null,"abstract":"<p>We establish a geometrization of the Breuil–Mézard conjecture for potentially Barsotti–Tate representations, as well as of the weight part of Serre’s conjecture, for moduli stacks of two-dimensional mod <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math> representations of the absolute Galois group of a <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>-adic local field. These results are first proved for the stacks of our earlier papers, and then transferred to the stacks of Emerton and Gee by means of a comparison of versal rings. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"54 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143071532","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":"Vanishing results for the coherent cohomology of automorphic vector bundles over the Siegel variety in positive characteristic","authors":"Thibault Alexandre","doi":"10.2140/ant.2025.19.143","DOIUrl":"https://doi.org/10.2140/ant.2025.19.143","url":null,"abstract":"<p>We prove vanishing results for the coherent cohomology of the good reduction modulo <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math> of the Siegel modular variety with coefficients in some automorphic bundles. We show that for an automorphic bundle with highest weight <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>λ</mi></math> near the walls of the antidominant Weyl chamber, there is an integer <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>e</mi>\u0000<mo>≥</mo> <mn>0</mn></math> such that the cohomology is concentrated in degrees <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mo stretchy=\"false\">[</mo><mn>0</mn><mo>,</mo><mi>e</mi><mo stretchy=\"false\">]</mo></math>. The accessible weights with our method are not necessarily regular and not necessarily <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>-small. Since our method is technical, we also provide an algorithm written in SageMath that computes explicitly the vanishing results. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"31 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142776368","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":"Picard rank jumps for K3 surfaces with bad reduction","authors":"Salim Tayou","doi":"10.2140/ant.2025.19.77","DOIUrl":"https://doi.org/10.2140/ant.2025.19.77","url":null,"abstract":"<p>Let <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>X</mi></math> be a K3 surface over a number field. We prove that <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>X</mi></math> has infinitely many specializations where its Picard rank jumps, hence extending our previous work with Shankar, Shankar and Tang to the case where <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>X</mi></math> has bad reduction. We prove a similar result for generically ordinary nonisotrivial families of K3 surfaces over curves over <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></math> which extends previous work of Maulik, Shankar and Tang. As a consequence, we give a new proof of the ordinary Hecke orbit conjecture for orthogonal and unitary Shimura varieties. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"215 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142776369","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 modification of the linear sieve, and the count of twin primes","authors":"Jared Duker Lichtman","doi":"10.2140/ant.2025.19.1","DOIUrl":"https://doi.org/10.2140/ant.2025.19.1","url":null,"abstract":"<p>We introduce a modification of the linear sieve whose weights satisfy strong factorization properties, and consequently equidistribute primes up to size <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>x</mi></math> in arithmetic progressions to moduli up to <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>x</mi></mrow><mrow><mn>1</mn><mn>0</mn><mo>∕</mo><mn>1</mn><mn>7</mn></mrow></msup></math>. This surpasses the level of distribution <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>x</mi></mrow><mrow><mn>4</mn><mo>∕</mo><mn>7</mn></mrow></msup></math> with the linear sieve weights from well-known work of Bombieri, Friedlander, and Iwaniec, and which was recently extended to <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>x</mi></mrow><mrow><mn>7</mn><mo>∕</mo><mn>1</mn><mn>2</mn></mrow></msup></math> by Maynard. As an application, we obtain a new upper bound on the count of twin primes. Our method simplifies the 2004 argument of Wu, and gives the largest percentage improvement since the 1986 bound of Bombieri, Friedlander, and Iwaniec. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"28 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142776367","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":"Ranks of abelian varieties in cyclotomic twist families","authors":"Ari Shnidman, Ariel Weiss","doi":"10.2140/ant.2025.19.39","DOIUrl":"https://doi.org/10.2140/ant.2025.19.39","url":null,"abstract":"&lt;p&gt;Let &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/math&gt; be an abelian variety over a number field &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/math&gt;, and suppose that &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;ℤ&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;[&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ζ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=\"false\"&gt;]&lt;/mo&gt;&lt;/math&gt; embeds in &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt; End&lt;/mi&gt;&lt;mo&gt; ⁡&lt;!--FUNCTION APPLICATION--&gt; &lt;/mo&gt;&lt;!--nolimits--&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mover accent=\"true\"&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mo accent=\"true\"&gt;¯&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/math&gt;, for some root of unity &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ζ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt; of order &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;n&lt;/mi&gt;\u0000&lt;mo&gt;=&lt;/mo&gt; &lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;. Assuming that the Galois action on the finite group &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;[&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;\u0000&lt;mo&gt;−&lt;/mo&gt; &lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ζ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=\"false\"&gt;]&lt;/mo&gt;&lt;/math&gt; is sufficiently reducible, we bound the average rank of the Mordell–Weil groups &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;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;/math&gt;, as &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;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt; varies through the family of &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;μ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;-twists of &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/math&gt;. Combining this result with the recently proved uniform Mordell–Lang conjecture, we prove near-uniform bounds for the number of rational points in twist families of bicyclic trigonal curves &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;\u0000&lt;mo&gt;=&lt;/mo&gt;\u0000&lt;mi&gt;f&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;/math&gt;, as well as in twist families of theta divisors of cyclic trigonal curves &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;\u0000&lt;mo&gt;=&lt;/mo&gt;\u0000&lt;mi&gt;f&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;/math&gt;. Our main technical result is the determination of the average size of a &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/math&gt;-isogeny Selmer group in a family of &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;msub&gt;&lt;mr","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"262 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142776372","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}