{"title":"Malle’s conjecture for fair counting functions","authors":"Peter Koymans, Carlo Pagano","doi":"10.2140/ant.2025.19.1007","DOIUrl":"https://doi.org/10.2140/ant.2025.19.1007","url":null,"abstract":"<p>We show that the naive adaptation of Malle’s conjecture to fair counting functions is not true in general. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"97 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143863043","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":"Syzygies of tangent-developable surfaces and K3 carpets via secant varieties","authors":"Jinhyung Park","doi":"10.2140/ant.2025.19.1029","DOIUrl":"https://doi.org/10.2140/ant.2025.19.1029","url":null,"abstract":"<p>We give simple geometric proofs of Aprodu, Farkas, Papadima, Raicu and Weyman’s theorem on syzygies of tangent-developable surfaces of rational normal curves and Raicu and Sam’s result on syzygies of K3 carpets. As a consequence, we obtain a quick proof of Green’s conjecture for general curves of genus <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>g</mi></math> over an algebraically closed field <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi> k</mi><mo> <!--FUNCTION APPLICATION--> </mo><!--nolimits--></math> with <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi> char</mi><mo> <!--FUNCTION APPLICATION--> </mo><!--nolimits--><mo stretchy=\"false\">(</mo><mi>k</mi><mo> <!--FUNCTION APPLICATION--> </mo><!--nolimits--><mo stretchy=\"false\">)</mo>\u0000<mo>=</mo> <mn>0</mn></math> or <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi> char</mi><mo> <!--FUNCTION APPLICATION--> </mo><!--nolimits--><mo stretchy=\"false\">(</mo><mi>k</mi><mo> <!--FUNCTION APPLICATION--> </mo><!--nolimits--><mo stretchy=\"false\">)</mo>\u0000<mo>≥</mo><mo stretchy=\"false\">⌊</mo><mo stretchy=\"false\">(</mo><mi>g</mi>\u0000<mo>−</mo> <mn>1</mn><mo stretchy=\"false\">)</mo><mo>∕</mo><mn>2</mn><mo stretchy=\"false\">⌋</mo></math>. Our approach provides a new way to study tangent-developable surfaces in general. Along the way, we show the arithmetic normality of tangent-developable surfaces of arbitrary smooth projective curves of large degree. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"219 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143863069","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 D-module of an isolated singularity","authors":"Thomas Bitoun","doi":"10.2140/ant.2025.19.763","DOIUrl":"https://doi.org/10.2140/ant.2025.19.763","url":null,"abstract":"<p>Let <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>Z</mi></math> be the germ of a complex hypersurface isolated singularity of equation <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>f</mi></math>, with <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>Z</mi></math> at least of dimension <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>2</mn></math>. We consider the family of analytic <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>D</mi></math>-modules generated by the powers of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>1</mn><mo>∕</mo><mi>f</mi></math> and describe it in terms of the pole order filtration on the de Rham cohomology of the complement of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mo stretchy=\"false\">{</mo><mi>f</mi>\u0000<mo>=</mo> <mn>0</mn><mo stretchy=\"false\">}</mo></math> in the neighbourhood of the singularity. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"18 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143695615","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":"Ribbon Schur functors","authors":"Keller VandeBogert","doi":"10.2140/ant.2025.19.771","DOIUrl":"https://doi.org/10.2140/ant.2025.19.771","url":null,"abstract":"<p>We investigate a generalization of the classical notion of a Schur functor associated to a ribbon diagram. These functors are defined with respect to an arbitrary algebra, and in the case that the underlying algebra is the symmetric/exterior algebra, we recover the classical definition of Schur/Weyl functors, respectively. In general, we construct a family of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>3</mn></math>-term complexes categorifying the classical concatenation/near-concatenation identity for symmetric functions, and one of our main results is that the exactness of these <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>3</mn></math>-term complexes is equivalent to the Koszul property of the underlying algebra <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>A</mi></math>. We further generalize these ribbon Schur functors to the notion of a multi-Schur functor and construct a canonical filtration of these objects whose associated graded pieces are described explicitly; one consequence of this filtration is a complete equivariant description of the syzygies of arbitrary Segre products of Koszul modules over the Segre product of Koszul algebras. Further applications to the equivariant structure of derived invariants, symmetric function identities, and Koszulness of certain classes of modules are explored at the end, along with a characteristic-free computation of the regularity of the Schur functor <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi mathvariant=\"double-struck\">𝕊</mi></mrow><mrow><mi>λ</mi></mrow></msup></math> applied to the tautological subbundle on projective space. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"28 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143695616","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":"Odd moments in the distribution of primes","authors":"Vivian Kuperberg","doi":"10.2140/ant.2025.19.617","DOIUrl":"https://doi.org/10.2140/ant.2025.19.617","url":null,"abstract":"<p>Montgomery and Soundararajan showed that the distribution of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>ψ</mi><mo stretchy=\"false\">(</mo><mi>x</mi>\u0000<mo>+</mo>\u0000<mi>H</mi><mo stretchy=\"false\">)</mo>\u0000<mo>−</mo>\u0000<mi>ψ</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo></math>, for <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>0</mn>\u0000<mo>≤</mo>\u0000<mi>x</mi>\u0000<mo>≤</mo>\u0000<mi>N</mi></math>, is approximately normal with mean <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\">\u0000<mo>∼</mo>\u0000<mi>H</mi></math> and variance <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\">\u0000<mo>∼</mo>\u0000<mi>H</mi><mi>log</mi><mo> <!--FUNCTION APPLICATION--> </mo><!--nolimits--><mo stretchy=\"false\">(</mo><mi>N</mi><mo>∕</mo><mi>H</mi><mo stretchy=\"false\">)</mo></math>, when <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>N</mi></mrow><mrow><mi>δ</mi></mrow></msup>\u0000<mo>≤</mo>\u0000<mi>H</mi>\u0000<mo>≤</mo> <msup><mrow><mi>N</mi></mrow><mrow><mn>1</mn><mo>−</mo><mi>δ</mi></mrow></msup> </math>. Their work depends on showing that sums <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>R</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>h</mi><mo stretchy=\"false\">)</mo></math> of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>k</mi></math>-term singular series are <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>μ</mi></mrow><mrow><mi>k</mi></mrow></msub><msup><mrow><mo stretchy=\"false\">(</mo><mo>−</mo><mi>h</mi><mi>log</mi><mo> <!--FUNCTION APPLICATION--> </mo><!--nolimits--><mi>h</mi>\u0000<mo>+</mo>\u0000<mi>A</mi><mi>h</mi><mo stretchy=\"false\">)</mo></mrow><mrow><mi>k</mi><mo>∕</mo><mn>2</mn></mrow></msup>\u0000<mo>+</mo> <msub><mrow><mi>O</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy=\"false\">(</mo><msup><mrow><mi>h</mi></mrow><mrow><mi>k</mi><mo>∕</mo><mn>2</mn><mo>−</mo><mn>1</mn><mo>∕</mo><mo stretchy=\"false\">(</mo><mn>7</mn><mi>k</mi><mo stretchy=\"false\">)</mo><mo>+</mo><mi>𝜀</mi></mrow></msup><mo stretchy=\"false\">)</mo></math>, where <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>A</mi></math> is a constant and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>μ</mi></mrow><mrow><mi>k</mi></mrow></msub></math> are the Gaussian moment constants. We study lower-order terms in the size of these moments. We conjecture that when <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>k</mi></math> is odd, <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>R</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>h</mi><mo stretchy=\"false\">)</mo>\u0000<mo>≍</mo> <msup><mrow><mi>h</mi></mrow><mrow><mo stretchy=\"false\">(</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo stretchy=\"false\">)</mo><mo>∕</mo><mn>2</mn></mrow></msup><msup><mrow><mo stretchy=\"false\">(</m","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"124 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143695760","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":"Efficient resolution of Thue–Mahler equations","authors":"Adela Gherga, Samir Siksek","doi":"10.2140/ant.2025.19.667","DOIUrl":"https://doi.org/10.2140/ant.2025.19.667","url":null,"abstract":"<p>A Thue–Mahler equation is a Diophantine equation of the form </p>\u0000<div><math display=\"block\" xmlns=\"http://www.w3.org/1998/Math/MathML\">\u0000<mi>F</mi><mo stretchy=\"false\">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi>\u0000<mo stretchy=\"false\">)</mo>\u0000<mo>=</mo>\u0000<mi>a</mi>\u0000<mo>⋅</mo> <msubsup><mrow><mi>p</mi></mrow><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>z</mi></mrow><mrow><mn>1</mn></mrow></msub>\u0000</mrow></msubsup><mo>⋯</mo><msubsup><mrow><mi>p</mi></mrow><mrow><mi>v</mi></mrow><mrow><msub><mrow><mi>z</mi></mrow><mrow><mi>v</mi></mrow></msub>\u0000</mrow></msubsup><mo>,</mo><mspace width=\"1em\"></mspace><mi>gcd</mi><mo> <!--FUNCTION APPLICATION--> </mo><mo stretchy=\"false\">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi>\u0000<mo stretchy=\"false\">)</mo>\u0000<mo>=</mo> <mn>1</mn>\u0000</math>\u0000</div>\u0000<p> where <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>F</mi></math> is an irreducible binary form of degree at least <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>3</mn></math> with integer coefficients, <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>a</mi></math> is a nonzero integer and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>p</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mi>…</mi><mo> <!--FUNCTION APPLICATION--></mo><mo>,</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>v</mi></mrow></msub></math> are rational primes. Existing algorithms for resolving such equations require computations in the field <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>L</mi>\u0000<mo>=</mo>\u0000<mi>ℚ</mi><mo stretchy=\"false\">(</mo><mi>𝜃</mi><mo>,</mo><msup><mrow><mi>𝜃</mi></mrow><mrow><mi>′</mi></mrow></msup><mo>,</mo><msup><mrow><mi>𝜃</mi></mrow><mrow><mi>′</mi><mi>′</mi></mrow></msup><mo stretchy=\"false\">)</mo></math>, where <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>𝜃</mi></math>, <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>𝜃</mi></mrow><mrow><mi>′</mi></mrow></msup></math>, <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>𝜃</mi></mrow><mrow><mi>′</mi><mi>′</mi></mrow></msup></math> are distinct roots of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>F</mi><mo stretchy=\"false\">(</mo><mi>X</mi><mo>,</mo><mn>1</mn><mo stretchy=\"false\">)</mo>\u0000<mo>=</mo> <mn>0</mn></math>. We give a new algorithm that requires computations only in <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>K</mi>\u0000<mo>=</mo>\u0000<mi>ℚ</mi><mo stretchy=\"false\">(</mo><mi>𝜃</mi><mo stretchy=\"false\">)</mo></math> making it far more suited for higher degree examples. We also introduce a lattice sieving technique reminiscent of the Mordell–Weil sieve that makes it practical to tackle Thue–Mahler equations of higher degree and with larger sets of primes than was previously possible. We give several examples including one of degree <math display=\"inline\" xmlns=\"http://www.w3.org/","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"57 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143695614","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":"Automorphisms of del Pezzo surfaces in characteristic 2","authors":"Igor Dolgachev, Gebhard Martin","doi":"10.2140/ant.2025.19.715","DOIUrl":"https://doi.org/10.2140/ant.2025.19.715","url":null,"abstract":"<p>We classify the automorphism groups of del Pezzo surfaces of degrees 1 and 2 over an algebraically closed field of characteristic 2. This finishes the classification of automorphism groups of del Pezzo surfaces in all characteristics. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"71 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143695617","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":"Fermat’s last theorem over ℚ(,)","authors":"Maleeha Khawaja, Frazer Jarvis","doi":"10.2140/ant.2025.19.457","DOIUrl":"https://doi.org/10.2140/ant.2025.19.457","url":null,"abstract":"<p>In this paper, we begin the study of the Fermat equation <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msup>\u0000<mo>+</mo> <msup><mrow><mi>y</mi></mrow><mrow><mi>n</mi></mrow></msup>\u0000<mo>=</mo> <msup><mrow><mi>z</mi></mrow><mrow><mi>n</mi></mrow></msup></math> over real biquadratic fields. In particular, we prove that there are no nontrivial solutions to the Fermat equation over <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>ℚ</mi><mo stretchy=\"false\">(</mo><msqrt><mrow><mn>2</mn></mrow></msqrt><mo>,</mo><msqrt><mrow><mn>3</mn></mrow></msqrt><mo stretchy=\"false\">)</mo></math> for <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>n</mi>\u0000<mo>≥</mo> <mn>4</mn></math>. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"6 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143452142","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":"Abelian varieties over finite fields and their groups of rational points","authors":"Stefano Marseglia, Caleb Springer","doi":"10.2140/ant.2025.19.521","DOIUrl":"https://doi.org/10.2140/ant.2025.19.521","url":null,"abstract":"<p>Over a finite field <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></math>, abelian varieties with commutative endomorphism rings can be described by using modules over orders in étale algebras. By exploiting this connection, we produce four theorems regarding groups of rational points and self-duality, along with explicit examples. First, when <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi> End</mi><mo> <!--FUNCTION APPLICATION--> </mo><!--nolimits--><mo stretchy=\"false\">(</mo><mi>A</mi><mo stretchy=\"false\">)</mo></math> is locally Gorenstein, we show that the group structure of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>A</mi><mo stretchy=\"false\">(</mo><msub><mrow><mi mathvariant=\"double-struck\">𝔽</mi></mrow><mrow><mi>q</mi></mrow></msub><mo stretchy=\"false\">)</mo></math> is determined by <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi> End</mi><mo> <!--FUNCTION APPLICATION--> </mo><!--nolimits--><mo stretchy=\"false\">(</mo><mi>A</mi><mo stretchy=\"false\">)</mo></math>. In fact, the same conclusion is attained if <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi> End</mi><mo> <!--FUNCTION APPLICATION--> </mo><!--nolimits--><mo stretchy=\"false\">(</mo><mi>A</mi><mo stretchy=\"false\">)</mo></math> has local Cohen–Macaulay type at most <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>2</mn></math>, under the additional assumption that <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>A</mi></math> is ordinary or <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>q</mi></math> is prime, although the conclusion is not true in general. Second, the description in the Gorenstein case is used to characterize cyclic isogeny classes in terms of conductor ideals. Third, going in the opposite direction, we characterize squarefree isogeny classes of abelian varieties with <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>N</mi></math> rational points in which every abelian group of order <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>N</mi></math> is realized as a group of rational points. Finally, we study when an abelian variety <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>A</mi></math> over <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></math> and its dual <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>A</mi></mrow><mrow><mo>∨</mo></mrow></msup></math> satisfy or fail to satisfy several interrelated properties, namely <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>A</mi><mi>≅</mi><mo> <!--FUNCTION APPLICATION--></mo><msup><mrow><mi>A</mi></mrow><mrow><mo>∨</mo></mrow>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"31 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143451709","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 Lyndon–Demushkin method and crystalline lifts of G2-valued Galois representations","authors":"Zhongyipan Lin","doi":"10.2140/ant.2025.19.415","DOIUrl":"https://doi.org/10.2140/ant.2025.19.415","url":null,"abstract":"<p>We develop obstruction theory for lifting characteristic-<math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math> local Galois representations valued in reductive groups of type <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>B</mi></mrow><mrow><mi>l</mi></mrow></msub></math>, <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>C</mi></mrow><mrow><mi>l</mi></mrow></msub></math>, <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>D</mi></mrow><mrow><mi>l</mi></mrow></msub></math> or <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub></math>. An application of the Emerton–Gee stack then reduces the existence of crystalline lifts to a purely combinatorial problem when <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math> is not too small. </p><p> As a toy example, we show for all local fields <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>K</mi><mo>∕</mo><msub><mrow><mi>ℚ</mi></mrow><mrow><mi>p</mi></mrow></msub></math>, with <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi>\u0000<mo>></mo> <mn>3</mn></math>, all representations <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mover accent=\"true\"><mrow><mi>ρ</mi></mrow><mo accent=\"true\">¯</mo></mover>\u0000<mo>:</mo> <msub><mrow><mi>G</mi></mrow><mrow><mi>K</mi></mrow></msub>\u0000<mo>→</mo> <msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub><mo stretchy=\"false\">(</mo><msub><mrow><mover accent=\"true\"><mrow><mi mathvariant=\"double-struck\">𝔽</mi></mrow><mo accent=\"true\">¯</mo></mover></mrow><mrow><mi>p</mi></mrow></msub><mo stretchy=\"false\">)</mo></math> admit a crystalline lift <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>ρ</mi>\u0000<mo>:</mo> <msub><mrow><mi>G</mi></mrow><mrow><mi>K</mi></mrow></msub>\u0000<mo>→</mo> <msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub><mo stretchy=\"false\">(</mo><msub><mrow><mover accent=\"true\"><mrow><mi>ℤ</mi></mrow><mo accent=\"true\">¯</mo></mover></mrow><mrow><mi>p</mi></mrow></msub><mo stretchy=\"false\">)</mo></math>, where <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub></math> is the exceptional Chevalley group of type <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub></math>. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"376 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143451707","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}
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