{"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}
Arul Shankar, Artane Siad, Ashvin A. Swaminathan, Ila Varma
{"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":"<p>We construct an explicit sequence <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>V</mi> </mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub></math> of crystalline representations of exceptional weights converging to a given irreducible two-dimensional semistable representation <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>V</mi> </mrow><mrow><mi>k</mi><mo>,</mo><mi mathvariant=\"bold-script\">ℒ</mi></mrow></msub></math> of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi> Gal</mi><mo> <!--FUNCTION APPLICATION--> </mo><!--nolimits--><mo stretchy=\"false\">(</mo><msub><mrow><mover accent=\"false\"><mrow><mi>ℚ</mi> </mrow><mo accent=\"true\">¯</mo></mover></mrow><mrow><mi>p</mi></mrow></msub></math>/<math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>ℚ</mi></mrow><mrow><mi>p</mi></mrow></msub><mo stretchy=\"false\">)</mo></math>. 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 <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi mathvariant=\"bold-script\">ℒ</mi></math>-invariant as a logarithmic derivative. </p><p> Our result can be used to compute the reduction of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>V</mi> </mrow><mrow><mi>k</mi><mo>,</mo><mi mathvariant=\"bold-script\">ℒ</mi></mrow></msub></math> in terms of the reductions of the <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>V</mi> </mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub></math>. 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 <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>V</mi> </mrow><mrow><mi>k</mi><mo>,</mo><mi mathvariant=\"bold-script\">ℒ</mi></mrow></msub></math> for weights <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>k</mi></math> at most <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi>\u0000<mo>−</mo> <mn>1</mn></math> (resp. <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi>\u0000<mo>+</mo> <mn>1</mn><mo stretchy=\"false\">)</mo><mo>,</mo></math> 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. </p><p> 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}
{"title":"The integral Chow ring of weighted blow-ups","authors":"Veronica Arena, Stephen Obinna","doi":"10.2140/ant.2025.19.1231","DOIUrl":"https://doi.org/10.2140/ant.2025.19.1231","url":null,"abstract":"<p>We give a formula for the Chow rings of weighted blow-ups. Along the way, we also compute the Chow rings of weighted projective stack bundles, a formula for the Gysin homomorphism of a weighted blow-up, and a generalization of the splitting principle. In addition, in the Appendix we compute the Chern class of a weighted blow-up. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"29 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144066528","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":"Explicit isogenies of prime degree over number fields","authors":"Barinder S. Banwait, Maarten Derickx","doi":"10.2140/ant.2025.19.1147","DOIUrl":"https://doi.org/10.2140/ant.2025.19.1147","url":null,"abstract":"<p>We provide an explicit and algorithmic version of a theorem of Momose classifying isogenies of prime degree of elliptic curves over number fields, which we implement in Sage and PARI/GP. Combining this algorithm with recent work of Box, Gajović and Goodman we obtain the first classifications of the possible prime degree isogenies of elliptic curves over cubic number fields, as well as for several quadratic fields not previously known. While the correctness of the general algorithm relies on the generalised Riemann hypothesis, the algorithm is unconditional for the restricted class of semistable elliptic curves. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"33 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144066728","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":"Presentations of Galois groups of maximal extensions with restricted ramification","authors":"Yuan Liu","doi":"10.2140/ant.2025.19.835","DOIUrl":"https://doi.org/10.2140/ant.2025.19.835","url":null,"abstract":"<p>Motivated by the work of Lubotzky, we use Galois cohomology to study the difference between the number of generators and the minimal number of relations in a presentation of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>G</mi></mrow><mrow><mi>S</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>k</mi><mo stretchy=\"false\">)</mo></math>, the Galois group of the maximal extension of a global field <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>k</mi></math> that is unramified outside a finite set <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>S</mi></math> of places, as <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>k</mi></math> varies among a certain family of extensions of a fixed global field <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>Q</mi></math>. We define a group <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>B</mi></mrow><mrow><mi>S</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>k</mi><mo>,</mo><mi>A</mi><mo stretchy=\"false\">)</mo></math>, for each finite simple <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>G</mi></mrow><mrow><mi>S</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>k</mi><mo stretchy=\"false\">)</mo></math>-module <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>A</mi></math>, to generalize the work of Koch and Shafarevich on the pro-<math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>ℓ</mi></math> completion of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>G</mi></mrow><mrow><mi>S</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>k</mi><mo stretchy=\"false\">)</mo></math>. We prove that <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>G</mi></mrow><mrow><mi>S</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>k</mi><mo stretchy=\"false\">)</mo></math> always admits a balanced presentation when it is finitely generated. In the setting of the nonabelian Cohen–Lenstra heuristics, we prove that the unramified Galois groups studied by the Liu–Wood–Zureick-Brown conjecture always admit a balanced presentation in the form of the random group in the conjecture. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"8 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143863037","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":"Motivic distribution of rational curves and twisted products of toric varieties","authors":"Loïs Faisant","doi":"10.2140/ant.2025.19.883","DOIUrl":"https://doi.org/10.2140/ant.2025.19.883","url":null,"abstract":"<p>This work concerns asymptotical stabilisation phenomena occurring in the moduli space of sections of certain algebraic families over a smooth projective curve, whenever the generic fibre of the family is a smooth projective Fano variety, or not far from being Fano. </p><p> We describe the expected behaviour of the class, in a ring of motivic integration, of the moduli space of sections of given numerical class. Up to an adequate normalisation, it should converge, when the class of the sections goes arbitrarily far from the boundary of the dual of the effective cone, to an effective element given by a motivic Euler product. Such a principle can be seen as an analogue for rational curves of the Batyrev–Manin–Peyre principle for rational points. </p><p> The central tool of this article is the property of equidistribution of curves. We show that this notion does not depend on the choice of a model of the generic fibre, and that equidistribution of curves holds for smooth projective split toric varieties. As an application, we study the Batyrev–Manin–Peyre principle for curves on a certain kind of twisted products.</p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"33 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143863038","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":"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":"Smooth cuboids in group theory","authors":"Joshua Maglione, Mima Stanojkovski","doi":"10.2140/ant.2025.19.967","DOIUrl":"https://doi.org/10.2140/ant.2025.19.967","url":null,"abstract":"<p>A smooth cuboid can be identified with a <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>3</mn><mo>×</mo><mn>3</mn></math> matrix of linear forms in three variables, with coefficients in a field <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>K</mi></math>, whose determinant describes a smooth cubic in the projective plane. To each such matrix one can associate a group scheme over <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>K</mi></math>. We produce isomorphism invariants of these groups in terms of their <span>adjoint algebras</span>, which also give information on the number of their maximal abelian subgroups. Moreover, when <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>K</mi></math> is finite, we give a characterization of the isomorphism types of the groups in terms of isomorphisms of elliptic curves and also describe their automorphism groups. We conclude by applying our results to the determination of the automorphism groups and isomorphism testing of finite <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>-groups of class <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>2</mn></math> and exponent <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math> arising in this way. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"71 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143863039","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}
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