{"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}
Antonio Cauchi, Francesco Lemma, Joaquín Rodrigues Jacinto
{"title":"Algebraic cycles and functorial lifts from G2 to PGSp6","authors":"Antonio Cauchi, Francesco Lemma, Joaquín Rodrigues Jacinto","doi":"10.2140/ant.2025.19.551","DOIUrl":"https://doi.org/10.2140/ant.2025.19.551","url":null,"abstract":"<p>We study instances of Beilinson–Tate conjectures for automorphic representations of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi> PGSp</mi><mo> <!--FUNCTION APPLICATION--> </mo><!--nolimits--></mrow><mrow><mn>6</mn></mrow></msub></math> whose spin <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>L</mi></math>-function has a pole at <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>s</mi>\u0000<mo>=</mo> <mn>1</mn></math>. We construct algebraic cycles of codimension 3 in the Siegel–Shimura variety of dimension 6 and we relate its regulator to the residue at <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>s</mi>\u0000<mo>=</mo> <mn>1</mn></math> of the <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>L</mi></math>-function of certain cuspidal forms of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi> PGSp</mi><mo> <!--FUNCTION APPLICATION--> </mo><!--nolimits--></mrow><mrow><mn>6</mn></mrow></msub></math>. Using the exceptional theta correspondence between the split 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> and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi> PGSp</mi><mo> <!--FUNCTION APPLICATION--> </mo><!--nolimits--></mrow><mrow><mn>6</mn></mrow></msub></math> and assuming the nonvanishing of a certain archimedean integral, this allows us to confirm a conjecture of Gross and Savin on rank-<math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>7</mn></math> motives 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":"15 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143451708","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}
Régis de la Bretèche, Daniel Fiorilli, Florent Jouve
{"title":"Moments in the Chebotarev density theorem: general class functions","authors":"Régis de la Bretèche, Daniel Fiorilli, Florent Jouve","doi":"10.2140/ant.2025.19.481","DOIUrl":"https://doi.org/10.2140/ant.2025.19.481","url":null,"abstract":"<p>We find lower bounds on higher moments of the error term in the Chebotarev density theorem. Inspired by the work of Bellaïche, we consider general class functions and prove bounds which depend on norms associated to these functions. Our bounds also involve the ramification and Galois theoretical information of the underlying extension <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>L</mi><mo>∕</mo><mi>K</mi></math>. Under a natural condition on class functions (which appeared in earlier work), we obtain that those moments are at least Gaussian. The key tools in our approach are the application of positivity in the explicit formula followed by combinatorics on zeros of Artin <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>L</mi></math>-functions (which generalize previous work), as well as precise bounds on Artin conductors. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"64 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143452144","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":"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}
Ilya Dumanski, Evgeny Feigin, Ievgen Makedonskyi, Igor Makhlin
{"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}
Ana Caraiani, Matthew Emerton, Toby Gee, David Savitt
{"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}
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