Pub Date : 2024-09-18DOI: 10.2140/ant.2024.18.1465
Shaunak V. Deo, Mladen Dimitrov, Gabor Wiese
We prove that the Galois pseudorepresentation valued in the mod cuspidal Hecke algebra for over a totally real number field , of parallel weight and level prime to , is unramified at any place above . The same is true for the noncuspidal Hecke algebra at places above whose ramification index is not divisible by . A novel geometric ingredient, which is also of independent interest, is the construction and study, in the case when ramifies in , of generalised -operators using Reduzzi and Xiao’s generalised Hasse invariants, including especially an injectivity criterion in terms of minimal weights.
我们证明,在完全实数域 F 上的 GL (2) 的 mod pn cuspidal Hecke 代数中,平行权重为 1 且级数为 p 的素数的伽罗瓦假呈现在 p 以上的任何位置都是无ramified 的。一个新颖的几何成分,也是一个独立的兴趣点,是在 p 在 F 中斜线化的情况下,利用 Reduzzi 和 Xiao 的广义哈塞不变式,特别是包括最小权重的注入性准则,构造和研究广义 Θ 运算符。
{"title":"Unramifiedness of weight 1 Hilbert Hecke algebras","authors":"Shaunak V. Deo, Mladen Dimitrov, Gabor Wiese","doi":"10.2140/ant.2024.18.1465","DOIUrl":"https://doi.org/10.2140/ant.2024.18.1465","url":null,"abstract":"<p>We prove that the Galois pseudorepresentation valued in the mod <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup></math> cuspidal Hecke algebra for <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi> GL</mi><mo> <!--FUNCTION APPLICATION--> </mo><!--nolimits--><mo stretchy=\"false\">(</mo><mn>2</mn><mo stretchy=\"false\">)</mo></math> over a totally real number field <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>F</mi></math>, of parallel weight <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>1</mn></math> and level prime to <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>, is unramified at any place above <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>. The same is true for the noncuspidal Hecke algebra at places above <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math> whose ramification index is not divisible by <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi><mo>−</mo><mn>1</mn></math>. A novel geometric ingredient, which is also of independent interest, is the construction and study, in the case when <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math> ramifies in <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>F</mi></math>, of generalised <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>Θ</mi></math>-operators using Reduzzi and Xiao’s generalised Hasse invariants, including especially an injectivity criterion in terms of minimal weights. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142236177","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}
Pub Date : 2024-09-18DOI: 10.2140/ant.2024.18.1403
Fu Liu, Brian Osserman, Montserrat Teixidor i Bigas, Naizhen Zhang
Building on recent work of the authors, we use degenerations to chains of elliptic curves to prove two cases of the Aprodu–Farkas strong maximal rank conjecture, in genus and . This constitutes a major step forward in Farkas’ program to prove that the moduli spaces of curves of genus and are of general type. Our techniques involve a combination of the Eisenbud–Harris theory of limit linear series, and the notion of linked linear series developed by Osserman.
{"title":"The strong maximal rank conjecture and moduli spaces of curves","authors":"Fu Liu, Brian Osserman, Montserrat Teixidor i Bigas, Naizhen Zhang","doi":"10.2140/ant.2024.18.1403","DOIUrl":"https://doi.org/10.2140/ant.2024.18.1403","url":null,"abstract":"<p>Building on recent work of the authors, we use degenerations to chains of elliptic curves to prove two cases of the Aprodu–Farkas strong maximal rank conjecture, in genus <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>2</mn><mn>2</mn></math> and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>2</mn><mn>3</mn></math>. This constitutes a major step forward in Farkas’ program to prove that the moduli spaces of curves of genus <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>2</mn><mn>2</mn></math> and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>2</mn><mn>3</mn></math> are of general type. Our techniques involve a combination of the Eisenbud–Harris theory of limit linear series, and the notion of linked linear series developed by Osserman. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142236169","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}
Pub Date : 2024-09-18DOI: 10.2140/ant.2024.18.1497
Asher Auel, V. Suresh
We prove the failure of the local-global principle, with respect to discrete valuations, for isotropy of quadratic forms in variables over function fields of transcendence degree