Daniel Barrera Salazar, Andrew Graham, Chris Williams
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We conjecture a <i>p</i>-refined analogue: that any <i>P</i>-parahoric <i>p</i>-refinement <math> <msup><mover><mi>π</mi> <mo>~</mo></mover> <mi>P</mi></msup> </math> is a functorial transfer from <math><msub><mtext>GSpin</mtext> <mrow><mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn></mrow> </msub> </math> if and only if a <i>P</i>-twisted version of <math><msub><mi>Z</mi> <mi>H</mi></msub> </math> is non-vanishing on the <math> <msup><mover><mi>π</mi> <mo>~</mo></mover> <mi>P</mi></msup> </math> -eigenspace in <math><mi>π</mi></math> . This twisted <math><msub><mi>Z</mi> <mi>H</mi></msub> </math> appears in all constructions of <i>p</i>-adic <i>L</i>-functions via Shalika models. We connect our conjecture to the study of classical symplectic families in the <math><msub><mtext>GL</mtext> <mrow><mn>2</mn> <mi>n</mi></mrow> </msub> </math> eigenvariety, and-by proving upper bounds on the dimensions of such families-obtain various results towards the conjecture.</p>","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"11 2","pages":"51"},"PeriodicalIF":0.8000,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12031854/pdf/","citationCount":"0","resultStr":"{\"title\":\"<ArticleTitle xmlns:ns0=\\\"http://www.w3.org/1998/Math/MathML\\\">On <i>p</i>-refined Friedberg-Jacquet integrals and the classical symplectic locus in the <ns0:math> <ns0:msub><ns0:mrow><ns0:mspace /> <ns0:mtext>GL</ns0:mtext> <ns0:mspace /></ns0:mrow> <ns0:mrow><ns0:mn>2</ns0:mn> <ns0:mi>n</ns0:mi></ns0:mrow> </ns0:msub> </ns0:math> eigenvariety.\",\"authors\":\"Daniel Barrera Salazar, Andrew Graham, Chris Williams\",\"doi\":\"10.1007/s40993-025-00631-z\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p><p>Friedberg-Jacquet proved that if <math><mi>π</mi></math> is a cuspidal automorphic representation of <math> <mrow><msub><mtext>GL</mtext> <mrow><mn>2</mn> <mi>n</mi></mrow> </msub> <mrow><mo>(</mo> <mi>A</mi> <mo>)</mo></mrow> </mrow> </math> , then <math><mi>π</mi></math> is a functorial transfer from <math><msub><mtext>GSpin</mtext> <mrow><mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn></mrow> </msub> </math> if and only if a global zeta integral <math><msub><mi>Z</mi> <mi>H</mi></msub> </math> over <math><mrow><mi>H</mi> <mo>=</mo> <msub><mtext>GL</mtext> <mi>n</mi></msub> <mo>×</mo> <msub><mtext>GL</mtext> <mi>n</mi></msub> </mrow> </math> is non-vanishing on <math><mi>π</mi></math> . We conjecture a <i>p</i>-refined analogue: that any <i>P</i>-parahoric <i>p</i>-refinement <math> <msup><mover><mi>π</mi> <mo>~</mo></mover> <mi>P</mi></msup> </math> is a functorial transfer from <math><msub><mtext>GSpin</mtext> <mrow><mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn></mrow> </msub> </math> if and only if a <i>P</i>-twisted version of <math><msub><mi>Z</mi> <mi>H</mi></msub> </math> is non-vanishing on the <math> <msup><mover><mi>π</mi> <mo>~</mo></mover> <mi>P</mi></msup> </math> -eigenspace in <math><mi>π</mi></math> . This twisted <math><msub><mi>Z</mi> <mi>H</mi></msub> </math> appears in all constructions of <i>p</i>-adic <i>L</i>-functions via Shalika models. We connect our conjecture to the study of classical symplectic families in the <math><msub><mtext>GL</mtext> <mrow><mn>2</mn> <mi>n</mi></mrow> </msub> </math> eigenvariety, and-by proving upper bounds on the dimensions of such families-obtain various results towards the conjecture.</p>\",\"PeriodicalId\":43826,\"journal\":{\"name\":\"Research in Number Theory\",\"volume\":\"11 2\",\"pages\":\"51\"},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2025-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12031854/pdf/\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Research in Number Theory\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1007/s40993-025-00631-z\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"2025/4/25 0:00:00\",\"PubModel\":\"Epub\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Research in Number Theory","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s40993-025-00631-z","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"2025/4/25 0:00:00","PubModel":"Epub","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
Friedberg-Jacquet证明了如果π是GSpin 2n (a)的倒自同构表示,则π是GSpin 2n + 1的泛函迁移,当且仅当全局zeta积分zh / H = GL n × GL n在π上不消失。我们推测了一个P-精化的类比:当且仅当π ~ P-本征空间上的Z - H的P-扭曲版本不消失时,任何P-逆P-精化π ~ P都是GSpin 2 n + 1的函子迁移。这种扭曲的zh通过Shalika模型出现在p进l函数的所有构造中。我们把我们的猜想与GL 2 n特征变中的经典辛族的研究联系起来,并通过证明这些辛族的维数上界,得到了关于这个猜想的各种结果。
On p-refined Friedberg-Jacquet integrals and the classical symplectic locus in the GL2n eigenvariety.
Friedberg-Jacquet proved that if is a cuspidal automorphic representation of , then is a functorial transfer from if and only if a global zeta integral over is non-vanishing on . We conjecture a p-refined analogue: that any P-parahoric p-refinement is a functorial transfer from if and only if a P-twisted version of is non-vanishing on the -eigenspace in . This twisted appears in all constructions of p-adic L-functions via Shalika models. We connect our conjecture to the study of classical symplectic families in the eigenvariety, and-by proving upper bounds on the dimensions of such families-obtain various results towards the conjecture.
期刊介绍:
Research in Number Theory is an international, peer-reviewed Hybrid Journal covering the scope of the mathematical disciplines of Number Theory and Arithmetic Geometry. The Mission of the Journal is to publish high-quality original articles that make a significant contribution to these research areas. It will also publish shorter research communications (Letters) covering nascent research in some of the burgeoning areas of number theory research. This journal publishes the highest quality papers in all of the traditional areas of number theory research, and it actively seeks to publish seminal papers in the most emerging and interdisciplinary areas here as well. Research in Number Theory also publishes comprehensive reviews.