{"title":"Iwasawa Theory for GU(2,1) at inert primes","authors":"Muhammad Manji","doi":"arxiv-2409.05664","DOIUrl":null,"url":null,"abstract":"Many problems of arithmetic nature rely on the computation or analysis of\nvalues of $L$-functions attached to objects from geometry. Whilst basic\nanalytic properties of the $L$-functions can be difficult to understand, recent\nresearch programs have shown that automorphic $L$-values are susceptible to\nstudy via algebraic methods linking them to Selmer groups. Iwasawa theory,\npioneered first by Iwasawa in the 1960s and later Mazur and Wiles provides an\nalgebraic recipe to obtain a $p$-adic analogue of the $L$-function. In this\nwork we aim to adapt Iwasawa theory to a new context of representations of the\nunitary group GU(2,1) at primes inert in the respective imaginary quadratic\nfield. This requires a novel approach using the Schneider--Venjakob regulator\nmap, working over locally analytic distribution algebras. Subsequently, we show\nvanishing of some Bloch--Kato Selmer groups when a certain $p$-adic\ndistribution is non-vanishing. These results verify cases of the Bloch--Kato\nconjecture for GU(2,1) at inert primes in rank 0.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"23 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Rings and Algebras","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.05664","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
Many problems of arithmetic nature rely on the computation or analysis of
values of $L$-functions attached to objects from geometry. Whilst basic
analytic properties of the $L$-functions can be difficult to understand, recent
research programs have shown that automorphic $L$-values are susceptible to
study via algebraic methods linking them to Selmer groups. Iwasawa theory,
pioneered first by Iwasawa in the 1960s and later Mazur and Wiles provides an
algebraic recipe to obtain a $p$-adic analogue of the $L$-function. In this
work we aim to adapt Iwasawa theory to a new context of representations of the
unitary group GU(2,1) at primes inert in the respective imaginary quadratic
field. This requires a novel approach using the Schneider--Venjakob regulator
map, working over locally analytic distribution algebras. Subsequently, we show
vanishing of some Bloch--Kato Selmer groups when a certain $p$-adic
distribution is non-vanishing. These results verify cases of the Bloch--Kato
conjecture for GU(2,1) at inert primes in rank 0.