A Shintani-type formula for Gross--Stark units over function fields

Journal of Mathematical Sciences

Dasgupta, Samit¹  Miller, Alison¹ 

关键词: asymptotic behavior;   

DOI  :  

学科分类：数学（综合）

来源: University of Tokyo * Department of Mathematical Sciences

【 摘 要 】

Let$F$beatotallyrealnumberfieldofdegree$n$,andlet$H$beafiniteabelianextensionof$F$.Let$p$denoteaprimeidealof$F$thatsplitscompletelyin$H$.FollowingBrumerandStark,Tateconjecturedtheexistenceofa$p$-unit$u$in$H$whose$p$-adicabsolutevaluesarerelatedinaprecisewaytothepartialzeta-functionsoftheextension$H/F$.Grosslaterrefinedthisconjecturebyproposingaformulaforthe$p$-adicnormoftheelement$u$.Recently,usingmethodsofShintani,thefirstauthorrefinedtheconjecturefurtherbyproposinganexactformulafor$u$inthe$p$-adiccompletionof$H$.InthisarticlewestateandproveafunctionfieldanalogueofthisShintani-typeformula.Theroleofthetotallyrealfield$F$isplayedbythefunctionfieldofacurveoverafinitefieldinwhich$n$placeshavebeenremoved.Theseplacesrepresentthe``realplaces"of$F$.OurmethodofprooffollowsthatofHayes,whoprovedGross'sconjectureforfunctionfieldsusingthetheoryofDrinfeldmodulesandtheirassociatedexponentialfunctions.

【 授权许可】

Unknown