Fake Congruence Subgroups and the Hurwitz Monodromy Group

Journal of Mathematical Sciences

Berger, Gabriel¹ 

关键词: Burau representation;    fake congruence subgroup;    Hurwitz space;    modular tower;   

DOI  :  

学科分类：数学（综合）

来源: University of Tokyo * Department of Mathematical Sciences

【 摘 要 】

Suppose$G$isafinitegroup,embeddedasatransitivesubgroupof$S_n$forsome$n$.Supposeinadditionthat$(mathcal{C}_1,dots,mathcal{C}_4)$isaquadrupleofconjugacyclassesof$G$.Inearlierpapers([F],[D-F],[B-F]),itwasshownthattothesedataonecancanonicallyassociateafiniteindexsubgroupof$PSL_2(mathbb{Z})$.Forexample,when$N$isanoddinteger,$G$isthedihedralgroup$D_N$andtheconjugacyclassesallconsistofinvolutions,theassociatedsubgroupis$Γ_0(N).$Inthispaperweinvestigatethecaseinwhich$G$isthesemidirectproductoftheabeliangroup$mathbb{Z}[ζ_d]/mathcal{N}$(where$ζ_d$isaprimitive$d$'throotofunityand$mathcal{N}$isanidealof$mathbb{Z}[ζ_d]$relativelyprimeto$d$)andthecyclicgroup$langleζ_dangle$.Werelatethecorrespondingsubgroupof$PSL_2(mathbb{Z})$tothe"fakecongruencesubgroups"describedincite{B2}.Specifically,ifwelet$mathcal{C}$denotetheconjugacyclassof$ζ_d$inthemultiplicativesubgroup$langleζ_dangle$andchooseourconjugacyclassestobe$(mathcal{C},mathcal{C},mathcal{C},mathcal{C}^{-3}),$thenthesubgroupisinfact$Γ_0(mathcal{N})$(definedoriginallyincite{B2};seesection2).

【 授权许可】

Unknown