Picard模曲面的剩余有限性与尖点上同调
Residual finiteness and cuspidal cohomology of Picard modular surfaces
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中文总结 AI 辅助
本文证明SU(2,1)中非均匀算术格在万有覆盖及连通有限覆盖中的逆像均剩余有限，关键新输入是每个可公度类含同余格使尖点第一上同调非零，证明基于Rogawski内窥镜分类及作者先前上同调判据。
中文摘要 AI 辅助
我们证明，对于$\mathrm{SU}(2,1)$中的每一个非均匀算术格，其在万有覆盖和所有连通有限覆盖中的逆像都是剩余有限的。关键的新输入是，此类格的每个可公度类都包含一个同余算术格$\Gamma$，使得$$H^1_{\mathrm{cusp}}(\Gamma\backslash\mathbb B^2,\mathbb C)\ne 0.$$特别地，该球商的第一内上同调非零。证明使用了Rogawski对$\mathrm{U}(3)$的内窥镜分类。作者先前证明的一个上同调判据随后给出了剩余有限性结果。剩余有限性还在合适的有限指数子群上产生任意分母的乘子系统。
英文摘要
We prove that, for every non-uniform arithmetic lattice in $\mathrm{SU}(2,1)$, its inverse images in the universal cover and in all connected finite covers are residually finite. The key new input is that every commensurability class of such lattices contains a congruence arithmetic lattice $Γ$ for which $$H^1_{\mathrm{cusp}}(Γ\backslash\mathbb B^2,\mathbb C)\ne 0.$$ In particular, the first inner cohomology of this ball quotient is non-zero. The proof uses Rogawski's endoscopic classification for $\mathrm{U}(3)$. A cohomological criterion proved previously by the author then gives the residual-finiteness result. Residual finiteness also yields multiplier systems of arbitrary denominator on suitable finite-index subgroups.
发表机构
- University College London（伦敦大学学院）
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