AI 中文总结
该研究针对非阿基米德局部域上中心可除代数的乘法群表示，计算其零深度等性质的Tate上同调群，验证了Treumann–Venkatesh关于模ℓ函子性的猜想。
AI 中文摘要
设F是具有剩余特征p的非阿基米德局部域，D是F上次数为d的中心可除代数，E/F是素数次数ℓ的有限伽罗瓦扩张，满足ℓ≠p且ℓ不整除d，记D_E=D⊗_F E，K是Q_ℓ在Q̄_ℓ中的极大非分歧扩张。本文在局部基变换提升的框架下，明确计算了D_E^×的绝对不可约、整系数、零深度、K-表示的Tate上同调群，验证了Treumann–Venkatesh关于模ℓ函子性的猜想。
英文摘要
Let $F$ be a non-Archimedean local field with residue characteristic $p$, and let $D$ be a central $F$-division algebra of degree $d$. Let $E/F$ be a finite Galois extension of prime degree $\ell$, where $\ell \ne p$ and $\ell$ does not divide $d$. Set $D_E=D\otimes_F E$. Let $\mathcal{K}$ be the maximal unramified extension of $\mathbb{Q}_\ell$ in $\overline{\mathbb{Q}}_\ell$. In this article, we explicitly compute the Tate cohomology groups of absolutely irreducible, integral, depth-zero, $\mathcal{K}$-representations of $D_E^\times$ in the context of local base change lifting, which verifies a conjecture of Treumann--Venkatesh on mod-$\ell$ functoriality.
Comments9 pages, Comments are welcome
