论文标题
Whittaker晶格的泰特共同体学和$ {\ rm gl} _n $的通用表示的基本变化
Tate cohomology of Whittaker lattices and base change of generic representations of ${\rm GL}_n$
论文作者
论文摘要
令$ p $和$ l $是独特的奇数，让$ n \ geq 2 $为正整数。让$ e $为$ p $ - 亚种菲尔德$ f $的$ l $ $ l $的有限galois扩展。令$ q $是$ f $的残留场的基础性。令$ \overlineπ_f$为$ {\ rm gl} _n（f）$的通用mod- $ l $表示，让$π_f$是$ \overlineπ_f$的$ l $ -ADIC LIFT。令$ \ mathbb {w}^0（π_e，ψ_e）$为$π_e$的积分模型，即，$ \ overline的晶格{\ mathbb {z}} _ l $ - l $ - l $ $ $ phalued函数$π_e$。假设$ l $不划分$ | {\ rm gl} _ {n-1}（\ mathbb {f} _q）| $，我们证明$ \overlineπ_f$的frobenius twist是$ g_n（f）$ g_n（f）$ cohomology Group $ up-sub-sub-sub-sub-sub-sub-sub-sub-sub-sub-of tate cohology $ \ wips { gal}（e/f），\ mathbb {w}^0（π_e，ψ_e））$。
Let $p$ and $l$ be distinct odd primes and let $n\geq 2$ be a positive integer. Let $E$ be a finite Galois extension of degree $l$ of a $p$-adic field $F$. Let $q$ be the cardinality of the residue field of $F$. Let $\overlineπ_F$ be a generic mod-$l$ representation of ${\rm GL}_n(F)$ and let $π_F$ be an $l$-adic lift of $\overlineπ_F$. Let $\mathbb{W}^0(π_E, ψ_E)$ be the integral Whittaker model of $π_E$, i.e., the lattice of $\overline{\mathbb{Z}}_l$-valued functions in the Whittaker model of $π_E$. Assuming that $l$ does not divide $|{\rm GL}_{n-1}(\mathbb{F}_q)|$, we prove that the Frobenius twist of $\overlineπ_F$ is a $G_n(F)$ sub-quotient of the Tate cohomology group $\widehat{H}^0({\rm Gal}(E/F), \mathbb{W}^0(π_E, ψ_E))$.
