Linkage principle and base change for ${\rm GL}_2$

Abstract: Let $l$ and $p$ be two distinct odd primes. Let $F$ be a finite extension of $Q_p$, and let $E$ be a finite Galois extension of $F$ with $[E: F]=l$. Let $(\pi, V)$ be a cuspidal representation of ${\rm GL}_2(F)$ with an integral central character. Let $(\pi_E, W)$ be the ${\rm GL}_2(E)$ representation obtained by base change of $\pi$. The Galois group of $E/F$, denoted by $G$, acts on $\pi_E$. We show that the zeroth Tate cohomology group of $\pi_E$, as a $G$-module, is isomorphic to the Frobenius twist of the mod-$l$ reduction of $\pi_F$. We use Kirillov model to prove this result. The first half of the lecture will be a review of some preliminary results on Kirillov model, and in the latter half, I will explain the proof of the above result.

number theory

Audience: researchers in the topic

Number theory during lockdown

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