Let $F$ be a finite extension of $\mathbb{Q}_p$, and let $E$ be a finite cyclic Galois extension of $F$. The theory of base change associates to an irreducible smooth $\overline{\mathbb{Q}}_l$-representation $(\pi_F, V)$ of ${\rm GL}_n(F)$ an irreducible $\overline{\mathbb{Q}}_l$-representation $(\pi_E, W)$ of ${\rm GL}_n(E)$. The ${\rm GL}_n(E)$-representation $\pi_E$ extends as a representation of ${\rm GL}_n(E)\rtimes {\rm Gal}(E/F)$. Assume that the central character of $\pi_F$ takes values in $\overline{\mathbb{Z}}_l^\times$, and $l\neq p$. When $\pi_E$ is cuspidal, for any ${\rm GL}_n(E)\rtimes {\rm Gal}(E/F)$ stable lattice $\mathcal{L}$ in $\pi_E$, Ronchetti supporting the linkage principle of Treumann and Venkatesh conjectured that the zeroth Tate cohomology of $\mathcal{L}$ with respect to ${\rm Gal}(E/F)$ is the Frobenius twist of mod-$l$ reduction of the representation $\pi_F$, i.e.,

\begin{equation} \widehat{H}^0 ({\rm Gal}(E/F), \mathcal{L}) \simeq \overline{\pi} _F^{(l)}. \end{equation}

This conjecture is verified by Ronchetti when $\pi_F$ is a depth-zero cuspidal representation using compact induction model. We will explain a proof in the case where $n=2$ and $\pi_F$ has arbitrary depth, using Kirillov model. If time permits, we will discuss the general case by local Rankin–Selberg convolutions.