English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

Let $E$ be a finite Galois extension of a $p$-adic field $F$ with degree ofextension $l$, where $l$ and $p$ are distinct primes with $p\ne 2$. Let $\pi_F$be an integral cuspidal $\overline{\mathbb{Q}}_l$-representation of ${\rmGL}_3(F)$ and let $\pi_E$ be the base change lifting of $\pi_F$ to the group${\rm GL}_3(E)$. Then for any ${\rm GL}_n(E)\rtimes{\rm Gal}(E/F)$-stablelattice $\mathcal{L}$ in the representation space of $\pi_E$, the Tatecohomology group $\widehat{H}^0({\rm Gal}(E/F),\mathcal{L})$ is defined as$\overline{\mathbb{F}}_l$-representation of ${\rm GL}_3(F)$, and itssemisimplification is denoted by $\widehat{H}^0(\pi_E)$. In this article, weshow that the Frobenius twist of mod-$l$ reduction of $\pi_F$ is isomorphic tothe Tate cohomology group $\widehat{H}^0({\rm Gal}(E/F),\mathcal{L})$. Thisresult is related to the notion of linkage under the local base change setting.Moreover, this notion of linkage under the local base change for non-cuspidalirreducible representations of ${\rm GL}_3$ is also investigated in thisarticle.

Linkage for irreducible representations of ${\rm GL}_3$ in local Base change