Characterisation of the poles of the $\mathrm{\ell }$-modular Asai $L$-factor

Let $E\mathrm{/}F$ be a quadratic extension of non-archimedean local fields, and let $\mathrm{\ell }$ be a prime number different from the residual characteristic of $F$. For a complex cuspidal representation $\pi$ of $GL(n,E)$, the Asai $L$-factor $L^{+}(X,\pi )$ has a pole at $X=1$ if and only if $\pi$ is $GL(n,F)$-distinguished. In this paper we solve the problem of characterising the occurrence of a pole at $X=1$ of $L^{+}(X,\pi )$ when $\pi$ is an $\mathrm{\ell }$-modular cuspidal representation of $GL(n,E)$: we show that $L^{+}(X,\pi )$ has a pole at $X=1$ if and only if $\pi$ is a relatively banal distinguished representation; namely $\pi$ is $GL(n,F)$-distinguished but not $\vert\det(~ )|_{F}$-distinguished. This notion turns out to be an exact analogue for the symmetric space $GL(n,E)\mathrm{/}GL(n,F)$ of M' inguez and S'echerre’s notion of banal cuspidal ${\bar{\mathbb{F}}}_{\mathrm{\ell }}$-representation of $GL(n,F)$. Along the way we compute the Asai $L$-factor of all cuspidal $\mathrm{\ell }$-modular representations of $GL(n,E)$ in terms of type theory, and prove new results concerning lifting and reduction modulo $\mathrm{\ell }$ of distinguished cuspidal representations. Finally, we determine when the natural $GL(n,F)$-period on the Whittaker model of a distinguished cuspidal representation of $GL(n,E)$ is nonzero.