Characterisation of the poles of the $ell$-modular Asai $L$-factor

Let $E/F$ be a quadratic extension of non-archimedean local fields, and let $ell$ 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 $ell$-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 $vertdet(~ )|_{F}$-distinguished. This notion turns out to be an exact analogue for the symmetric space $GL(n,E)/GL(n,F)$ of Mu0027 inguez and Su0027echerreu0027s notion of banal cuspidal $overline{mathbb{F}}_ell$-representation of $GL(n,F)$. Along the way we compute the Asai $L$-factor of all cuspidal $ell$-modular representations of $GL(n,E)$ in terms of type theory, and prove new results concerning lifting and reduction modulo $ell$ 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.