Characterisation Of The Poles Of The L-Modular Asai L-Factor

Let F/F-o be a quadratic extension of non-archimedean local fields of odd residual characteristic, set G = GL(n) (F), G(o) = GL(n)z (F-o) and let l be a prime number different from the residual characteristic of F. For a complex cuspidal representation pi of G, the Asai L-factor L-As (X, pi) has a pole at X = 1, if and only if pi is G(o)-distinguished. In this paper, we solve the problem of characterising the occurrence of a pole at X = 1 of L-As (X, pi) when pi is an l-modular cuspidal representation of G; we show that L-As (X, pi) has a pole at X = 1, if and only if pi is a relatively banal distinguished representation, namely pi is G(o)-distinguished but not vertical bar det( )vertical bar(Fo)-distinguished. This notion turns out to be an exact analogue for the symmetric space G/G(o) of Minguez and Secherre's notion of banal cuspidal (F-l) over bar -representation of G(o). Along the way, we compute the Asai L-factor of all cuspidal .e-modular representations of G in terms of type theory and prove new results concerning lifting and reduction modulo l of distinguished cuspidal representations. Finally, we determine when the natural G(o)-period on the Whittaker model of a distinguished cuspidal representation of G is non-zero.