2019/03/06 by Robert Kurinczuk, Kurinczuk, Robert, Nadir Matringe +1
Mathematics · #11F70 #22E50 #Advanced Algebra and Geometry #Advanced Topics in Algebra #FOS: Mathematics #Geometry and complex manifolds #Number Theory (math.NT) #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.1903.02427
openalex publication_date 2019/03/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let E/F be a quadratic extension of non-archimedean local fields, and let ℓ be a prime number different from the residual characteristic of F. For a complex cuspidal representation π of GL(n,E), the Asai L-factor L+(X,π) has a pole at X=1 if and only if π 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,π) when π is an ℓ-modular cuspidal representation of GL(n,E): we show that L+(X,π) has a pole at X=1 if and only if π is a relatively banal distinguished representation; namely π 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 M' inguez and Sécherre's notion of banal cuspidal \mathbbF_ℓ-representation of GL(n,F). Along the way we compute the Asai L-factor of all cuspidal ℓ-modular representations of GL(n,E) 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 GL(n,F)-period on the Whittaker model of a distinguished cuspidal representation of GL(n,E) is nonzero.