Let F be a nonarchimedean local field of odd characteristic p>0. We consider local exterior square L-functions L(s, π,∧2),Bump–Friedberg L-functions L(s, π, BF), and Asai L-functions L(s, π, As) of an irreducible admissible representation π of GLm(F). In particular, we establish that those Lfunctions, via the theory of integral representations, are equal to their corresponding Artin L-functions (Formula Presented) and L(s, As(φ(π))) of the associated Langlands parameter φ(π) under the local Langlands correspondence. These are achieved by proving the identity for irreducible supercuspidal representations, exploiting the local-to-global argument due to Henniart and Lomelí.
Bibliographical noteFunding Information:
As this paper is a conclusion of my Ph.D. work, I would like to thank my advisor, James Cogdell, for countless discussions and for proposing to express L-factors in terms of those for supercuspidal representations. I sincerely thank Nadir Matringe for patiently answering questions about linear and Shalika periods. I also thank Shantanu Agarwal for drawing my attention to positive characteristic. In particular, I am indebted to Muthu Krishnamurthy for fruitful mathematical communications over the years and describing a whole picture of the Langlands–Shahidi method to me. I am grateful to the University of Iowa and the University of Maine for their hospitality and support while the article was being written. Lastly, I would like to thank the referee for many valuable remarks and suggestions, which significantly improved the exposition and organization of this paper. This work was supported by a National Research Foundation of Korea (NRF) grant funded by the Korean government (RS-2023-00209992).
© 2023 MSP (Mathematical Sciences Publishers).
- Bernstein–Zelevinsky derivatives
- local exterior square and Asai L-functions in positive characteristic
- Rankin–Selberg methods