A two variable Rankin-Selberg integral for $\mathrm{GU}(2,2)$ and the degree 5 $L$-function of ${\mathrm{GSp}}_4$

Abstract

We give a two-variable Rankin--Selberg integral for generic cusp forms on ${\mathrm{PGL}}_4$ and ${\mathrm{PGU}}_{2,2}$ which represents a product of exterior square $L$-functions. As a residue of our integral, we obtain an integral representation on ${\mathrm{PGU}}_{2,2}$ of the degree 5 $L$-function of ${\mathrm{GSp}}_4$ twisted by the quadratic character of $E/F$ of cuspidal automorphic representations which contribute to the theta correspondence for the pair $({\mathrm{PGSp}}_4,{\mathrm{PGU}}_{2,2})$.

22 pages. To appear in Mathematische Zeitschrift. arXiv admin note: text overlap with arXiv:2311.05759