Algebraicity of $L$-values for $\text{GSp}_4 \times \text{GL}_2$ and $\text{GSp}_4 \times \text{GL}_2 \times \text{GL}_2$

David Loeffler, Óscar Rivero

Published 2023-03-28Version 1

We prove algebraicity results for critical $L$-values attached to the group $\text{GSp}_4 \times \text{GL}_2$, and for Gan--Gross--Prasad periods which are conjecturally related to central $L$-values for $\text{GSp}_4 \times \text{GL}_2 \times \text{GL}_2$. A key aspect is the computation of certain archimedean zeta integrals, whose $p$-adic counterparts are also studied in this note. This is crucially used in a sequel to this paper in order to construct a new $p$-adic $L$-function for $\text{GSp}_4 \times \text{GL}_2$.