A construction of $p$-adic Asai $L$-functions for ${\mathrm{G}\mathrm{L}}_2$ over CM fields

We give a construction of $p$-adic Asai $L$-functions for cohomological cuspidal automorphic representations of ${\mathrm{G}\mathrm{L}}_2$ over CM fields. If the base field is imaginary quadratic, Loeffler-Williams recently constructed the $p$-adic Asai $L$-functions. We generalize their construction to the case that the base fields are general CM fields.