We give a construction of $p$-adic Asai $L$-functions for cohomological cuspidal automorphic representations of ${\rm GL}_2$ 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.
Adjoint $L$-functions, congruence ideals, and Selmer groups over $\mathrm{GL}_n$
On Shalika models and p-adic L-Functions
On $p$-adic Asai $L$-functions of Bianchi modular forms at non-ordinary primes and their decomposition into bounded $p$-adic $L$-functions
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