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

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