Abstract
We give a new proof of the norm relations for the Asai–Flach Euler system built by Lei–Loeffler–Zerbes. More precisely, we redefine Asai–Flach classes in the language used by Loeffler–Skinner–Zerbes for Lemma–Eisenstein classes and prove both the vertical and the tame norm relations using local zeta integrals. These Euler system norm relations for the Asai representation attached to a Hilbert modular form over a quadratic real field [Formula: see text] have been already proved by Lei–Loeffler–Zerbes for primes which are inert in [Formula: see text] and for split primes satisfying some assumption; with this technique we are able to remove it and prove tame norm relations for all inert and split primes.
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