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.

中文翻译：

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关于 Asai-Flach 类的范数关系

我们对 Lei-Loeffler-Zerbes 建立的 Asai-Flach Euler 系统的范数关系给出了新的证明。更准确地说，我们用 Loeffler-Skinner-Zerbes 为 Lemma-Eisenstein 类使用的语言重新定义了 Asai-Flach 类，并使用局部 zeta 积分证明了垂直和驯服范数关系。Lei-Loeffler-Zerbes 已经证明了这些在二次实场 [公式：见文本] 上附加到希尔伯特模形式的 Asai 表示的欧拉系统范数关系对于在 [公式：见文本] 中惰性的素数和分裂素数满足一些假设；使用这种技术，我们能够移除它并证明所有惰性和分裂素数的温和范数关系。