On norm relations for Asai-Flach classes

TL;DR

This paper provides a new proof of the norm relations for the Asai-Flach Euler system, extending the validity to all inert and split primes by redefining classes and using local zeta integrals.

Contribution

It introduces a new proof technique for the norm relations of Asai-Flach classes, removing previous restrictions on primes and unifying the approach.

Findings

Proves norm relations for all inert and split primes.

Redefines Asai-Flach classes in a new language.

Uses local zeta integrals to establish the relations.

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 $F$ have been already proved by Lei-Loeffler-Zerbes for primes which are inert in $F$ 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.