On norm relations for Asai–Flach classes

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.

Gross–Prasad periods for reducible representations

“…We now give a global application, a strengthening of some results from [9] and [6] on Euler systems for quadratic Hilbert modular forms. Let K/Q be a real quadratic field and write G =…”

“…We conclude with an application to global arithmetic. For π a Hilbert modular form over a real quadratic field, the constructions of [6,8,9] give rise to a family of cohomology classes taking values in the 4-dimensional Asai Galois representation associated to π. We show that if π is not of CM type and not a base-change from Q, then these elements all lie in a 1-dimensional subspace.…”

“…Then we have dim Hom H (π ⊠ Σ F , ) = 1.Remark Note that the case when E/F is unramified, and π is unramified and tempered, is part of [6, Theorem 4.1.1]. However, the proof of this statement given in[6] has a minor error which means the argument does not work when π is the normalised induction of the trivial character of B E . So the argument below fixes this small gap.…”

An Euler system for GU(2, 1)

“…Part (i) is proved in the same way as [KLZ17, Prop 5.2.3] in the Rankin-Selberg case. Part (ii) is proved in [LLZ18] in the special case when the primes above ℓ are trivial in the narrow class group; the general case is proved in [Gro20] (although in fact we shall shortly restrict to the case when all the primes dividing m are inert F , so the arguments in [LLZ18] suffice). Part (iii) is [LLZ18, Corollary 9.2.3].…”

Higher Hida theory for Hilbert modular varieties in the totally split case

“…cit., where the authors had to rely on the overconvergent results. In order to obtain the application for the Bloch-Kato conjecture (for F a real quadratic field), we plan to prove an explicit reciprocity law, linking such p-adic L-functions with the Euler system classes studied in [LLZ18,Gro20].…”