On Rankin-Selberg integral structures and Euler systems for $\mathrm{GL}_2\times \mathrm{GL}_2$

TL;DR

This paper investigates the interaction of Rankin-Selberg periods with integral structures in representations, proving the local Euler factors in a motivic Euler system are integrally optimal and establishing related integrality results.

Contribution

It settles Loeffler's conjecture by demonstrating the integrally optimal nature of local Euler factors in the Rankin-Selberg Euler system for modular forms.

Findings

Local Euler factors are integrally optimal in the motivic Rankin-Selberg Euler system.

Any construction with integral input data yields local factors divisible by the Euler factor modulo p-1.

An interpretative result on the integrality of the unramified period part for Rankin-Selberg convolution.

Abstract

We study how Rankin-Selberg periods and distinction problems interact with integral structures in spherical Whittaker type representations. Using this representation-theoretic framework, we settle a conjecture of Loeffler by showing that the local Euler factors appearing in the construction of the motivic Rankin-Selberg Euler system for a product of modular forms are integrally optimal; i.e. any construction of this type with any choice of integral input data in the recipe of Loeffler-Skinner-Zerbes, would give local factors appearing in tame norm relations at $p$, which are integrally divisible by the Euler factor $\mathcal{P}_p^{'}(\mathrm{Frob}_p^{-1})$ modulo $p-1$. We also interpret this as an integrality result on the unramified part of the period associated to the Rankin-Selberg convolution of two modular forms.