Integral structures in smooth ${\mathrm{G}\mathrm{L}}_2({\mathbf{Q}}_p)$-representations and zeta integrals

Using zeta-integrals and lattices of functions on a spherical variety, we study integral structures in spherical representations of ${\mathrm{G}\mathrm{L}}_2({\mathbf{Q}}_p)$ and their interaction with the unique linear functional invariant under an unramified maximal torus. Within this framework, we reformulate and prove the first instance of optimality of abstract integral norm-relations as proposed by Loeffler. We also interpret this as a form of integrality for toric periods associated to modular forms, where part of it can be regarded as an arithmetic integral analogue of Waldspurger’s multiplicity one in the unramified setting.