On Multilinear Forms for Mod $p$ Representations of $\mathrm{GL}_2(\mathbb{Q}_p)$

Motivated by the study of trilinear forms for complex representations, we investigate the space of $G$-invariant linear forms on tensor products of irreducible admissible representations of $G = \mathrm{GL}_2(\mathbb{Q}_p)$ over $\overline{\mathbb{F}}_p$. Our main result is a complete vanishing theorem: for any $n \ge 1$ and $n$ infinite-dimensional irreducible admissible representations $\pi_1,\dots,\pi_n$ of $G$, $$\operatorname{Hom}_G(\pi_1 \otimes \cdots \otimes \pi_n, \mathbb{1}) = 0.$$ A refined version holds for $B^+ := \begin{pmatrix} p^{\mathbb{Z}} & \mathbb{Q}_p \\ 0 & 1 \end{pmatrix}$-invariant forms when at least one $\pi_i$ is supersingular. The proof proceeds by a detailed analysis of certain subgroups, reducing the problem from $G$ to $B^+$ and ultimately to the representation theory of $\mathbb{Z}_p$. We also deduce partial extensions of the result to $\mathrm{GL}_2(F)$ for finite extensions $F/\mathbb{Q}_p$.