# Test vectors for trilinear forms, when two representations are unramified and one is special

## TL;DR

This paper investigates explicit test vectors for trilinear forms on tensor products of representations of GL(2,F), focusing on cases where two are unramified and one is special, completing the understanding for low conductor cases.

## Contribution

It extends the explicit construction of test vectors to new cases where two representations are unramified and one is special, especially for low conductor representations.

## Key findings

- Explicit test vectors are identified for cases with low conductors.
- Almost complete classification for when each representation has conductor at most 1.
- Results generalize previous work by Gross and Prasad.

## Abstract

Let F be a finite extension of Qp and G be GL(2,F). When V is the tensor product of three admissible, irreducible, finite dimensional representations of G, the space of G-invariant linear forms has dimension at most one. When a non zero linear form exists, one wants to find an element of V which is not in its kernel: this is a test vector. Gross and Prasad found explicit test vectors for some triple of representations. In this paper, others are found, and they almost complete the case when the conductor of each representation is at most 1.