Abstract: We give an explicit construction of test vectors for $T$-equivariant linear functionals on representations $\\Pi$ of $GL_2$ of a local field\, where $T$ is a non-split torus. Of particular interest is the case when both the representations are ramified\; we completely so lve this problem for principal series and Steinberg representations of $GL _2$\, as well as for depth zero supercuspidals over $\\qq_p$. A key ingred ient is a theorem of Casselman and Sillberger\, which allows us to quickly reduce almost all cases to that of the principal series\, which can be an alyzed directly. Our method shows that the only genuinely difficult cases are the characters of $T$ which occur in the primitive part (or ``type") o f $\\Pi$ when $\\Pi$ is supercuspidal\, and we resolve the depth zero case . The method to handle the depth zero case is based on modular representat ion theory\, motivated by considerations from Deligne-Lusztig theory and t he de Rham cohomology of Deligne-Lusztig-Drinfeld curves. The proof also r eveals some interesting features related to the Langlands correspondence i n characteristic $p$. We show in particular that the test vector problem h as an obstruction in characteristic $p$ beyond the root number criterion o f Waldspurger and Tunnell\, and exhibits an unexpected dichotomy related t o the weights in Serre's conjecture and the signs of standard Gauss sums.< /p> LOCATION:PH H3 31 https://plan.epfl.ch/?room=PHH331 STATUS:CONFIRMED END:VEVENT END:VCALENDAR