Rost injectivity for classical groups over function fields of curves over local fields

Let F 0 F₀ be a complete discretely valued field with residue field a global field with no real places or a local field. Let F / F 0 F/F₀ be a separable quadratic extension and 𝐴 a central simple algebra over 𝐹 with an F / F 0 F/F₀ -involution 𝜏. Let ℎ be a hermitian form over ( A , τ ) (A,τ) . If 2 ⁢ ind ⁡ ( A ) 2ind(A) is coprime to the characteristic of the residue field of F 0 F₀ , then we prove that the Rost invariant R SU ⁡ ( h ) : H 1 ⁢ ( F 0 , SU ⁡ ( h ) ) → H 3 ⁢ ( F 0 , Q / Z ⁢ ( 2 ) ) RSU(h) H¹(F₀,SU(h))→ H³(F₀,Q/Z(2)) is injective. Let 𝐾 be a local field and 𝐿 the function field of a curve over 𝐾. Let 𝐺 be an absolutely simple simply connected group over 𝐿 of classical type. Suppose that the characteristic of the residue field of 𝐾 is a good prime for 𝐺. As a consequence of our result and some known results, we conclude that the Rost invariant of 𝐺 is injective.