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

Abstract

Let

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/F_{0}

be a separable quadratic extension and 𝐴 a central simple algebra over 𝐹 with an

F / F 0 F/F_{0}

-involution 𝜏. Let ℎ be a hermitian form over

( A , τ ) (A,\tau)

. If

2 ⁢ ind ⁡ ( A ) 2\operatorname{ind}(A)

is coprime to the characteristic of the residue field of

F 0 F_{0}

, then we prove that the Rost invariant

R SU ⁡ ( h ) : H 1 ⁢ ( F 0 , SU ⁡ ( h ) ) → H 3 ⁢ ( F 0 , Q / Z ⁢ ( 2 ) ) R_{\operatorname{SU}(h)}\colon H^{1}(F_{0},\operatorname{SU}(h))\to H^{3}(F_{0},\mathbb{Q}/\mathbb{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.