Rost Injectivity and Local Global Principle for Classical Groups Over Function Fields of Arithmetic Surfaces

Let K be a nonarchimedean local field and A a central simple algebra over K with ind(A) coprime to char(K). Let F be the function field of a curve over K. Then we show that the natural map \(H^1(F, SL_1(A)) \rightarrow \prod _\nu H^1(F_\nu , SL_1(A))\) has trivial kernel, where \(\nu \) is running over all divisorial discrete valuations of F. We also show that the Rost invariant \(R_{SL_1(A)} : H^1(F, SL_1(A)) \rightarrow H^3(F, \mathbb Q/\mathbb Z(2))\) has trivial kernel. These results are new if the characteristic of the residue field K divides the index of A. Let K be a number field and G an absolutely simple simply connected linear algebraic group over K. Using our result, we discuss the triviality of the kernel of the Rost invariant map \(R_G : H^1(K(t), G) \rightarrow H^3(K(t), \mathbb Q/\mathbb Z(2))\) .