On Generators of Arithmetic Groups over Function Fields

Fq = the finite field with q elements; throughout the paper q is assumed to be odd. A = Fq[T ], T indeterminate. F = Fq(T ) = the fraction field of A. |F | = the set of places of F . For x ∈ |F |, Fx = the completion of F at x. Ox = {z ∈ Fx | ordx(z) ≥ 0} = the ring of integers of Fx. Fx = the residue field of Ox; deg(x) = [Fx : Fq]. For 0 = f ∈ A, deg(f) = the degree of f as a polynomial in T , and deg(0) = +∞. For f/g ∈ F , deg(f/g) = deg(f)− deg(g). ord = − deg defines a valuation on F ; the corresponding place is denoted by ∞. K = F∞. O = O∞. π = T−1 = uniformizer at infinity.