Hyperelliptic Jacobians and simple groups U₃(2^m)

In a previous paper, the author proved that in characteristic zero the jacobian J(C) of a hyperelliptic curve C : y² = f(x) has only trivial endomorphisms over an algebraic closure K_a of the ground field K if the Galois group Gal(f) of the irreducible polynomial f(x) ∈ K[x] is either the symmetric group S_n or the alternating group A_n. Here n > 4 is the degree of f. In another paper by the author this result was extended to the case of certain "smaller" Galois groups. In particular, the infinite series n = 2^r + 1, Gal(f) = L₂(2_r) := PSL₂(F_2r) and n = 2^4r+2 + 1, Gal(f) = Sz(2^2r+1) were treated. In this paper the case of Gal(f) = U₃(2^m) := PSU₃(F_2m) and n = 2^3m + 1 is treated.