Families of absolutely simple hyperelliptic jacobians

We prove that the jacobian of a hyperelliptic curve y² = (x - t)h(x) has no non-trivial endomorphisms over an algebraic closure of the ground field K of characteristic zero if t ∈ K and the Galois group of the polynomial h(x) over K is an alternating or symmetric group on deg(h) letters and deg(h) is an even number greater than 8. (The case of odd deg(h) > 3 follows easily from previous results of the author.)