Семейства гиперэллиптических якобианов без комплексного умножения

We prove that the jacobian of a hyperelliptic curve $y^2=(x-t)h(x)$ has no nontrivial endomorphisms over an algebraic closure of the ground field $K$ of characteristic zero if a parameter $t$ lies in $K$, the polynomial $h(x)$ has coefficients in $K$ and its Galois group over $K$ is “very big” while $\deg(h)$ is an even number $>8$.