Families of Absolutely Simple Hyperelliptic Jacobians

As usual, Z, Q and C stand for the ring of integers, the field of rational numbers and the field of complex numbers respectively. If l is a prime then we write Fl,Zl and Ql for the l-element (finite) field, the ring of l-adic integers and field of l-adic numbers respectively. If A is a finite set then we write #(A) for the number of its elements. Let K be a field of characteristic different from 2, let K̄ be its algebraic closure and Gal(K) = Aut(K̄/K) its absolute Galois group. Let n ≥ 5 be an integer, f(x) ∈ K[x] a degree n polynomial without multiple roots, Rf ⊂ K̄ the n-element set of its roots, K(Rf ) ⊂ K̄ the splitting field of f(x) and Gal(f) = Gal(K(Rf )/K) the Galois group of f(x) over K. One may view Gal(f) as a certain group of permutations of Rf . Let Cf : y 2 = f(x) the corresponding hyperelliptic curve of genus [n/2]. Let J(Cf ) be the jacobian of Cf ; it is a [(n − 1)/2]-dimensional abelian variety that is defined over K. We write End(J(Cf )) for the ring of all K̄endomorphisms of J(Cf ). As usual, we write End (J(Cf )) for the corresponding (finite-dimensional semisimple) Q-algebra End(J(Cf ))⊗ Q. In [33, 36] (see also [37]) the author proved the following statement.