Endomorphism Algebras of Superelliptic Jacobians

As usual, we write Z,Q,Fp,C for the ring of integers, the field of rational numbers, the finite field with p elements and the field of complex numbers respectively. If Z is a smooth algebraic variety over an algebraically closed field then we write Ω(Z) for the space of differentials of the first kind on Z. If Z is an abelian variety then we write End(Z) for its ring of (absolute) endomorphisms and End(Z) for its endomorphism algebra End(Z) ⊗Q. If Z is defined over a (not necessarily algebraically closed) field K then we write EndK(Z) ⊂ End(Z) for the (sub)ring of K-endomorphisms of Z. Let p be a prime, q = p an integral power of p, ζq ∈ C a primitive qth root of unity, Q(ζq) ⊂ C the qth cyclotomic field and Z[ζq] the ring of integers in Q(ζq). If q = 2 then Q(ζq) = Q. It is well-known that if q > 2 then Q(ζq) is a CM-field of degree (p− 1)pr−1. Let us put