Vanishing of Eigenspaces and Cyclotomic Fields

Let p > 3 be a prime, ζp a pth primitive root of 1, and ∆ the Galois group of Q(ζp) over Q. Let q 6= p be a prime and n the order of q modulo p. Assume q 6≡ 1 mod p and so n ≥ 2, p(q− 1)|qn − 1, and n|p− 1. Set f = (q − 1)/p and e = (p− 1)/n. Let Q be a prime ideal of Z[ζp] above q and let F = Z[ζp]/Q. Thus F ∼= Fqn , the finite field with q elements. Let α ∈ Z[ζp] be a generator of F such that α ≡ ζp mod Q. Now let A be the p-Sylow subgroup of the ideal class group Q(ζp), Zp the ring of p-adic integers, ω : ∆ → Z×p is the Teichmüller character defined by ω(k) ≡ k mod p, and er, 0 ≤ r ≤ p− 2, the idempotents 1 p− 1 ∑