On the Coates–Sinnott Conjecture

Let E signify a totally real Abelian number field with a prime power conductor and ring of p-integers R_E for a prime p. Let G denote the Galois group of E over the rationals, and let χ be a p-adic character of G of order prime to p. Theorem A calculates, under a minor restriction on χ, the Fitting ideals of H²_ét(R_E;Z_p(n/2+1))(χ) over Z_p[G](χ). Here we require that n≡2 mod 4. These Fitting ideals are principal and generated by a Stickelberger element. This gives a partial verification and also a strong indication of the Coates–Sinnott conjecture.