The Eisenstein ideal at prime-square level has constant rank

Abstract

Let N and p be prime numbers with p ≥ 5 such that p ∣ ∣ ( N + 1 ) . In a previous paper, we showed that there is a cuspform f of weight 2 and level Γ 0 ( N 2 ) whose ℓ -th Fourier coefficient is congruent to ℓ + 1 modulo a prime above p for all primes ℓ . In this paper, we prove that this form f is unique up to Galois conjugacy, and the extension of Z p generated by the coefficients of f is exactly Z p [ ζ p + ζ p − 1 ] . We also prove similar results when a higher power of p divides N + 1 .