The Eisenstein ideal at prime-square level has constant rank

Abstract

Let $N$ and $p$ be prime numbers with $p\ge 5$ such that $p||(N+1)$. In a previous paper, we showed that there is a cuspform $f$ of weight 2 and level $\hat{I}{“}_0(N^2)$ whose $\ell$-th Fourier coefficient is congruent to $\ell +1$ modulo a prime above $p$ for all primes $\ell$. In this paper, we prove that this form $f$ is unique up to Galois conjugacy, and the extension of ${\mathbb{Z}}_p$ generated by the coefficients of $f$ is exactly ${\mathbb{Z}}_p[\hat{I}{\P }_p+\hat{I}{\P }_p^{-1}]$. We also prove similar results when a higher power of $p$ divides $N+1$.

15 pages