The Eisenstein ideal at prime-square level has constant rank

TL;DR

This paper proves the uniqueness and describes the coefficient field of a specific cuspform of weight 2 and level N^2, related to Eisenstein ideals at prime-square levels, extending previous results on Fourier coefficient congruences.

Contribution

It establishes the uniqueness of a cuspform with certain congruence properties and determines its coefficient field at prime-square levels, extending prior work on Eisenstein ideals.

Findings

The cuspform is unique up to Galois conjugacy.

The coefficient field is exactly bp[bp + bp^{-1}].

Results extend to higher powers of p dividing N+1.

Abstract

Let $N$ and $p$ be prime numbers with $p \geq 5$ such that $p || (N + 1)$. In a previous paper, we showed that there is a cuspform $f$ of weight 2 and level $\Gamma_0(N^2)$ 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 $\mathbb{Z}_p[\zeta_p + \zeta_p^{-1}]$. We also prove similar results when a higher power of $p$ divides $N + 1$.