The Eisenstein ideal at prime-square level has constant rank
Jaclyn Lang, Preston Wake
TL;DR
The paper addresses the Eisenstein ideal at prime-square level by studying deformations of the residual representation $ar{ ho}=oldsymbol{ u}oxplus 1$ in weight $2$ at level $Γ_0(N^2)$ under $pigm| (N+1)$. It develops a pseudodeformation framework, reduces to a local problem at $N$, and computes the local deformation ring $R_N$, identifying it with $oldsymbol{ ext{Λ}}^+$; this local model is then globalized to obtain an isomorphism $oldsymbol{ ext{Λ}}^+ o oldsymbol{ ext{T}}$, sending the augmentation to the Eisenstein ideal. Consequently, when $pigm| (N+1)$ with $r=v_p(N+1)$, the minimal primes of $oldsymbol{ ext{T}}$ correspond to eigenforms, and for $r=1$ there is a unique cuspform (up to Galois conjugacy) with coefficients in $oldsymbol{ ext{Z}}_p[oldsymbol{ ext{ζ}}_p+oldsymbol{ ext{ζ}}_p^{-1}]$, satisfying $a_ u(f)modoldsymbol{ rak{p}}=1+ u$ for all primes $ u$. The work also explains the $r>1$ situation with multiple conjugacy classes and relates the analysis to Massey products, showing the obstructions are simple, which accounts for the constant rank phenomenon of the Eisenstein component. Overall, the paper provides a complete local-to-global description of the Eisenstein component at prime-square level and clarifies the arithmetic structure of the associated Hecke algebra.
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 $Γ_0(N^2)$ 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 $\mathbb{Z}_p[ζ_p + ζ_p^{-1}]$. We also prove similar results when a higher power of $p$ divides $N + 1$.