[Paper Review] On Eisenstein ideals and the cuspidal group of $J_0(N)$

Hwajong Yoo|arXiv (Cornell University)|Feb 5, 2015

Algebraic Geometry and Number Theory10 references4 citations

TL;DR

This paper proves Ribet's conjecture that every Eisenstein maximal ideal of the Hecke algebra of level $N$ (square-free, $N > 6$) acts nontrivially on the cuspidal subgroup $\mathcal{C}_N$ of the Jacobian $J_0(N)$. By computing the index of Eisenstein ideals via the order of specific cuspidal divisors $C_{M,N}$, the authors show $\mathcal{C}_N[\mathfrak{m}] \neq 0$ for all Eisenstein maximal ideals $\mathfrak{m}$, confirming the conjecture in full generality.

ABSTRACT

Let $\mathcal{C}_N$ be the cuspidal subgroup of the Jacobian $J_0(N)$ for a square-free integer $N>6$. For any Eisenstein maximal ideal $\mathfrak{m}$ of the Hecke ring of level $N$, we show that $\mathcal{C}_N[\mathfrak{m}] eq 0$. To prove this, we calculate the index of an Eisenstein ideal $\mathcal{I}$ contained in $\mathfrak{m}$ by computing the order of a cuspidal divisor annihilated by $\mathcal{I}$.

Motivation & Objective

• To prove Ribet's conjecture that every Eisenstein maximal ideal $\mathfrak{m}$ of the Hecke algebra $\mathbb{T}(N)$ acts nontrivially on the cuspidal subgroup $\mathcal{C}_N$ of $J_0(N)$.
• To classify all Eisenstein maximal ideals of $\mathbb{T}(N)$ via the structure of ideals $I_{M,N}$ generated by $U_p - 1$, $U_q - q$, and the classical Eisenstein ideal $\mathcal{I}_0(N)$.
• To compute the exact order of the cuspidal divisor $C_{M,N} = \sum_{d\mid M} (-1)^{\omega(d)} P_d$ in terms of $\varphi(N)$, $\psi(N/M)$, and a correction factor $h \in \{1,2\}$.
• To establish a precise link between the index of the Eisenstein ideal $I_{M,N}$ and the order of $C_{M,N}$, showing equality up to powers of 2.
• To complete the proof of the main theorem by analyzing all cases of $\ell$-adic Eisenstein maximal ideals, especially for $\ell = 2$, using level-lowering and mod $\ell$ modular form techniques.

Proposed method

• Classify Eisenstein maximal ideals $\mathfrak{m}$ of $\mathbb{T}(N)$ as those containing $I_{M,N} = (U_p - 1, U_q - q, \mathcal{I}_0(N))$ for some proper divisor $M$ of $N$, $M \neq 1$.
• Define the cuspidal divisor $C_{M,N} = \sum_{d\mid M} (-1)^{\omega(d)} P_d$, where $P_d$ is the cusp corresponding to $1/d \in \mathbb{P}^1(\mathbb{Q})$.
• Compute the order of $C_{M,N}$ as $\mathrm{num}\left( \frac{\varphi(N)\psi(N/M)}{24} \times h \right)$, where $h = 2$ iff $N = M \equiv 1 \pmod{8}$ or $N = 2M$, $M \equiv 1 \pmod{8}$.
• Establish that the index of the Eisenstein ideal $I_{M,N}$ equals the order of $C_{M,N}$ when $N/M$ is odd, and equals it up to powers of 2 when $N/M$ is even.
• Use mod $\ell$ modular forms and level-lowering techniques to rule out maximality of certain ideals $\mathfrak{m} = (\ell, I_{1,N})$ when $\ell \nmid \varphi(N)$ and $\ell$ odd.
• Analyze the $\ell = 2$ case by case, using known results on mod 2 modular forms and the structure of $J_0(N)$, especially for $N$ prime or $N = 2M$.

Experimental results

Research questions

• RQ1 Does every Eisenstein maximal ideal $\mathfrak{m}$ of $\mathbb{T}(N)$ act nontrivially on the cuspidal subgroup $\mathcal{C}_N$ of $J_0(N)$?
• RQ2 What is the exact order of the cuspidal divisor $C_{M,N}$ in terms of arithmetic functions of $N$ and $M$?
• RQ3 How does the index of the Eisenstein ideal $I_{M,N}$ relate to the order of $C_{M,N}$, especially in the presence of 2-adic torsion?
• RQ4 Can the maximality of $\mathfrak{m} = (\ell, I_{1,N})$ be ruled out for odd $\ell$ not dividing $\varphi(N)$ using mod $\ell$ modular forms?
• RQ5 What conditions ensure that $\mathcal{C}_N[\mathfrak{m}] \neq 0$ when $\ell = 2$, particularly for $N = M$ or $N = 2M$?

Key findings

• The main theorem confirms Ribet's conjecture: $\mathcal{C}_N[\mathfrak{m}] \neq 0$ for all Eisenstein maximal ideals $\mathfrak{m}$ of $\mathbb{T}(N)$.
• The order of the cuspidal divisor $C_{M,N}$ is $\mathrm{num}\left( \frac{\varphi(N)\psi(N/M)}{24} \times h \right)$, where $h = 2$ if $N = M \equiv 1 \pmod{8}$ or $N = 2M$, $M \equiv 1 \pmod{8}$, and $h = 1$ otherwise.
• For $N/M$ odd, the index of the Eisenstein ideal $I_{M,N}$ is exactly equal to the order of $C_{M,N}$.
• For $N/M$ even, the index of $I_{M,N}$ is equal to the order of $C_{M,N}$ up to a power of 2.
• The case $\ell = 2$ is resolved by showing $\mathcal{C}_N[\mathfrak{m}] \neq 0$ via the order of $C_{p,N}$ being divisible by 2 when $N = M$ or $N = 2M$ with $\omega(M) \geq 1$.
• The proof uses level-lowering via mod $\ell$ modular forms to rule out maximality of $\mathfrak{m} = (\ell, I_{1,N})$ when $\ell$ is odd and $\ell \nmid \varphi(N)$.