This is a note prepared for the Harvard Mazur's torsion theorem seminar (see the references listed there). This talk will tie up several loose ends from the previous talks. We will recall the construction of the Eisenstein prime quotient of $J_0(N)$ and show that the $p$-torsion of its Neron model is an admissible group scheme. This allows us to run fppf descent and bound the Mordell-Weil rank of the Eisenstein prime quotient. As an immediate consequence, it follows that the number of rational points on $X_0(N)$ is finite whenever it is not obviously infinite, for $N$ a prime.

TopEisenstein quotients

We fix a prime number $N$ throughout this talk. Recall that the Eisenstein ideal $\mathfrak{I}\subseteq \mathbb{T}=\End(J_0(N)/\mathbb{Q})$ is defined to be the ideal generated by $T_\ell-(1+\ell),\ell\ne N$, and $w_N+1$ and we have seen from Cheng-Chiang's talk that $\mathbb{T}/\mathfrak{I}\cong \mathbb{Z}/n \mathbb{Z}$, where $n$ is the numerator of $\frac{N-1}{12}$. So the maximal ideals of $\mathbb{T}$ containing $\mathfrak{I}$ are exactly the Eisenstein primes $\mathfrak{P}=(\mathfrak{I},p)$, $p\mid n$, with residue fields $\mathbb{T}/\mathfrak{P}=\mathbb{F}_p$.

Remark 1 Notice that there exists an Eisenstein prime if and only if $n>1$ (i.e., $N\ne2,3,5,7,13$), if and only if the genus $g(X_0(N))\ge1$.
Remark 2 We can visualize the Eisenstein primes intuitively as follows. Consider the Hecke algebra $\mathbb{T}'$ generated by $T_\ell$ and $w_N$ as endomorphisms on all holomorphic modular forms $\mathcal{M}_2(\Gamma_0(N))$ of weight 2. Since $\mathcal{M}_2(\Gamma_0(N))=\mathcal{S}_2(\Gamma_0(N))\oplus\mathcal{E}_2(\Gamma_0(N))$ decomposes as the direct sum of the space of cusp forms and the space of Eisenstein series (which is one-dimensional, generated by $E_2(q)-N\cdot E_2(q^N)$), restricting the action of $\mathbb{T}'$ on $\mathcal{S}_2(\Gamma_0(N))$ and $\mathcal{E}_2(\Gamma_0(N))$ gives two inclusions $\Spec \mathbb{T}\hookrightarrow\Spec \mathbb{T}'$ and $\Spec \mathbb{Z}\hookrightarrow \Spec \mathbb{T}'$. The latter $\Spec \mathbb{Z}$ is naturally called the Eisenstein line. So an Eisenstein prime $\mathfrak{P}$ can be viewed as the intersection of $\Spec \mathbb{T}$ with the Eisenstein line, reflecting the congruence relation between the Eisenstein series and a certain cusp form modulo $p$: there exists a cusp form whose $q$-expansion modulo $p$ coincides with the Hecke eigenvector $\delta=\sum\sigma'(m)q^m$ (whose eigenvalues are exactly $T_\ell\delta=(1+\ell)\delta$ and $w_N\delta=-\delta$). This congruence relation is the key input for the existence of an Eisenstein prime $\mathfrak{P}$.
Eisenstein primes
Eisenstein primes
Remark 3 Associated to an Eisenstein prime $\mathfrak{P}$ we can construct a canonical quotient of $J_0(N)/\mathbb{Q}$. Since $\mathbb{T}$ is a free $\mathbb{Z}$-module of finite rank, it is a flat extension of $\mathbb{Z}$ and has Krull dimension 1 by going-down (so our picture above is accurate in this sense). Since $\mathbb{T}\otimes \mathbb{Q}$ is a finite dimensional vector space, it is Artinian and we have already seen that it is actually a product of totally real fields $K_\alpha$ (due to the fact that $J_0(N)$ has semistable reduction). The embedding $\mathbb{T}\hookrightarrow \mathbb{T}_\mathbb{Q}$ then gives a bijection between minimal prime ideals of $\mathbb{T}$ and prime ideals of $\mathbb{T}_\mathbb{Q}$ (which are all maximal), and in turn a bijection with the totally real fields $K_\alpha$, and the isogenous factors of $J_0(N)$. We define the Eisenstein prime quotient $J^{(\mathfrak{P})}$ to be the unique optimal quotient (i.e., quotient by an abelian subvariety of $J_0(N)$) whose isogenous factors correspond to minimal prime ideals contained in $\mathfrak{P}$. Then $\mathbb{T}$ also acts on $J^{(\mathfrak{P})}$. To visualize, each isogenous factor of $J_0(N)$ corresponds to an irreducible component of $\Spec \mathbb{T}$ and $J^{(\mathfrak{P})}$ simply corresponds to all the components passing through the Eisenstein prime $\mathfrak{P}$.
Remark 4 Let $\mathbb{T}_p=\mathbb{T}\otimes \mathbb{Z}_p$ be the completion of $\mathbb{T}$ at $p$. Then $\mathbb{T}_p$ is a free $\mathbb{Z}_p$-module of finite rank, hence is a semilocal ring and $\mathbb{T}_p\cong \prod_{\mathfrak{m}\mid p}\mathbb{T}_\mathfrak{m}$, where $\mathfrak{m}$ runs over all maximal ideals containing $p$. Then $\mathbb{T}_p$ acts on the $p$-adic Tate module $T_p J^{(\mathfrak{P})}$. Moreover, the rational $p$-adic Tate module $V_p J^{(\mathfrak{P})}$ can be identified as the direct product of $V_p J_\mathfrak{q}$, where $\mathfrak{q}$ runs over all minimal primes contained in $\mathfrak{P}$. The action of $\mathbb{T}\otimes \mathbb{Q}_p$ on each $V_pJ_\mathfrak{q}$ factors through $K_\alpha\otimes \mathbb{Q}_p$, where $K_\alpha$ is the totally real field corresponding to $\mathfrak{q}$. It follows that the action of $\mathbb{T}_p$ on $T_p J^{(\mathfrak{P})}$ factors through $\mathbb{T}_\mathfrak{P}$.

Our main goal is to prove the following

Theorem 1 Let $\mathfrak{P}$ be an Eisenstein prime and $J^{(\mathfrak{P})}$ be the Eisenstein prime quotient of $J_0(N)$. Then $J^{(\mathfrak{P})}$ has rank 0, i.e., $J^{(\mathfrak{P})}(\mathbb{Q})$ is finite.

As an immediate application, we obtain a "conceptual" proof of the following interesting result.

Theorem 2 Let $N$ be a prime, $N\ne2,3,5,7,13$. Then $X_0(N)(\mathbb{Q})$ is finite.
Remark 5 When $N=2,3,5,7,13$, $g(X_0(N))=0$ and the two cusps are rational, hence $X_0(N)(\mathbb{Q})$ is clearly infinite in these cases. In the case $g=1$, we have seen an example of explicitly computing rational points for $N=11$ in Erick's talk. One can also quote the big theorem of Faltings for $g\ge2$, but Mazur's method can certainly say more about $X_0(N)(\mathbb{Q})$.
Proof Under the assumption, $n>1$, so there exists a nontrivial Eisenstein prime quotient $J^{(\mathfrak{P})}$ (Remark 2). Since $X_0$ is an irreducible projective curve, the composite map $f: X_0(N)\hookrightarrow J_0(N)\twoheadrightarrow J^{(\mathfrak{P})}$ is either constant or a finite morphism onto a curve. Because the image of $X_0(N)\hookrightarrow J_0(N)$ generates $J_0(N)$ as a group, the image of $f$ generates $J^{(\mathfrak{P})}$ as a group (which has positive dimension) and hence $f$ cannot be constant. So $f: X_0(N)\rightarrow f(X_0(N))\subseteq J^{(\mathfrak{P})}$ is a finite morphism of curves. By Theorem 1, we know that $f(X_0(N))$ has only finite rational points, thus so does $X_0(N)$.

TopEisenstein descent

Recall that a quasi-finite flat separated group scheme $G/_{S}=\Spec \mathbb{Z}$ (finite flat over $S'{}=\Spec \mathbb{Z}[1/N]$) is called ($p$)-admissible if $G$ is killed by a power of $p$ and $G/_{S'}$ admits a filtration by finite flat group schemes such that the successive quotients are either $\mathbb{Z}/p$ or $\mu_p$. Bao has explained in his talk that the admissibility can be detected on the associated $\Gal(\overline{\mathbb{Q}}/\mathbb{Q})$-module $G(\overline{\mathbb{Q}})$ and also proved the following easy but crucial estimate.

Theorem 3 Let $G/_{S}$ be an admissible group scheme. Then $$h^1(G)-h^0(G)\le\delta(G)-\alpha(G),$$ where $h^i(G)=\log_p\# H^i_\mathrm{fppf}(S,G)$, $\delta(G)=\log_p\# G/_{S'}-\log_p\# G/_{\mathbb{F}_N}$ is the defect and $\alpha(G)$ is the number of $\mathbb{Z}/p$'s in the successive quotients of $G/_{S'}$.

The proof of Theorem 1 relies on the following admissibility result, which allows one to bound the Mordell-Weil rank of $J^{(\mathfrak{P})}$ via Theorem 3 and is the major motivation to introduce the notion of the Eisenstein quotients.

Theorem 4 Let $J/_{S}$ be the Neron model of the Eisenstein prime quotient $J^{(\mathfrak{P})}$. Then its $p$-torsion $J[p]/_{S}$ is an admissible group scheme.

Assuming Theorem 4, we can finish the proof of main Theorem 1 using a standard descent argument.

Proof (Proof of Theorem 1) Let $J^0$ be the fiberwise connected component of $J/_{S}$. We know from George's talk that $J_0(N)$ has good reduction outside $N$ and toric reduction at $N$, hence so does $J^{(\mathfrak{P})}$. Then $[p]:J^0\rightarrow J^0$ is a surjective morphism of schemes (which can be checked on geometric points, Lemma 0487) as there is no unipotent part in $J/_{\mathbb{F}_p}$. Since $[p]$ is also fppf (but not etale since $p$ is not invertible on $S$), $J^0\rightarrow J^0$ is a surjection as fppf sheaves (Lemma 05VM, but not as etale sheaves). So we obtain an exact sequence of fppf sheaves $$0\rightarrow J^0[p]\rightarrow J^0\rightarrow J^0\rightarrow 0,$$ which induces an exact sequence in fppf cohomology $$0\rightarrow J^0(\mathbb{Z})/p J^0(\mathbb{Z})\rightarrow H^1_\mathrm{fppf}(S, J^0[p])\rightarrow H^1_\mathrm{fppf}(S, J^0)[p]\rightarrow 0.$$ The first inclusion implies that $$r+h^0=\log_p(\# J^0(\mathbb{Z})/p J^0(\mathbb{Z}))\le h^1,$$ where $r$ is the rank of the abelian group $J^0(\mathbb{Z})$.

On the other hand, by Theorem 4, $J^0[p]$ is also admissible. So Theorem 3 gives $$h^1-h^0\le\delta-\alpha.$$ Let $g=\dim J^{(\mathfrak{P})}$. Using the toric reduction at $N$, we can compute $\delta(J^0[p])=2g-g=g$. Replacing $J^{(\mathfrak{P})}$ by $J^{(\mathfrak{P})}\times (J^{(\mathfrak{P})})^\vee$ (notice this does not change the reduction type), we may assume that $J^0[p]$ is its own Cartier dual, then $\alpha(J^0[p])=2g/2=g$. Then we have $$r\le h^1-h^0\le \delta-\alpha=0,$$ hence $r=0$. Finally, by then Neron mapping property, $J^{(\mathfrak{P})}(\mathbb{Q})=J(\mathbb{Z})$, which has the same rank as $J^0(\mathbb{Z})$. This completes the proof.

Remark 6 The information about the toric reduction at $N$ is crucial: the defect $\delta$ gets bigger when there are more unipotent parts and so the bound on the rank $r$ gets higher. Also etale cohomology does not fulfill our purpose since $[p]$ is not invertible over the base $S$. This point was already illustrated in Tom's talk when performing 19-descent on $J_1(13)$.
Remark 7 Mazur called this descent argument "geometric descent" in the sense that the fppf cohomology group $H^1_\mathrm{fppf}(S, J^0[p])$ has a purely geometric description as $J^0[p]$-torsors without using cocycles and coboundaries: since $J^0[p]$ itself is fppf over $S$, every $J^0[p]$-torsor is automatically fppf over $S$.

TopAdmissibility

It remains to prove the key admissibility result for $J[p]/_{S}$ (Theorem 4), where $J/_{S}$ is the Neron model of $J^{(\mathfrak{P})}$. It suffices to check that the finite Galois module $J[p](\overline{\mathbb{Q}})$ has composition factors $\mathbb{Z}/p$ or $\mu_p$. We will utilize the classical theorem of Brauer-Nesbitt.

Theorem 5 (Brauer-Nesbitt) Let $k$ be any field and $A$ be a $k$-algebra. Let $M$, $N$ be two $A$-modules which are finite-dimensional as $k$-vector spaces. If for all $a\in A$, the characteristic polynomials of $a$ on $M$ and $N$ are equal, then $M$ and $N$ have the same composition factors.
Remark 8 In our mind $A=\mathbb{F}_p[G]$ will be the group algebra of a finite Galois group $G$. Since we are in characteristic $p$, we need the full characteristic polynomials rather than merely the character of $G$.
Lemma 1 $J[p]\subseteq J_0(N)[\mathfrak{P}^r]$ for some $r>0$.
Proof Since the action of $\mathbb{T}_p$ on $T_pJ[p]$ factors through $\mathbb{T}_\mathfrak{P}$ (Remark 4), we know that the action of $\mathbb{T}_p$ on $J[p]$ factors through $\mathbb{T}_\mathfrak{P}/p \mathbb{T}_\mathfrak{P}$, which is a finite $\mathbb{F}_p$-vector space, hence is an Artinian local ring and the maximal ideal $\mathfrak{P}$ is nilpotent in $\mathbb{T}_\mathfrak{P}/p \mathbb{T}_\mathfrak{P}$. In other words, $\mathfrak{P}^r\subseteq pT_\mathfrak{P}$ for some $r>0$.

So to finish the proof of Theorem 4, it suffices to prove

Theorem 6 $J_0(N)[\mathfrak{P}^r]$ is an admissible group scheme for any $r>0$.
Proof Consider the filtration $$J_0(N)[\mathfrak{P}^r]\supseteq \mathfrak{P}J_0(N)[\mathfrak{P}^r]\supseteq \cdots\supseteq \mathfrak{P}^rJ_0(N)[\mathfrak{P}^r]=0.$$ It suffices to show that $V=\mathfrak{P}^iJ_0(N)[\mathfrak{P}^r]/\mathfrak{P}^{i+1} J_0(N)[\mathfrak{P}^r]$ is admissible. Let $W=V(\overline{\mathbb{Q}})\oplus V^\vee(\overline{\mathbb{Q}})$, then $W$ is a finite $\Gal(\overline{\mathbb{Q}}/\mathbb{Q})$-module and the action of $\Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ on $W$ factors through a finite quotient $G$, i.e., $W$ is a $\mathbb{F}_p[G]$-module. For any element $\sigma\in G$, we can find some $\ell\ne N$ such that $\Frob_\ell$ and $\sigma$ are in the same conjugacy class by Chebotarev. By the Eichler-Shimura relation, we also know that $\Frob_\ell^2-T_\ell\Frob_\ell+\ell=0$. Since $\mathfrak{P}$ kills $W$, $T_\ell$ acts as $1+\ell$ on $W$, hence the eigenvalue of $\Frob_\ell$ must be either 1 or $\ell$. Since taking the Cartier dual interchanges the eigenvalues 1 and $\ell$, we know that the characteristic polynomial of $\Frob_\ell$ on $W$ is equal to $(X-1)^k(X-\ell)^k$, where $2k$ is the dimension of $W$ as an $\mathbb{F}_p$-vector space. So the characteristic polynomial of $\sigma$ on $W$ is also $(X-1)^k(X-\ell)^k$. On the other hand, the characteristic polynomial of $\sigma$ on the admissible group $\mathbb{Z}/p^k\oplus \mu_p^k$ (we can always choose $\ell\ne p$ and enlarge $G$ such that $G$ acts on it) is also equal to $(X-1)^k(X-\ell)^k$. We now apply Brauer-Nesbitt's Theorem 5 to $A=\mathbb{F}_p[G]$ and $M=W$, $N=\mathbb{Z}/p^k\oplus \mu_p^k$ to conclude that $W$ is admissible, and thus $V$ is admissible as required.