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 and show that the -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 is finite whenever it is not obviously infinite, for a prime.

We fix a prime number throughout this talk. Recall that the *Eisenstein ideal* is defined to be the ideal generated by , and and we have seen from Cheng-Chiang's talk that , where is the numerator of . So the maximal ideals of containing are exactly the *Eisenstein primes* , , with residue fields .

Remark 1
Notice that there exists an Eisenstein prime if and only if (i.e., ), if and only if the genus .

Remark 2
We can visualize the Eisenstein primes intuitively as follows. Consider the Hecke algebra generated by and as endomorphisms on all holomorphic modular forms of weight 2. Since decomposes as the direct sum of the space of cusp forms and the space of Eisenstein series (which is one-dimensional, generated by ), restricting the action of on and gives two inclusions and . The latter is naturally called the *Eisenstein line*. So an Eisenstein prime can be viewed as the intersection of with the Eisenstein line, reflecting the congruence relation between the Eisenstein series and a certain cusp form modulo : there exists a cusp form whose -expansion modulo coincides with the Hecke eigenvector (whose eigenvalues are exactly and ). This congruence relation is the key input for the existence of an Eisenstein prime .

Eisenstein primes |

Remark 3
Associated to an Eisenstein prime we can construct a canonical quotient of . Since is a free -module of finite rank, it is a flat extension of and has Krull dimension 1 by going-down (so our picture above is accurate in this sense). Since is a finite dimensional vector space, it is Artinian and we have already seen that it is actually a product of totally real fields (due to the fact that has semistable reduction). The embedding then gives a bijection between minimal prime ideals of and prime ideals of (which are all maximal), and in turn a bijection with the totally real fields , and the isogenous factors of . We define the *Eisenstein prime quotient* to be the unique *optimal quotient* (i.e., quotient by an abelian subvariety of ) whose isogenous factors correspond to minimal prime ideals contained in . Then also *acts* on . To visualize, each isogenous factor of corresponds to an irreducible component of and simply corresponds to all the components passing through the Eisenstein prime .

Remark 4
Let be the completion of at . Then is a free -module of finite rank, hence is a semilocal ring and , where runs over all maximal ideals containing . Then acts on the -adic Tate module . Moreover, the rational -adic Tate module can be identified as the direct product of , where runs over all minimal primes contained in . The action of on each factors through , where is the totally real field corresponding to . It follows that the action of on factors through .

Our main goal is to prove the following

Theorem 1
Let be an Eisenstein prime and be the Eisenstein prime quotient of . Then has rank 0, i.e., is finite.

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

Remark 5
When , and the two cusps are rational, hence is clearly infinite in these cases. In the case , we have seen an example of explicitly computing rational points for in Erick's talk. One can also quote the big theorem of Faltings for , but Mazur's method can certainly say more about .

Proof
Under the assumption, , so there exists a nontrivial Eisenstein prime quotient (Remark 2). Since is an irreducible projective curve, the composite map is either constant or a finite morphism onto a curve. Because the image of generates as a group, the image of generates as a group (which has positive dimension) and hence cannot be constant. So is a finite morphism of curves. By Theorem 1, we know that has only finite rational points, thus so does .
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Recall that a quasi-finite flat separated group scheme (finite flat over ) is called *()-admissible* if is killed by a power of and admits a filtration by finite flat group schemes such that the successive quotients are either or . Bao has explained in his talk that the admissibility can be detected on the associated -module and also proved the following easy but crucial estimate.

Theorem 3
Let be an admissible group scheme. Then where , is the defect and is the number of 's in the successive quotients of .

The proof of Theorem 1 relies on the following admissibility result, which allows one to bound the Mordell-Weil rank of via Theorem 3 and is the major motivation to introduce the notion of the Eisenstein quotients.

Theorem 4
Let be the Neron model of the Eisenstein prime quotient . Then its -torsion 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 be the fiberwise connected component of . We know from George's talk that has good reduction outside and toric reduction at , hence so does . Then is a surjective morphism of schemes (which can be checked on geometric points, Lemma 0487) as there is no unipotent part in . Since is also fppf (but not etale since is not invertible on ), is a surjection as fppf sheaves (Lemma 05VM, but not as etale sheaves). So we obtain an exact sequence of fppf sheaves which induces an exact sequence in fppf cohomology The first inclusion implies that where is the rank of the abelian group .

On the other hand, by Theorem 4, is also admissible. So Theorem 3 gives Let . Using the toric reduction at , we can compute . Replacing by (notice this does not change the reduction type), we may assume that is its own Cartier dual, then . Then we have hence . Finally, by then Neron mapping property, , which has the same rank as . This completes the proof. ¡õ

Remark 6
The information about the toric reduction at is crucial: the defect gets bigger when there are more unipotent parts and so the bound on the rank gets higher. Also etale cohomology does not fulfill our purpose since is not invertible over the base . This point was already illustrated in Tom's talk when performing 19-descent on .

Remark 7
Mazur called this descent argument "geometric descent" in the sense that the fppf cohomology group has a purely geometric description as -torsors without using cocycles and coboundaries: since itself is fppf over , every -torsor is automatically fppf over .

It remains to prove the key admissibility result for (Theorem 4), where is the Neron model of . It suffices to check that the finite Galois module has composition factors or . We will utilize the classical theorem of Brauer-Nesbitt.

Theorem 5 (Brauer-Nesbitt)
Let be any field and be a -algebra. Let , be two -modules which are finite-dimensional as -vector spaces. If for all , the characteristic polynomials of on and are equal, then and have the same composition factors.

Remark 8
In our mind will be the group algebra of a finite Galois group . Since we are in characteristic , we need the full characteristic polynomials rather than merely the character of .

Proof
Since the action of on factors through (Remark 4), we know that the action of on factors through , which is a finite -vector space, hence is an Artinian local ring and the maximal ideal is nilpotent in . In other words, for some .
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So to finish the proof of Theorem 4, it suffices to prove

Proof
Consider the filtration It suffices to show that is admissible. Let , then is a finite -module and the action of on factors through a finite quotient , i.e., is a -module. For any element , we can find some such that and are in the same conjugacy class by Chebotarev. By the Eichler-Shimura relation, we also know that . Since *kills* , acts as on , hence the eigenvalue of must be either 1 or . Since taking the Cartier dual interchanges the eigenvalues 1 and , we know that the characteristic polynomial of on is equal to , where is the dimension of as an -vector space. So the characteristic polynomial of on is also . On the other hand, the characteristic polynomial of on the admissible group (we can always choose and enlarge such that acts on it) is also equal to . We now apply Brauer-Nesbitt's Theorem 5 to and , to conclude that is admissible, and thus is admissible as required.
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