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 .
Our main goal is to prove the following
be an Eisenstein prime and
be the Eisenstein prime quotient of
has rank 0, i.e.,
As an immediate application, we obtain a "conceptual" proof of the following interesting result.
be a prime,
Under the assumption,
, so there exists a nontrivial Eisenstein prime quotient
is an irreducible projective curve, the composite map
is either constant or a finite morphism onto a curve. Because the image of
as a group, the image of
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
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.
be an admissible group scheme. Then
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.
be the Neron model of the Eisenstein prime quotient
. Then its
is an admissible group scheme.
Assuming Theorem 4, we can finish the proof of main Theorem 1 using a standard descent argument.
(Proof of Theorem 1)
be the fiberwise connected component of
. We know from George's talk that
has good reduction outside
and toric reduction at
, hence so does
is a surjective morphism of schemes (which can be checked on geometric points, Lemma 0487
) as there is no unipotent part in
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
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.
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.
be any field and
-modules which are finite-dimensional as
-vector spaces. If for all
, the characteristic polynomials of
are equal, then
have the same composition factors.
Since the action of
), we know that the action of
, which is a finite
-vector space, hence is an Artinian local ring and the maximal ideal
is nilpotent in
. In other words,
So to finish the proof of Theorem 4, it suffices to prove
is an admissible group scheme for any
Consider the filtration
It suffices to show that
is admissible. Let
is a finite
-module and the action of
factors through a finite quotient
-module. For any element
, we can find some
are in the same conjugacy class by Chebotarev. By the Eichler-Shimura relation, we also know that
. Since kills
, 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
is equal to
is the dimension of
-vector space. So the characteristic polynomial of
. On the other hand, the characteristic polynomial of
on the admissible group
(we can always choose
acts on it) is also equal to
. We now apply Brauer-Nesbitt's Theorem 5
to conclude that
is admissible, and thus
is admissible as required.