We complete the proof that every elliptic curve over the rational numbers is modular.
We will first remind the reader of the content of these results and then briefly outline the method of proof.
If is a positive integer, then we let denote the subgroup of consisting of matrices that modulo are of the form
The quotient of the upper half plane by acting by fractional linear transformations, is the complex manifold associated to an affine algebraic curve , This curve has a natural model . which for , is a fine moduli scheme for elliptic curves with a point of exact order We will let . denote the smooth projective curve which contains as a dense Zariski open subset.
Recall that a cusp form of weight and level is a holomorphic function on the upper half complex plane such that
for all matrices
and all we have ,;
and is bounded on .
The space of cusp forms of weight and level is a finite-dimensional complex vector space. If then it has an expansion ,
and we define the of -series to be
For each prime there is a linear operator on defined by
with and The operators . for can be simultaneously diagonalised on the space and a simultaneous eigenvector is called an eigenform. If is an eigenform, then the corresponding eigenvalues, are algebraic integers and we have ,.
Let be a place of the algebraic closure of in above a rational prime and let denote the algebraic closure of thought of as a algebra via If . is an eigenform, then there is a unique continuous irreducible representation
such that for any prime , is unramified at and The existence of . is due to Shimura if Reference Sh2, to Deligne if Reference De and to Deligne and Serre if Reference DS. Its irreducibility is due to Ribet if Reference Ri and to Deligne and Serre if Reference DS. Moreover is odd in the sense that of complex conjugation is Also, . is potentially semi-stable at in the sense of Fontaine. We can choose a conjugate of which is valued in and reducing modulo the maximal ideal and semi-simplifying yields a continuous representation ,
which, up to isomorphism, does not depend on the choice of conjugate of .
Now suppose that is a continuous representation which is unramified outside finitely many primes and for which the restriction of to a decomposition group at is potentially semi-stable in the sense of Fontaine. To we can associate both a pair of Hodge-Tate numbers and a Weil-Deligne representation of the Weil group of We define the conductor . of to be the product over of the conductor of and of the conductor of the Weil-Deligne representation associated to We define the weight . of to be plus the absolute difference of the two Hodge-Tate numbers of It is known by work of Carayol and others that the following two conditions are equivalent: .
for some eigenform and some place ;
for some eigenform of level and weight and some place .
When these equivalent conditions are met we call modular. It is conjectured by Fontaine and Mazur that if is a continuous irreducible representation which satisfies
is unramified outside finitely many primes,
is potentially semi-stable with its smaller Hodge-Tate number ,
and, in the case where both Hodge-Tate numbers are zero, is odd,
then is modular Reference FM.
Next consider a continuous irreducible representation Serre .Reference Se2 defines the conductor and weight of We call . modular if for some eigenform and some place We call . strongly modular if moreover we may take to have weight and level It is known from work of Mazur, Ribet, Carayol, Gross, Coleman, Voloch and others that for . , is strongly modular if and only if it is modular (see Reference Di1). Serre has conjectured that all odd, irreducible are strongly modular Reference Se2.
Now consider an elliptic curve Let . (resp. denote the representation of ) on the Tate module (resp. the -adic of -torsion) Let . denote the conductor of It is known that the following conditions are equivalent: .
The -function of equals the -function for some eigenform .
The -function of equals the -function for some eigenform of weight and level .
For some prime the representation , is modular.
For all primes the representation , is modular.
There is a non-constant holomorphic map for some positive integer .
There is a non-constant morphism which is defined over .
The implications (2) (1), (4) (3) and (6) (5) are tautological. The implication (1) (4) follows from the characterisation of in terms of The implication (3) . (2) follows from a theorem of Carayol Reference Ca1. The implication (2) (6) follows from a construction of Shimura Reference Sh2 and a theorem of Faltings Reference Fa. The implication (5) (3) seems to have been first noticed by Mazur Reference Maz. When these equivalent conditions are satisfied we call modular.
It has become a standard conjecture that all elliptic curves over are modular, although at the time this conjecture was first suggested the equivalence of the conditions above may not have been clear. Taniyama made a suggestion along the lines (1) as one of a series of problems collected at the Tokyo-Nikko conference in September 1955. However his formulation did not make clear whether should be a modular form or some more general automorphic form. He also suggested that constructions as in (5) and (6) might help attack this problem at least for some elliptic curves. In private conversations with a number of mathematicians (including Weil) in the early 1960’s, Shimura suggested that assertions along the lines of (5) and (6) might be true (see Reference Sh3 and the commentary on [1967a] in Reference We2). The first time such a suggestion appears in print is Weil’s comment in Reference We1 that assertions along the lines of (5) and (6) follow from the main result of that paper, a construction of Shimura and from certain “reasonable suppositions” and “natural assumptions”. That assertion (1) is true for CM elliptic curves follows at once from work of Hecke and Deuring. Shimura Reference Sh1 went on to check assertion (5) for these curves.
If is irreducible, we show that is modular.
If is reducible, but is absolutely irreducible, we show that is modular.
If is reducible and is absolutely reducible, then we show that is isogenous to an elliptic curve with -invariant , or , and so (from tables of modular elliptic curves of low conductor) is modular.
In each of cases (1) and (2) there are two steps. First we prove that is modular and then that is modular. In case (1) this first step is our Theorem B and in case (2) it is a celebrated theorem of Langlands and Tunnell Reference L, Reference T. In fact, in both cases obtains semi-stable reduction over a tame extension of and the deduction of the modularity of from that of was carried out in Reference CDT by an extension of the methods of Reference Wi and Reference TW. In the third case we have to analyse the rational points on some modular curves of small level. This we did, with Elkies’ help, in Reference CDT.
is unramified at .
has order .
has order .
has order and has conductor .
has order .
is induced from a character such that and
where we use the Artin map (normalised to take uniformisers to arithmetic Frobenius) to identify with a character of .
We will refer to these as the and cases respectively.
Using the technique of Minkowski and Klein (i.e. the observation that the moduli space of elliptic curves with full level structure has genus see for example ;Reference Kl), Hilbert irreducibility and some local computations of Manoharmayum Reference Man, we find an elliptic curve with the following properties (see §2.2):
is surjective onto ,
in the case, either or
and is peu ramifié;
in the case,
in the case, ;
in the case,
and is très ramifié;
in the case,
and is très ramifié;
in the case,
is non-split over and is très ramifié.
(We are using the terms très ramifié and peu ramifié in the sense of Serre Reference Se2. We are also letting denote the cyclotomic character and the second fundamental character i.e. ,
We will often use the equality without further remark.) We emphasise that for a general elliptic curve over with the representation , does not have the above form, rather we are placing a significant restriction on .
In each case our strategy is to prove that is modular and so deduce that is modular. Again we use the Langlands-Tunnell theorem to see that is modular and then an analogue of the arguments of Reference Wi and Reference TW to conclude that is modular. This was carried out in Reference Di2 in the cases and and in ,Reference CDT in the case (In these cases the particular form of . is not important.) This leaves the cases , and which are complicated by the fact that , now only obtains good reduction over a wild extension of In these cases our argument relies essentially on the particular form we have obtained for . (depending on We do not believe that our methods for deducing the modularity of ). from that of would work without this key simplification. It seems to be a piece of undeserved good fortune that for each possibility for we can find a choice for for which our methods work.
Following Wiles, to deduce the modularity of from that of we consider certain universal deformations of , and identify them with certain modular deformations which we realise over certain Hecke algebras. The key problem is to find the right local condition to impose on these deformations at the prime As in .Reference CDT we require that the deformations lie in the closure of the characteristic zero points which are potentially Barsotti-Tate (i.e. come from a group over the ring of integers of a finite extension of -divisible and for which the associated representation of the Weil group (see for example Appendix B of )Reference CDT) is of some specified form. That one can find suitable conditions on the representation of the Weil group at for the arguments of Reference TW to work seems to be a rare phenomenon in the wild case. It is here we make essential use of the fact that we need only treat our specific pairs .
Our arguments follow closely the arguments of Reference CDT. There are two main new features. Firstly, in the case, we are forced to specify the restriction of our representation of the Weil group completely, rather than simply its restriction to the inertia group as we have done in the past. Secondly, in the key computation of the local deformation rings, we now make use of a new description (due to Breuil) of finite flat group schemes over the ring of integers of any field in terms of certain (semi-)linear algebra data (see -adicReference Br2 and the summary Reference Br1). This description enables us to make these computations. As the persistent reader will soon discover they are lengthy and delicate, particularly in the case It seems miraculous to us that these long computations with finite flat group schemes in § .7, §8 and §9 give answers completely in accord with predictions made from much shorter computations with the local Langlands correspondence and the modular representation theory of in §3. We see no direct connection, but cannot help thinking that some such connection should exist.
If is a field we let denote a separable closure, the maximal subextension of which is abelian over and the Galois group If . is a field (i.e. a finite extension of -adic and ) a (possibly infinite) Galois extension, then we let denote the inertia subgroup of We also let . denote , denote the arithmetic Frobenius element and denote the Weil group of i.e. the dense subgroup of , consisting of elements which map to an integer power of We will normalise the Artin map of local class field theory so that uniformisers and arithmetic Frobenius elements correspond. (We apologise for this convention, which now seems to us a bad choice. However we feel it is important to stay consistent with .Reference CDT.) We let denote the ring of integers of , the maximal ideal of and the residue field We write simply . for , for and for We also let . denote the unique unramified degree extension of in If . is any perfect field of characteristic we also use to denote the automorphism of -power and its canonical lift to the Witt vectors .
We write for the cyclotomic character and sometimes -adic for the reduction of modulo We write . for the second fundamental character i.e. ,
We also use and to denote the Teichmuller lifts of and .
We let denote the trivial character of a group. We will denote by the dual of a vector space .
If is a homomorphism of rings and if is an then we sometimes write -scheme, for the pullback of by We adopt this notation so that . Similarly if . is a morphism of schemes over we will sometimes write for the pullback of by .
By finite flat group scheme we always mean commutative finite flat group scheme. If is a field of characteristic with fixed algebraic closure we use without comment the canonical identification of finite flat schemes with finite discrete -group and we will say that such objects correspond. If -modules, is a Dedekind domain with field of fractions of characteristic then by a model of a finite flat , scheme -group we mean a finite locally free scheme -group and an isomorphism As in Proposition 2.2.2 of .Reference Ra the isomorphism classes of models for form a lattice ( if there exists a map of finite flat group schemes compatible with and and we can talk about the inf and sup of two such models. If ) is also local we call the model local-local if its special fibre is local-local. When the ring is understood we sometimes simply refer to or even just , as an integral model of ,.
We use Serre’s terminology peu ramifié and très ramifié; see Reference Se2.
1.1. Types of local deformations
By an we mean an equivalence class of two-dimensional representations -type
over which have open kernel and which can be extended to a representation of By an extended . we shall simply mean an equivalence class of two-dimensional representations -type