12.1. The goal🔗

At the time of writing, mathlib still does not have a proof that the space of classical modular forms of a fixed weight, level and character is finite-dimensional. The main result in this miniproject is to prove that certain spaces of quaternionic modular forms are finite-dimensional. We need finiteness results like this in order to control the Hecke algebras which act on these spaces; these Hecke algebras are the Ts which are isomorphic to the Rs in the R = T theorem which is the big first target for the FLT project.

12.2. Initial definitions🔗

Our first goal in this miniproject is the definition of these spaces of quaternionic modular forms. We start with some preliminary work towards this.

Let K be a field. Recall that a quaternion algebra over K is a central simple K-algebra of K-dimension 4.

A fundamental fact about central simple algebras is that if D/K is a central simple K-algebra and L/K is an extension of fields, then D \otimes_K L is a central simple L-algebra. In particular, if D is a quaternion algebra over K then D \otimes_K L is a quaternion algebra over L. Some Imperial undergraduate students have established this fact in ongoing project work.

A totally real field is a number field F such that the image of every ring homomorphism F \to \bbC is a subset of \R. We fix once and for all a totally real field F and a quaternion algebra D over F. We furthermore assume that D is totally definite, that is, that for all field embeddings \tau : F \to \R we have D \otimes_{F,\tau} \R \cong \bbH. Because F has at least one real place, the totally definite hypothesis is enough to show that D is not a matrix algebra and thus must be a division algebra. Thus Fujisaki's theorem Theorem 11.3.15 from the Fujisaki miniproject applies, and we know that D^\times\backslash (D\otimes_F\A_F)^{(1)} is compact.

The high-falutin' explanation of what is about to happen is that the units D^\times of D can be regarded as a connected reductive algebraic group over F, and we are going to define certain spaces of automorphic forms for this algebraic group. For a general connected reductive algebraic group, part of the definition of an automorphic form is that it is satisfies some differential equations (for example modular forms are automorphic forms for the algebraic group GL_2 over Q, and the definition of a modular form involves holomorphic functions, which are solutions to the Cauchy-Riemann equations). However under the assumption that F is totally real and D/F is totally definite, the "associated symmetric space is a 0-dimensional manifold", meaning in practice that the part of the definition of an automorphic form involving differential equations is vacuously satisfied in this setting.

As a consequence, the definitions we are about to give have a far more algebraic flavour. Crucially, the fact that we do not need to talk about differential equations at all means that one does not need to assume that our automorphic forms are ℂ-valued; our definition makes sense for forms valued in an arbitrary additive commutative group. In particular, it is possible to talk about mod p^n and p-adic automorphic forms in this setting without doing any complicated algebraic geometry.

12.3. Brief introduction to automorphic forms in this setting🔗

Having made assumptions on D which make the theory of automorphic forms over D^\times far less technical, we will now make it a little more technical by using the modern adelic approach to the theory. Note that many results about the adeles of a number field are proved in the adele miniproject. Our automorphic forms will be certain functions on the units of the ring D_{\A^\infty} := D \otimes_F \A_F^\infty \cong D \otimes_{\Q} \A_{\Q}^\infty.

To prove Fermat's Last Theorem it suffices to work in weight 2 and trivial character, and for simplicity we shall, at least for now, bake these assumptions into our definitions, even though they would be easy to remove. We remark again that there is no analogue of the holomorphicity condition that one sees in the classical theory, because the symmetric space is a finite set of points rather than the upper half plane. Also there is no analogue of the cuspidality condition because there are no cusps in this setting. Other than the number field F and the quaternion algebra D, the other variable we will see in our situation will be the level of the forms. The main result in this miniproject will be that the space of weight 2 automorphic forms of level U is finite-dimensional.

12.4. Definition of spaces of automorphic forms🔗

Let us now give some precise definitions. Recall that by \A_F^\infty we mean the finite adeles of the totally real number field F.

A level is a compact open subgroup U of (D \otimes_F \A_F^\infty)^\times. These are plentiful. The ring D_f := D \otimes_F \A_F^\infty is a topological ring, and hence the units D_f^\times of this ring are a topological group. This group is locally profinite, and hence has many compact open subgroups; we will see explicit examples later on.

We regard \A_F^\infty as a subring of D_{\A^\infty} := D \otimes_F \A_F^\infty, which is possible because F is a subring of D. More precisely, we embed \A_F^\infty into D \otimes_F \A_F^\infty via the map sending g to 1 \otimes g. Because F is in the centre of D, we have that \A_F^\infty is in the centre of D_{\A^\infty}. As a consequence we can identify (\A_F^\infty)^\times as a subgroup of (D \otimes_F \A_F^\infty)^\times. We may also regard D as a subring of D \otimes_F \A_F^\infty via the map d \mapsto d \otimes 1, and hence we can think of D^\times as a subgroup of (D \otimes_F \A_F^\infty)^\times.

Let R be an additive commutative group. Later on R will be a commutative ring but we will not need this for the definition.

Let us now give some precise definitions. Recall that by \A_F^\infty we mean the finite adeles of the totally real number field F.

A level is a compact open subgroup U of (D\otimes_F\A_F^\infty)^\times. These are plentiful. The ring D_f:=D\otimes_F\A_F^\infty is a topological ring, and hence the units D_f^\times of this ring are a topological group. This group is locally profinite, and hence has many compact open subgroups; we will see explicit examples later on.

We regard \A_F^\infty as a subring of D_{\A^\infty}:=D\otimes_F\A_F^\infty, which is possible because F is a subring of D. More precisely, we embed \A_F^\infty into D\otimes_F\A_F^\infty via the map sending g to 1\otimes g. Because F is in the centre of D, we have that \A_F^\infty is in the centre of D_{\A^\infty}. As a consequence we can identify (\A_F^\infty)^\times as a subgroup of (D\otimes_F\A_F^\infty)^\times. We may also regard D as a subring of D\otimes_F\A_F^\infty via the map d\mapsto d\otimes 1, and hence we can think of D^\times as a subgroup of (D\otimes_F\A_F^\infty)^\times.

Let R be an additive commutative group. Later on R will be a commutative ring but we will not need this for the definition.

Let S^D(R) denote the set of automorphic forms for D^\times. The space S^D(R) is sometimes referred to as a space of "quaternionic modular forms" over R. Three basic observations about S^D(R) are as follows.

If R is a commutative ring then the action of D_{\A^\infty}^\times commutes with the R-action.

Now let U be a level, that is, a compact open subgroup of D_{\A^\infty}^\times.

The Hecke algebras involved in the main modularity lifting theorems needed in the FLT project will be endomorphisms of these spaces S^D(U;R).

12.5. The main result🔗

The point of this miniproject is the finite-dimensionality result below. This is an analogue of the result that classical modular forms of a fixed level, weight and character are finite-dimensional. In fact, by delicate results of Jacquet and Langlands this result, in the case k = \bbC, implies many cases of that classical claim, although of course the Jacquet--Langlands theorem is much harder to prove than the classical proof of finite-dimensionality.