11 Miniproject: Quaternion algebras

11.1 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 “\(T\)”s which are isomorphic to the “\(R\)”s in the \(R=T\) theorem which is the big first target for the FLT project.

11.2 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 \(D\otimes _KL\) is a central simple \(L\)-algebra. In particular if \(D\) is a quaternion algebra over \(K\) then \(D\otimes _KL\) 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 \(F\to \mathbb {C}\) is a subset of \(\mathbb {R}\). 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 \(\tau :F\to \mathbb {R}\) we have \(D\otimes _{F,\tau }\mathbb {R}\cong \mathbb {H}\). 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 10.7 from the Fujisaki miniproject) applies, and we know that \(D^\times \backslash (D\otimes _F\mathbb {A}_F)^{(1)}\) 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 \(\operatorname{GL}_2\) over \(\mathbb {Q}\), 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’re about to give have a far more algebaic flavour. Crucially, in stark contrast to the general theory, 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 \(\mathbb {C}\)-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.

11.3 Brief introduction to automorphic forms in this setting

Having made assumptions on \(D\) which makes 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 (section 8). Our automorphic forms will be certain functions on the units of the ring \(D_{\mathbb {A}^\infty }:=D\otimes _F\mathbb {A}_F^\infty \cong D\otimes _{\mathbb {Q}}\mathbb {A}_{\mathbb {Q}}^\infty \). 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 (at the expense of having to write “of weight 2 and trivial character” throughout the proof). 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.

11.4 Definition of spaces of automorphic forms

Let us now give some precise definitions. Recall that by \(\mathbb {A}_F^\infty \) we mean the finite adeles of the totally real number field \(F\).

A level is a compact open subgroup \(U\) of \((D\otimes _F\mathbb {A}_F^\infty )^\times \). These are plentiful. The ring \(D_f:=D\otimes _F\mathbb {A}_F^\infty \) 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 \(\mathbb {A}_F^\infty \) as a subring of \(D_{\mathbb {A}^\infty }:=D\otimes _F\mathbb {A}_F^\infty \), which is possible because \(F\) is a subring of \(D\). More precisely we embed \(\mathbb {A}_F^\infty \) into \(D\otimes _F\mathbb {A}_F^\infty \) via the map sending \(g\) to \(1\otimes g\). Because \(F\) is in the centre of \(D\), we have that \(\mathbb {A}_F^\infty \) is in the centre of \(D_{\mathbb {A}^\infty }\) (in fact it is the centre, but we do not need this). As a consequence we can identify \((\mathbb {A}_F^\infty )^\times \) as a subgroup of \((D\otimes _F\mathbb {A}_F^\infty )^\times \). We may also regard \(D\) as a subring of \(D\otimes _F\mathbb {A}_F^\infty \) via the map \(d\mapsto d\otimes 1\), and hence we can think of \(D^\times \) as a subgroup of \((D\otimes _F\mathbb {A}_F^\infty )^\times \).

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

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

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

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

11.5 Statement of the main result of the miniproject

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 \(k=\mathbb {C}\)) implies many cases of that classical claim, although of course the Jacquet–Langlands theorem is much much harder to prove than the classical proof of finite-dimensionality.