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 of a fixed weight, level and character 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 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 definiteness hypothesis is enough to show that \(D\) is not a matrix algebra and thus must be a division algebra. Thus Fujisaki’s theorem 10.7 from the Fujisaki miniproject applies, and we know that \(D^\times \backslash (D\otimes _F\mathbb {A}_F)^{(1)}\) is compact. We fix once and for all a maximal compact subgroup \(K_\infty \) of \((D\otimes _{\mathbb {Q}}\mathbb {R})^\times \); all maximal compact subgroups are conjugate so changing this choice just amounts to some bookkeeping. Because \(D\otimes _{\mathbb {Q}}\mathbb {R}\) is isomorphic to \(\prod _{v\mid \infty }\mathbb {H}\), the maximal compact subgroup is isomorphic to \(\prod _{v\mid \infty }S^3\), where \(S^3\) denotes the unit quaternions in \(\mathbb {H}\).

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 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 0-dimensional”, meaning in practice that no differential equations are involved in the definition of an automorphic form in this setting. As a consequence, the definitions we’re about to give have a far more algebaic flavour and, as we shall see, it is possible to construct mod \(p\) and \(p\)-adic variants of these spaces without too much trouble.

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 8. Our automorphic forms will be certain functions on the units of the ring \(D_{\mathbb {A}}:=D\otimes _F\mathbb {A}_F\cong D\otimes _{\mathbb {Q}}\mathbb {A}_{\mathbb {Q}}\) and there are adelic analogues of a weight, level and character in this setting, corresponding to the classical notions which one sees in the theory of modular forms. 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. We will fix a weight \(\rho _\infty \), a level \(U\) and a character \(\chi \), and define a complex vector space \(S_{\rho _\infty }^D(U;\chi )\) of quaternionic modular forms of level \(U\), weight \(\rho _\infty \) and character \(\chi \). The main theorem of this miniproject is the claim that this space is finite-dimensional.

11.4 Definition of spaces of automorphic forms

Let us now give precise definitions of weights, levels, characters, and these spaces of quaternionic modular forms.

A weight \(\rho _\infty \) is a finite-dimensional \(\mathbb {C}\)-vector space \(W\) (with the usual vector space topology) equipped with a continuous action of \(K_\infty \), our fixed maximal compact subgroup. For applications to Fermat’s Last Theorem we only need to consider the case where \(W=\mathbb {C}\) and \(\rho _\infty \) is the trivial representation, but there is no harm in setting things up in more generality. The case \(W=\mathbb {C}\) corresponds (via the highly non-trivial Jacquet-Langlands correspondence) when \(F=\mathbb {Q}\) to the case of modular forms of weight 2, and for general \(F\) corresponds to Hilbert modular forms of parallel weight 2.

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 this fact is currently in the process of being PRed to mathlib. 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.

A character is a continuous group homomorphism \(\chi \) from \(F^\times \backslash \mathbb {A}_F^\times \) to \(\mathbb {C}^\times \). For many of the applications, \(\chi \) can also be trivial, although we will crucially have to allow certain nontrivial characters of \(p\)-power order when we are doing the patching argument needed to prove the modularity lifting theorem which is the big first target of the FLT project. We regard \(\mathbb {A}_F\) as a subring of \(D_{\mathbb {A}}:=D\otimes _F\mathbb {A}_F\), which is possible because \(F\) is a subring of \(D\). More precisely we embed \(\mathbb {A}_F\) into \(D\otimes _F\mathbb {A}_F\) via the map sending \(g\) to \(1\otimes g\). Because \(F\) is in the centre of \(D\), we have that \(\mathbb {A}_F\) is in the centre of \(D_{\mathbb {A}}\) and thus \(F^\times \backslash \mathbb {A}_F^\times \) is a central subgroup of \(D_{\mathbb {A}}^\times \).

We fix a weight \((W,\rho _{\infty })\), a level \(U\) and a character \(\chi \).

Let \(S_{\rho _\infty }^D(U;\chi )\) denote the space of automorphic forms of weight \(W\), level \(U\) and character \(\chi \). Two basic observations are

These spaces \(S_{\rho _\infty }^D(U;\chi )\) are sometimes referred to as spaces of “quaternionic modular forms”. The Hecke algebras involved in the main modularity lifting theorems needed in the FLT project will be endomorphisms of these spaces.

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.