10 Miniproject: Quaternion algebras
10.1 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.
10.2 Initial definitions
Our first goal is the definition of these spaces of quaternionic modular forms. We start with some preliminary work towards this.
Let \(K\) be a field. A central simple \(K\)-algebra is a \(K\)-algebra \(D\) with centre \(K\) such that \(D\) has no nontrivial two-sided ideals. A quaternion algebra over \(K\) is a central simple \(K\)-algebra of \(K\)-dimension 4.
Matrix algebras \(M_n(K)\) are examples of central simple \(K\)-algebras, so \(2\times 2\) matrices \(M_2(K)\) are an example of a quaternion algebra over \(K\). If \(K=\mathbb {C}\) then \(M_2(\mathbb {C})\) is the only example, up to isomorphism, but there are two examples over the reals, the other being Hamilton’s quaternions \(\mathbb {H}:=\mathbb {R}\oplus \mathbb {R}i\oplus \mathbb {R}j\oplus \mathbb {R}k\) with the usual rules \(i^2=j^2=k^2=-1\), \(ij=-ji=k\) etc. For a general field \(K\) one can make an analogue of Hamilton’s quaternions \(K\oplus Ki\oplus Kj\oplus Kk\) with these same rules to describe the multiplication, and if the characteristic of \(K\) isn’t 2 then this is a quaternion algebra (which may or may not be isomorphic to \(M_2(K)\) in this generality). If \(K\) is a number field then there are infinitely many isomorphism classes of quaternion algebras over \(K\).
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 \(D\otimes _{F,\tau }\mathbb {R}\cong \mathbb {H}\).
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 the solution to 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 that no differential equations are involved in the definition of an automorphic form in this setting. As a consequence, the definition we’re about to give makes sense not just over the complex numbers but over any commutative ring \(R\), which will be crucial for us as we will need to think about, for example, mod \(p\) and more generally mod \(p^n\) automorphic forms in this setting.
10.3 The adelic viewpoint
Having made assumptions on \(D\) which makes the theory far less technical, we will now make it a little more technical by using the modern adelic approach to the theory. The adeles of a number field are discussed in far more detail in the adele miniproject 8. We just recall here that if \(K\) is a number field then there are two huge topological \(K\)-algebras called the finite adeles \(\mathbb {A}_K^\infty \) and the adeles \(\mathbb {A}_K\) of \(K\).
Let us now fix a weight, a level and a character, and we will define a space of automorphic forms for \(D^\times \) of this weight, level and character. If you know something about the theory of classical modular forms you will have seen these words used in that theory. In the adelic set-up these words have slightly different interpretations. We will define automorphic forms over \(R\), an arbitrary commutative base ring. If you are thinking about analogies with spaces of classical modular forms then you could imagine the case \(R=\mathbb {C}\).
A weight is a finitely-generated \(R\)-module \(W\) with an action of \(D^\times \). For applications to Fermat’s Last Theorem we only need to consider the case where \(W=R\) and the action is trivial, but there is no harm in setting things up in more generality. The case \(W=R=\mathbb {C}\) 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 \(D\otimes _F\mathbb {A}_F^\infty \) is a topological ring, and this fact is currently in the process of being PRed to mathlib. Hence the units \((F\otimes _F\mathbb {A}_F^\infty )^\times \) 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 group homomorphism \(\chi \) from \((\mathbb {A}_F^\infty )^\times \) to \(R^\times \). 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 \(\mathbb {A}_F^\infty \) as a subring of \(D\otimes _F\mathbb {A}_F^\infty \), 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 \(D\otimes _F\mathbb {A}_F^\infty \) and thus \((\mathbb {A}_F^\infty )^\times \) is a central subgroup of \((D\otimes _F\mathbb {A}_F^\infty )^\times \).
We fix a base commutative ring \(R\), a weight \(W\), a level \(U\) and a character \(\chi \).
An automorphic form of weight \(W\), level \(U\) and character \(\chi \) for \(D\) is a function \(f:(D\otimes _F\mathbb {A}_F^\infty )^\times \to W\) satisfying the following axioms:
\(f(\delta g)=\delta \cdot f(g)\) for all \(\delta \in D^\times \) and \(g\in (D\otimes _F\mathbb {A}_F^\infty )^\times \) (this makes sense because \(W\) has an action of \(D^\times \)).
\(f(gz)=\chi (z)f(g)\) for all \(g\in (D\otimes _F\mathbb {A}_F^\infty )^\times \) and \(z\in (\mathbb {A}_F^\infty )^\times \) (this makes sense because \(W\) is an \(R\)-module).
\(f(gu)=f(g)\) for all \(g\in (D\otimes _F\mathbb {A}_F^\infty )^\times \) and \(u\in U\).
Let \(S_{W,\chi }(U;R)\) denote the space of automorphic forms of weight \(W\), level \(U\) and character \(\chi \). Two basic observations are
Pointwise addition \((f_1+f_2)(g):=f_1(g)+f_2(g)\) makes \(S_{W,\chi }(U;R)\) into an additive abelian group.
Pointwise scalar multiplication \((r\cdot f)(g):= r\cdot (f(g))\) makes \(S_{W,\chi }(U;R)\) into an \(R\)-module.
These spaces \(S_{W,\chi }(U;R)\) 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.
10.4 Statement of the main result of the miniproject
The point of this miniproject is the following goal:
If \(R\) is a field \(k\) and the weight \(W\) is a finite-dimensional \(k\)-vector space then the space \(S_{W,\chi }(U;k)\) is a finite-dimensional \(k\)-vector space.
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.
The finite-dimensionality theorem is in fact an easy consequence of a finiteness assertion which is valid in far more generality, namely for division algebras over number fields. We state and prove this result in this generality. Let \(K\) be a number field and let \(B/K\) be a finite-dimensional central simple \(K\)-algebra. Assume furthermore that \(B\) is a division algebra, that is, that every nonzero element of \(B\) is a unit. The finiteness theorem we want is this.
If \(U\subseteq (B\otimes _K\mathbb {A}_K^\infty )^\times \) is a compact open subgroup, then the double coset space \(B^\times \backslash (B\otimes _K\mathbb {A}_K^\infty )^\times /U\) is a finite set.
I (kmb) had always imagined that this latter finiteness statement was “folklore” or “a standard consequence of results about automorphic forms”, but in John Voight’s book [ 16 ] this is Main Theorem 27.6.14(b) and Voight calls it Fujisaki’s lemma.
Let’s prove Theorem 10.4, the finite-dimensionality of the space of quaternionic modular forms, assuming Fujisaki’s lemma.
Choose a set of coset representative \(g_1,g_2,\ldots ,g_n\) for \(D^\times \backslash (D\otimes _F\mathbb {A}_F^\infty )^\times /U\). My claim is that the function \(S_{W,\chi }(U;k)\to W^n\) sending \(f\) to \((f(g_1),f(g_2),\ldots ,f(g_n))\) is injective and \(k\)-linear, which suffices by finite-dimensionality of \(W\). \(k\)-linearity is easy, so let’s talk about injectivity.
Say \(f_1\) and \(f_2\) are two elements of \(S_{W,\chi }(U;k)\) which agree on each \(g_i\). It suffices to prove that \(f_1(g)=f_2(g)\) for all \(g\in (D\otimes _F\mathbb {A}_F^\infty )^\times \). So say \(g\in (D\otimes _F\mathbb {A}_F^\infty )^\times \), and write \(g=\delta g_iu\) for \(\delta \in D^\times \) and \(u\in U\). Then \(f_1(g)=f_1(\delta g_iu)=\delta \cdot f_1(g_i)\) (by hypotheses (i) and (iii) of the definition of \(S_{W,\chi }(U;k)\)), and similarly \(f_2(g)=\delta \cdot f_2(g_i)\) and because \(f_1(g_i)=f_2(g_i)\) by assumption, we deduce \(f_1(g)=f_2(g)\) as required.
It thus remains to prove Fujisaki’s lemma, which is what we do in the rest of this miniproject. Note that we will be assuming the results from the adele mini-project, chapter 8 of this blueprint, and also from the Haar character mini-project, chapter 9.
10.5 Proof of Fujisaki’s lemma
TODO.
See Main Theorem 27.6.14 of Voight.