7 Automorphic forms and the Langlands Conjectures
This chapter came from discussions between Patrick, Mario and myself, all currently visiting the Hausdorff Research Institute for Mathematics in Bonn. The ultimate goal is to formally state some version of the global Langlands reciprocity conjectures for \(\operatorname{GL}_n\) over \(\mathbb {Q}\).
7.1 Definition of an automorphic form for \(\operatorname{GL}_n\) over \(\mathbb {Q}\).
The global Langlands reciprocity conjectures relate automorphic forms to Galois representations. The statements for a general connected reductive group involve the construction of the Langlands dual group, and we do not have quite enough Lie algebra theory to push this definition through in general. However if we restrict the special case of the group \(\operatorname{GL}_n/\mathbb {Q}\), the dual group is just \(\operatorname{GL}_n(\mathbb {C})\) and several other technical obstructions are also removed. In this section we will explain the definition of an automorphic form for the group \(\operatorname{GL}_n/\mathbb {Q}\), following the exposition by Borel and Jacquet in Corvallis.
7.2 The finite adeles of the rationals.
Mathlib already has the definition of the finite adeles \(\mathbb {A}_{\mathbb {Q}}^f\) of the rationals as a commutative \(\mathbb {Q}\)-algebra. It does not yet have the topology; work on this is on PR 13703 to mathlib. See also the prerequisite PR 13705, which is ready for review. Once these PRs are merged, we will have
The finite adeles are a topological ring.
All this is done in the slightly technical mathlib PR 13703.
7.3 The group \(\operatorname{GL}_n(\mathbb {A}_{\mathbb {Q}})\).
The adeles \(\mathbb {A}_{\mathbb {Q}}\) of \(\mathbb {Q}\) are the product \(\mathbb {A}_{\mathbb {Q}}^f\times \mathbb {R}\), with the product topology. They are a topological ring. Hence \(\operatorname{GL}_n(\mathbb {A}_{\mathbb {Q}})=\operatorname{GL}_n(\mathbb {A}_{\mathbb {Q}}^f)\times \operatorname{GL}_n(\mathbb {R})\) is a topological group, where we are being a bit liberal with our use of the equality symbol.
7.4 Smooth functions
A function \(f:\operatorname{GL}_n(\mathbb {A}_{\mathbb {Q}}^f)\times \operatorname{GL}_n(\mathbb {R})\to \mathbb {C}\) is smooth if it has the following three properties.
\(f\) is continuous.
For all \(x\in \operatorname{GL}_n(\mathbb {A}_{\mathbb {Q}}^f)\), the function \(y\mapsto f(x,y)\) is smooth.
For all \(y\in \operatorname{GL}_n(\mathbb {R})\), the function \(x\mapsto f(x,y)\) is locally constant.
Current state of this definition: I’ve half-formalised it; I don’t know how to say the the function is smooth on the infinite part, because I have never used the manifold library before and I have no idea what my model with corners is supposed to be.
7.5 Slowly-increasing functions
Automorphic representations satisfy a growth condition which we may as well factor out into a separate definition.
We define the following temporary “size” function \(s:\operatorname{GL}_n(\mathbb {R})\to \mathbb {R}\) by \(s(M)=trace(MM^T+M^{-1}M^{-T})\) where \(M^{-T}\) denotes inverse-transpose. Note that \(s(M)\) is always positive, and is large if \(M\) has a very large or very small (in absolute value) eigenvalue.
We say that a function \(f:\operatorname{GL}_n(\mathbb {R})\to \mathbb {C}\) is slowly-increasing if there’s some real constant \(C\) and positive integer \(n\) such that \(|f(M)|\leq Cs(M)^n\) for all \(M\in \operatorname{GL}_n(\mathbb {R})\).
Note: the book says \(n\) is positive, but \(\{ M|s(M)\leq 1\} \) is compact so I don’t think it makes any difference.
7.6 Weights at infinity
The weight of an automorphic form for \(\operatorname{GL}_n/\mathbb {Q}\) can be thought of as a finite-dimensional continuous complex representation \(\rho \) of a maximal compact subgroup of \(\operatorname{GL}_n(\mathbb {R})\), and it’s convenient to choose one (they’re all conjugate) so we choose \(O_n(\mathbb {R})\).
The Lean definition is incomplete right now – I don’t demand irreducibility (I wasn’t sure whether I was doing this the right way; if I used category theory then I might have struggled to say that the representation was continuous).
7.7 The action of the universal enveloping algebra.
There is a natural action of the real Lie algebra of \(\operatorname{GL}_n(\mathbb {R})\) on the complex vector space of smooth complex-valued functions on \(\operatorname{GL}_n(\mathbb {R})\).
This extends to is a natural complex Lie algebra action of the complexification of the real Lie algebra, on the smooth complex functions on \(\operatorname{GL}_n(\mathbb {R})\).
By functoriality, we get an action of the universal enveloping algebra of this complexified Lie algebra on the smooth complex functions.
Thus the centre \(\mathbb {Z}_n\) of this universal enveloping algebra also acts on the smooth complex functions.
The centre we just defined is a commutative ring which contains a copy of \(\mathbb {C}\). Note that Harish-Chandra, or possibly this was known earlier, showed that it is a polynomial ring in \(n\) variables over the complexes. We shall not need this.
7.8 Automorphic forms
From here on there is no more Lean right now, only LaTeX.
A smooth function \(f:\operatorname{GL}_n(\mathbb {A}_{\mathbb {Q}}^f)\times \operatorname{GL}_n(\mathbb {R})\to \mathbb {C}\) is an \(O_n(\mathbb {R})\)-automorphic form on \(\operatorname{GL}_n(\mathbb {A}_{\mathbb {Q}})\) if it satisfies the following five conditions.
(periodicity) For all \(g\in \operatorname{GL}_n(\mathbb {Q})\), we have \(f(gx,gy)=f(x,y)\).
(has a finite level) There exists a compact open subgroup \(U\subseteq \operatorname{GL}_n(\mathbb {A}_{\mathbb {Q}}^f)\) such that \(f(xu,y)=f(x,y)\) for all \(u\in U\), \(x\in \operatorname{GL}_n(\mathbb {A}_{\mathbb {Q}}^f)\) and \(y\in \operatorname{GL}_n(\mathbb {R})\).
(weight \(\rho \)) There exists a continuous finite-dimensional irreducible complex representation \(\rho \) of \(O_n(\mathbb {R})\) such that for every \((x,y)\in \operatorname{GL}_n(\mathbb {A}_{\mathbb {Q}})\), the set of functions \(k\mapsto f(x,yk)\) span a finite-dimensional complex vector space isomorphic as \(O_n(\mathbb {R})\)-representation to a direct sum of copies of \(\rho \).
(has an infinite level) There is an ideal \(I\) of the centre \(Z_n\) described in the previous section, which has finite complex codimension, and which annihiliates the function \(y \mapsto f(x,y)\) for all \(x\in \operatorname{GL}_n(\mathbb {A}_{\mathbb {Q}}^f)\). Note that this is a very fancy way of saying “the function satisfies some natural differential equations”. In the case of modular forms, the differential equations are the Cauchy-Riemann equations, which is why modular forms are holomorphic.
(growth condition) For every \(x\in \operatorname{GL}_n(\mathbb {A}_{\mathbb {Q}}^f)\), the function \(y\mapsto f(x,y)\) on \(\operatorname{GL}_n(\mathbb {R})\) is slowly-increasing.
Automorphic forms of a fixed weight \(\rho \) form a complex vector space, and if we also fix the finite level \(U\) and the infinite level \(I\) then we get a subspace which is finite-dimensional; this is a theorem of Harish-Chandra. There is also the concept of a cusp form, meaning an automorphic form for which furthermore some adelic integrals vanish.
7.9 Hecke operators
The group \(\operatorname{GL}_n(\mathbb {A}_{\mathbb {Q}}^f)\) acts (on the left) on the space of automorphic forms for \(\operatorname{GL}_n(\mathbb {A}_{\mathbb {Q}})\) by the formula \((g\cdot f)(x,y)=f(xg,y)\).
This is obvious. Note that the conjugate of a compact open subgroup is still compact and open.
A formal development of the theory of Hecke operators looks like the following.
Let \(U\) be a fixed compact open subgroup of \(\operatorname{GL}_n(\mathbb {A}_{\mathbb {Q}}^f)\), and let’s also fix a weight \(\rho \), and let \(M_\rho (n)\) denote the complex vector space of automorphic forms for \(\operatorname{GL}_n/\mathbb {Q}\) of weight \(\rho \). The level \(U\) forms \(M_\rho (n,U)\) are just the \(U\)-invariants of this space. If \(g\in \operatorname{GL}_n(\mathbb {A}_{\mathbb {Q}}^f)\), then I claim that the double coset space \(UgU\) can be written as a finite disjoint union of single cosets \(g_iU\); one way of sesing this is that the double coset space is certainly a disjoint union of left cosets, but the double coset space is compact and the left cosets are open.
Define the Hecke operator \(T_g:M_\rho (n,U)\to M_\rho (n,U)\) by \(T_g(f)=\sum g_i\cdot f\).
This function is well-defined, i.e., it sends a \(U\)-invariant form to a \(U\)-invariant form which is independent of the choice of \(g_i\).
Easy group theory.