Theorem 14.1 The topological group described above is called the Weil group of K.

If X is an affine scheme of finite type over K, and if R is a K-algebra which is also a topological ring, then we define a topology on the R-points X(R) of X by embedding the K-algebra homomorphisms from A to R into the set-theoretic maps from A to R with its product topology, and giving it the subspace topology.

Let K be a field equipped with an isomorphism to the reals, complexes, or a finite extension of the p-adic numbers. Let X be a smooth affine algebraic variety over K. Then the points X(K) naturally inherit the structure of a manifold over K.

An affine algebraic group G of finite type over a field k is said to be connected if it is connected as a scheme, and reductive if G_{\overline{k}} has no nontrivial smooth connected unipotent normal k-subgroup.

A function f : G(N_\infty)\to\bbC is slowly-increasing if there exist some C>0 and n\geq1 such that |f(x)|\leq C||x||_\rho^n.

Theorem 14.18 Definition 14.16 Corollary 14.15 Theorem 14.12 An automorphic form is a function \phi:G(\A_N)\to\bbC satisfying the following conditions.

1. \phi is locally constant on G(\A_N^f) and C^\infty on G(N_\infty). In other words, for every g_\infty, \phi(-,g_\infty) is locally constant, and for every g_f, \phi(g_f,-) is smooth.

2. \phi is left-invariant under G(N).

3. \phi is right-U_\infty-finite, that is, the space spanned by x\mapsto \phi(xu) as u varies over U_\infty is finite-dimensional.

4. \phi is right K_f-finite, where K_f is one, or equivalently all, compact open subgroups of G(\A_N^f).

5. \phi is \mathcal{z}-finite, where \mathcal{z} is the centre of the universal enveloping algebra of the Lie algebra of G(N_\infty), acting via differential operators. Equivalently \phi is annihilated by a finite index ideal of this centre, so morally \phi satisfies many differential equations of a certain type.

6. For all g_f, the function g_\infty\mapsto \phi(g_f g_\infty) is slowly-increasing in the sense above.

Definition 14.19 An automorphic form is cuspidal, or a cusp form, if it furthermore satisfies \int_{U(N)\backslash U(\A_N)}\phi(ux)\,du=0, where P runs through all the proper parabolic subgroups of G defined over N and U is the unipotent radical of P, and the integral is with respect to the measure coming from Haar measure.

Definition 14.19 The group G(\A_N) acts on itself on the right, and this induces a left action of its subgroup G(\A_N^f)\times U_\infty on the spaces of automorphic forms and cusp forms. The Lie algebra \mathfrak{g} of G(N_\infty) also acts, via differential operators. Furthermore the actions of \mathfrak{g} and U_\infty are compatible in the sense that the differential of the U_\infty action is the action of its Lie algebra considered as a subalgebra of \mathfrak{g}. We say that the spaces are (G(\A_N^f)\times U_\infty,\mathfrak{g})-modules.

Definition 14.21 An automorphic representation is an irreducible (G(\A_N^f)\times U_\infty,\mathfrak{g})-module isomorphic to an irreducible subquotient of the space of automorphic forms.

We need the definition of the canonical model over F of the Shimura curve attached to an inner form of \GL_2 with precisely one split infinite place, and likewise the Shimura surface associated to an inner form split at two infinite places and ramified elsewhere.

Theorem 14.26 A cuspidal automorphic representation is an irreducible (G(\A_N^f)\times U_\infty,\mathfrak{g})-module isomorphic to an irreducible summand of the space of cusp forms.

Let N be a number field. A compatible family of d-dimensional Galois representations over N is a finite set of finite places S of N, a number field E, a monic degree d polynomial F_{\mathfrak p}(X)\in E[X] for each finite place \mathfrak p of K not in S, and, for each prime number \ell and field embedding \phi : E \to \overline{\mathbf{Q}}_\ell (or essentially equivalently for each finite place of E), a continuous homomorphism \rho:\operatorname{Gal}(\overline{K}/K)\to\GL_2(\overline{\mathbf{Q}}_\ell) unramified outside S and the primes of K above \ell, such that \rho(\operatorname{Frob}_{\mathfrak p}) has characteristic polynomial P_\pi(X) if \pi lies above a prime number p\neq \ell with p\not\in S.