By Robert P. Langlands

ISBN-10: 0691082634

ISBN-13: 9780691082639

ISBN-10: 0691082723

ISBN-13: 9780691082721

The matter of base switch or of lifting for automorphic representations will be brought in different methods. It emerges in a short time while one pursues the formal rules expounded within the article 20 that could in reality be diminished to at least one, viz., the functoriality of automorphic varieties with appreciate to what's now referred because the L-group.

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**Extra resources for Base change for GL(2)**

**Sample text**

If z ∈ Z(F ) lies close to 1 then φ(g −1 zσ(g))dg f (z) = G(F )\G (F ) and if ε=z 1 1 0 1 then Ψφ (ε , µ) = Φf (ε, µ) and N ε = ε . The asserted formulae for Φ (z, T, µ) and for Φ (z, T, µ) follow. Before we discuss the case that δ is not σ -conjugate to a scalar we comment on the manner in which one shows that an HCS family {Φ(γ, T )} for which Φ(γ, T ) = 0 when γ ∈ / N T (E) is a Shintani family.

The apartment A is a line; every vertex lies on two edges. If p 1 , p2 are two points in X there is a g in G(F ) and a t in A(F ) such that p1 = gtp0 , p2 = gp0 . If λ(t) = (k , k) then |k − k| is uniquely determined, and is just the distance from p2 to p1 . We may also associate a simplicial complex X to GL(2, F ) = G(F ). The points are lattices, two lattices M1 and M2 being joined by an edge if M1 ⊃ M2 ⊃ = = M1 or M2 ⊃ M1 ⊃ = = M2 . We may define an apartment A and the type of an ordered pair (p1 , p2 ).

We may suppose < α, µ > ≥ 0 and < α, λ > ≥ 0. 1) is q <α,λ> 2 if λ = µ, q <α,λ> 2 1− 1 q if λ = µ + nα, n > 0, and 0 otherwise. There are two possibilities which have to be treated in different fashions. Suppose the following picture = 1. Then we have γp p .............. b a ... ... ... ... ......... ... ... .......... p0 .......... .......... . ... . . ........ .. .... . . ... . ......... γ p0 .......... A The distance between p0 and γp0 is m − m if µ = (m , m). If the distance of p from p0 is k then the type of γp, p is 2k + m − m, provided d(p, p0 ) = d(p, A), and the type of (γp , p ) is (m + k, m − k).

### Base change for GL(2) by Robert P. Langlands

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