By Robert P. Langlands

ISBN-10: 0691082723

ISBN-13: 9780691082721

The matter of base swap or of lifting for automorphic representations might be brought in different methods. It emerges in a short time whilst one pursues the formal rules expounded within the article 20 that may actually be diminished to 1, viz., the functoriality of automorphic types with appreciate to what's now referred because the L-group.

**Read Online or Download Base change for GL(2) PDF**

**Similar group theory books**

Within the final 20 years Cohen-Macaulay jewelry and modules were imperative themes in commutative algebra. This booklet meets the necessity for a radical, self-contained creation to the homological and combinatorial points of the speculation of Cohen-Macaulay jewelry, Gorenstein earrings, neighborhood cohomology, and canonical modules.

**Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups**

This atlas covers teams from the households of the category of finite uncomplicated teams. lately up-to-date incorporating corrections

**Classical Theory of Algebraic Numbers**

The exposition of the classical thought of algebraic numbers is apparent and thorough, and there is a huge variety of routines in addition to labored out numerical examples. A cautious learn of this publication will supply an outstanding heritage to the training of more moderen issues.

- p-Automorphisms of Finite p-Groups
- Algebra Vol 1. Groups
- Subgroups of Teichmuller modular groups
- Compact Semitopological Semigroups and Weakly Almost Periodic Functions

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

**Example text**

A1 (γ, fλ ) is certainly 0 unless λ = (m + r, m − r), r ≥ 0. If this condition is satisfied it equals 2 n| |(r + α)q r 1− 1 q meas G(O) , meas A(O) r > 0, or 2q −α n| | meas G(O) meas A(O) α j=0 jq j 1 q , 1− 1 q 1− The contribution 2 n| | meas G(O) r rq meas A(O) r = 0. q r meas G(O) Base change 38 is accounted for by the first of the three summands in the lemma. The contribution 2 n| | meas G(O) r αq meas A(O) 1− 1 q by the second, and the remainder, which is 0 for r > 0 and 2 n| | meas G(O) meas A(O) α−1 j=0 jq j−α 1− 1 q for r = 0, by the third.

P) ........ p0 X Since X is the set of fixed points of σ in X(E), the paths from p0 to p and from p0 to σ(p) must start off in different directions. In other words the initial edge of the path from p0 to p does not lie in X . This shows that there are q r (1 − q 1− ) possibilities for the p or, what is the same, the p occurring in the above sum if r > 0 and just 1 if r = 0. 3. Since λ(γ) = λ(δ), the integral is certainly 0 unless λ = (m + r, m − r). If this condition is satisfied it equals mE (λ) 1 + q −1 −r q 2 meas G(O) 1 + q − times 1 − z −1 1−z 1 − q− z− · · z − −1 −1 1−z 1−q z 1 − q −1 z 1 2πi r + 1 − q− z 1 − z −1 1−z · · z− −1 −1 1−z 1−q z 1 − q −1 z r dz .

We have already seen that H is isomorphic to the representation ring of GL(2, C). With G = G(E/F ) we form the direct product L G = GL(2, C) × G which is the L-group of G. Let Φ be the Frobenius element in G. The representation ring of GL(2, C) is isomorphic, by means of the map g → g × Φ from GL(2, C) to GL(2, C) × Φ ⊆LG, to the algebra H obtained by restricting to GL(2, C) × Φ the representation ring of LG, which is the algebra of functions on LG formed by linear combinations of characters of finite-dimensional complex analytic representations of LG.