1549380232-Automorphic_Forms_and_Applications__Sarnak_

Lecture 2. Eisenstein series and £-functions 2.1. Eisenstein Series and Intertwining Operators; The Constant Term Let 7f = 0v1fv ...

308 FREYDOON SHAHID!, LANGLANDS-SHAHID! METHOD They both converge for Re( s) > > 0 and have a finite number of simple pole ...

LECTURE 2. EISENSTEIN SERIES AND L--FUNCTIONS 309 Here 8M,M' is the Kronecker 8-function. Its analytic properties and therefore ...

310 FREYDOON SHAHID!, LANGLANDS-SHAHID! METHOD where 'T/v is the character of F* (with cusp form rJ = ®v'T/v on A'F) defining t ...

LECTURE 2. EISENSTEIN SERIES AND £ -FUNCTIONS Av E X*(T). Now take t such that A(t) = a.v, where a.v E X*(Tv)(= Hom(X*(T), Z) = ...

312 FREYDOON SHAHID!, LANGLANDS-SHAHID! METHOD Using the standard formula where (2.30) then equals (2.31) f(l/2)f(s/2)/r((s + 1) ...

LECTURE 2. EISENSTEIN SERIES AND L-FUNCTIONS 313 2.3. Examples We shall now give a number of important examples of £-functions w ...

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LECTURE 3 Functional equations and rnultiplicativity 3.1. Local Coefficients, Non-constant Term and the Crude Func- tional Equat ...

316 FREYDOON SHAHID!, LANGLANDS-SHAHID! METHOD In fact, if W1Jgv) = AxJs, 7rv)(Iv(gv)fv) is the Whittaker function attached to f ...

LECTURE 3. FUNCTIONAL EQUATIONS AND MULTIPLICATIVITY 317 Proposition 3.5 [Shl]. Given 1 < i:::; m , there exists a split grou ...

318 FREYDOON SHAHID!, LANGLANDS-SHAHID! METHOD an irreducible admissible x - generic representation i:J of M = M(F), these exist ...

LECTURE 3. FUNCTIONAL EQUATIONS AND MULTIPLICATIVITY 319 £-function and root number £ - functions are now defined using 1 - func ...

,. 320 FREYDOON SHAHID!, LANGLANDS-SHAHID! METHOD and (3.20) c:(s,7r,ri) =II c(s,7rv,ri,'l/Jv)· v We then have: Theorem 3.11 (Fu ...

LECTURE 4 Holomorphy and boundedness; applications 4.1. Twists by Highly Ramified Characters, Holomorphy and Boundedness Since o ...

322 FREYDOON SHAHID!, LANGLANDS-SHAHID! METHOD We refer to [Kl,K2,K3] concerning the progress made on this assumption. It should ...

LECTURE 4. HOLOMORPHY AND BOUNDEDNESS; APPLICATIONS 323 L(s, n.,,, ri) is entire. Then, given a finite real interval I, each L(s ...

324 FREYDOON SHAHIDI, LANGLANDS-SHAHIDI METHOD for almost all v. Similarly for root numbers. We in fact prove these equalities f ...

LECTURE 4. HOLOMORPHY AND BOUNDEDNESS; APPLICATIONS 325 Corollary [K4]. The representation Sym^4 (n) is automorphic, where Sym^4 ...

326 FREYDOON SHAHID!, LANGLANDS-SHAHID! METHOD For the next proposition we refer the reader to the discussion before and after t ...