By Hida H., et al. (eds.)

In Contributions to Automorphic kinds, Geometry, and quantity conception, Haruzo Hida, Dinakar Ramakrishnan, and Freydoon Shahidi collect a distinct team of specialists to discover automorphic kinds, mostly through the linked L-functions, illustration concept, and geometry. simply because those subject matters are on the innovative of a relevant quarter of contemporary arithmetic, and are concerning the philosophical base of Wiles' facts of Fermat's final theorem, this e-book may be of curiosity to operating mathematicians and scholars alike. by no means formerly released, the contributions to this quantity divulge the reader to a bunch of adverse and thought-provoking problems.Each of the extreme and remarkable mathematicians during this quantity makes a different contribution to a box that's at present seeing explosive development. New and strong effects are being proved, noticeably and constantly altering the field's make up. Contributions to Automorphic varieties, Geometry, and quantity idea will most likely result in very important interplay between researchers and likewise support arrange scholars and different younger mathematicians to go into this interesting region of natural mathematics.Contributors: Jeffrey Adams, Jeffrey D. Adler, James Arthur, Don Blasius, Siegfried Boecherer, Daniel Bump, William Casselmann, Laurent Clozel, James Cogdell, Laurence Corwin, Solomon Friedberg, Masaaki Furusawa, Benedict Gross, Thomas Hales, Joseph Harris, Michael Harris, Jeffrey Hoffstein, Hervé Jacquet, Dihua Jiang, Nicholas Katz, Henry Kim, Victor Kreiman, Stephen Kudla, Philip Kutzko, V. Lakshmibai, Robert Langlands, Erez Lapid, Ilya Piatetski-Shapiro, Dipendra Prasad, Stephen Rallis, Dinakar Ramakrishnan, Paul Sally, Freydoon Shahidi, Peter Sarnak, Rainer Schulze-Pillot, Joseph Shalika, David Soudry, Ramin Takloo-Bigash, Yuri Tschinkel, Emmanuel Ullmo, Marie-France Vignéras, Jean-Loup Waldspurger.

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If ρ = ρ0 , the above expression reduces to the identity N (B) = N (b)(n − 1). Reversing the steps and using the uniqueness of the Fourier expansion proves the lemma. 4. Let V be a quadratic extension field of . Let A denote the orthogonal group of the quadratic form N V / . Then the image of the group algebra, L 1 ( A), in the commuting algebra C of the representation D( , V ) is dense. The kernel of the homomorphism L 1 ( A) −→ C is one-dimensional. ) Proof. The proof will be divided into two parts: (A) Decomposition of D( , V ) into irreducible components (B) Classification of those components according to unitary equivalence ˆ 1 , let (L ρ h)(x) = 1 ρ −1 ( )h( x) d .

10 has K-spectrum 1≤m≤n n µ(2m, 2n); 9. cls December 6, 2003 10:10 theta-10 43 10. 10 is the direct summand of Aq(0) containing µ(2, 2); equivalently it is the tempered summand of this representation; 11. 10 is tempered, has infinitesimal character χ (1, 0) and wave-front set O [17]. 10 is unitary, tempered, and has Gelfand-Kirillov dimension 2 (cf. [35]). 2. There is an outer automorphism of Sp(4, R) that takes 10 to its contragredient ∗10 . There is no intrinsic way to choose one of these two representations.

Also consider the symplectic similitude group G Sp(2n). This has center Z G m and the adjoint group G Sp(2n)/Z is isomorphic to P Sp(2n). Furthermore the set of real points of the adjoint group is isomorphic to G Sp(2n, R)/Z (R). So the notation P G Sp(2n, R) is unambiguous, and perhaps for this reason some authors prefer to use this group. Simliar remarks hold over any local field. 4. Θ10 . We give a number of descriptions of see Section 3. 10 . 1. There is a unique irreducible representation 10 of Sp(4, R) satisfying the following equivalent conditions: c 1.