Contributions to Automorphic Forms, Geometry, and Number by Haruzo Hida, Dinakar Ramakrishnan, Freydoon Shahidi

Number Theory

By Haruzo Hida, Dinakar Ramakrishnan, Freydoon Shahidi

In Contributions to Automorphic kinds, Geometry, and quantity Theory, Haruzo Hida, Dinakar Ramakrishnan, and Freydoon Shahidi compile a uncommon team of specialists to discover automorphic varieties, largely through the linked L-functions, illustration thought, and geometry. simply because those topics are on the innovative of a vital quarter of contemporary arithmetic, and are regarding the philosophical base of Wiles' facts of Fermat's final theorem, this booklet could be of curiosity to operating mathematicians and scholars alike. by no means formerly released, the contributions to this quantity reveal the reader to a number of tricky and thought-provoking difficulties.

Each of the intense and remarkable mathematicians during this quantity makes a special contribution to a box that's presently seeing explosive development. New and robust effects are being proved, considerably and consistently altering the field's make up. Contributions to Automorphic types, Geometry, and quantity Theory will most likely result in very important interplay between researchers and likewise support arrange scholars and different younger mathematicians to go into this fascinating zone of natural arithmetic.

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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Extra resources for Contributions to Automorphic Forms, Geometry, and Number Theory: A Volume in Honor of Joseph Shalika

Example text

Let V be a four-dimensional real vector space with a non-degenerate symplectic form < , >. Let G = Sp(V ) = Sp(V , < , >) be the isometry group of < , >. Choose a basis of V such that < , > is given by J = ( −I0 2 I02 ). Then Sp(V ) is isomorphic to G = Sp(4, R), the group of matrices g satisfying g J tg = J . Thus B ) where G consists of the matrices ( CA D A tB = t( A tB) C tD = t(C tD) A tD − B tC = I2 . Let g0 be the Lie algebra of G. , X = ( CA −BtA ) where B and C are symmetric. We will make repeated use of the Cartan subgroups of G.

3 we have Nk (ηx ) = TX˜ ,n Nk . This completes the result for even n. For n odd, n = 2k + 1 we may write n ∈ Nk+1 uniquely in the form n = 1 + π k+1 (again, is taken mod pk ), Tr( ) ≡ 0 mod pk . As before, Nk+1 , gk and g∗k are isomorphic as K n -modules. In this case however for x ∈ gk , x primitive, X a pre-image in g, Nk+1 (ηx ) = TX˜ ,n Nk . 0 ∗ ), where λ ∈ Op , In what follows X will denote an element of g of the form ( λ0 −λ 0 ζ ∗ or one of the form ( ω˜ 0 ), where ζ ∈ Op and ω ˜ ∈ p. In particular X is primitive.

H. Rohrbach, Die character der bin¨aren konguence gruppen mod p 2 , Schriften d. math. Seminars d. Univ. Berlin 1 (1932), 33–94. J. Schur, Undtersuchungen u¨ ber die darstellung der endlichen gruppen durch gebrochene lineare substitution, J. Reine Angew. Math. 132 (1907), 85–137. H. Spies, Die darstellung der inhomogen modulargruppen mod q n durch die ganzen modulformen gerader dimension, Math. Ann. 111 (1935), 329–354. A. Weil, Sur certains groupes d’op´erateurs unitaires, Acta Math. 111 (1964), 143–211.

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