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

In Contributions to Automorphic kinds, Geometry, and quantity idea, Haruzo Hida, Dinakar Ramakrishnan, and Freydoon Shahidi assemble a individual workforce of specialists to discover automorphic types, largely through the linked L-functions, illustration concept, and geometry. simply because those subject matters are on the leading edge of a valuable sector of contemporary arithmetic, and are with regards to the philosophical base of Wiles' evidence of Fermat's final theorem, this booklet might be of curiosity to operating mathematicians and scholars alike. by no means formerly released, the contributions to this quantity reveal the reader to a bunch of inauspicious and thought-provoking problems.Each of the intense and remarkable mathematicians during this quantity makes a different contribution to a box that's at present seeing explosive development. New and robust effects are being proved, appreciably and regularly altering the field's make up. Contributions to Automorphic varieties, Geometry, and quantity concept will most likely result in important interplay between researchers and in addition aid organize scholars and different younger mathematicians to go into this fascinating 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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Extra resources for Contributions to automorphic forms, geometry, and number theory: in honor of Joseph Shalika

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For (10) we compute the K-spectrum of Aq(0) by the Blattner formula. We see it has K-types µ(1, 1), µ(2, 2), µ(3, 1), . . , and as in (8) any constituent of this representation has a highest weight. Considering infinitesimal characters as in (2) we immediately see Aq(0) has two constituents, with highest weights (1, 1) and (2, 2). By (2) the term with highest weight (1, 1) is not tempered. On the other hand Aq(0) is completely reducible. This proves (10) is equivalent to (1). For (11) 10 has the indicated wave-front set, and any representation with this wave-front set is a highest weight module.

Gelfand and M. I. Graev, The group of matrices of second order with coefficients in a locally compact fields, Uspehi Mati. Nauk, 1963, pp. 29–99. I. M. Gelfand, M. I. Graev and I. I. Pyatetskii-Shapiro, Representation theory and automorphic functions, Translated from the Russian by K. A. Hirsch, W. B. , 1969. cls December 10, 2003 7:5 joseph a. shalika Harish-Chandra, Harmonic analysis on reductive p-adic groups, Springer-Verlag, Berlin, 1970, Notes by G. van Dijk, Lecture Notes in Mathematics, vol.

97 (1926), 210–242. R. Howe, θ -series and invariant theory Automorphic forms, representations and L-functions (Proc. Sympos. , 1977), Part 1, Proc. Sympos. , XXXIII, 275–285, Amer. Math. , 1979. H. D. Kloosterman, The behaviour of general theta functions under the modular group and the characters of binary modular congruence groups, i–ii, Annals of Mathematics, 47 (1949), 317–447. ¨ H. Maass, Uber eine neue Art von nicht analytischen automorphen Funktionen und die bestimmung von dirichletscher reihen durch funktional gleichunger, Math.

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