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Hecke Algebras and Automorphic Forms

Published online by Cambridge University Press:  04 December 2007

Joshua Lansky  and
David Pollack
Joshua Lansky
Affiliation:
Department of Mathematics, University of Rochester, Rochester, NY 14627, U.S.A. E-mail: lansky@math.rochester.edu
David Pollack
Affiliation:
Department of Mathematics, Ohio State University, Columbus, OH 43210, U.S.A. E-mail: pollack@math.ohio-state.edu
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Abstract

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The goal of this paper is to carry out some explicit calculations of the actions of Hecke operators on spaces of algebraic modular forms on certain simple groups. In order to do this, we give the coset decomposition for the supports of these operators. We present the results of our calculations along with interpretations concerning the lifting of forms. The data we have obtained is of interest both from the point of view of number theory and of representation theory. For example, our data, together with a conjecture of Gross, predicts the existence of a Galois extension of Q with Galois group G2(F5) which is ramified only at the prime 5. We also provide evidence of the existence of the symmetric cube lifting from PGL2 to PGSp4.

Keywords

automorphic formsmodular formsHecke algebrasp-adic groups
Type
Research Article
Information
Compositio Mathematica , Volume 130 , Issue 1 , January 2002 , pp. 21 - 48
Copyright
© 2002 Kluwer Academic Publishers