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Some Remarks on Eisenstein Series for Metaplectic Coverings

Published online by Cambridge University Press:  20 November 2018

Lawrence Morris
Lawrence Morris*
Affiliation:
Clark University, Worcester, Massachusetts
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In recent years the harmonic analysis of n-fold (n > 2) metaplectic coverings of GL 2 has played an increasingly important role in certain aspects of algebraic number theory. In large part this has been inspired by the pioneering work of Kubota (see [3] for example); as an application one could cite the solution by Heath-Brown and Patterson [3] to a question of Kummer's on the distribution of the arguments of cubic Gauss sums. In that paper, Eisenstein series on the 3-fold metaplectic cover of GL 2(A) play a crucial role.

The object of this note is to point out that the theory of Eisenstein series can be made to work for a wide class of finite central coverings. Indeed, once the assumptions are made, the usual theory carries over readily, and one obtains a spectral decomposition of the appropriate L 2-space of functions; this is done in Section 2 of this paper.

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 35 , Issue 6 , 01 December 1983 , pp. 974 - 985
Copyright
Copyright © Canadian Mathematical Society 1983

References

1. Arthur, J. G., Eisenstein series and the trace formula, Proc. Symp. Pure Math. 33 (Amer. Math. Soc, Providence, R.I., 1979), 253–274.Google Scholar
2. Borel, A. and Jacquet, H., Automorphic forms and automorphic representations, ibid, 189–202.Google Scholar
3. Flicker, Y., Inv. Math. 57 (1980).CrossRefGoogle Scholar
4. Gelbart, S. and Jacquet, H., Ann. Scient. Ec. Norm. Sup. 4e serie, t. 11 (1978).Google Scholar
5. Gelbart, S. and Sally, P., Intertwining operators and automorphic forms for the metaplectic group, Proc. Nat. Acad. Sci., USA 72 (1975).Google Scholar
6. Heath-Brown, D. R. and Patterson, S. J., The distribution of Kummer sums at prime arguments, Crelle 310 (1979), 111–130.Google Scholar
7. Kazhdan, D. and Patterson, S., Metaplectic forms (1981) preprint.Google Scholar
8. Langlands, R. P., On the functional equations satisfied by Eisenstein series, Springer Lecture Notes 544 (1976).CrossRefGoogle Scholar
9. Moore, C. C., Group extensions of p-adic and adelic linear groups, Publ. Math. I. H. E. S. 35 (1968).Google Scholar
10. Morris, L. E., Eisenstein series for reductive groups over global function fields Part I: The cusp form case, Can. J. Math. 34 (1982), 91–168; Part II, ibid, 1112–1182.Google Scholar
11. Steinberg, R., Générateurs, relations et revêtements de groupes algébriques, Colloque sur la Théorie des Groupes Algébriques. Bruxelles (1962), 113–127.Google Scholar
12. Tits, J., Reductive groups over local fields, Proc. Symp. Pure Math. 33 (Amer. Math. Soc., Providence R.I., 1979), 29–69.Google Scholar