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NOTE ON A PAPER OF J. HOFFSTEIN

Published online by Cambridge University Press:  09 August 2007

S. J. PATTERSON*
Affiliation:
Mathematisches Institut, Bunsenstr. 3–5, 37073 Göttingen, Germany e-mail: sjp@uni-math.gwdg.de
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Abstract

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The concept of a metaplectic form was introduced about 40 years ago by T. Kubota. He showed how Jacobi-Legendre symbols of arbitrary order give rise to characters of arithmetic groups. Metaplectic forms are the automorphic forms with these characters. Kubota also showed how higher analogues of the classical theta functions could be constructed using Selberg's theory of Eisenstein series. Unfortunately many aspects of these generalized theta series are still unknown, for example, their Fourier coefficients. The analogues in the case of function fields over finite fields can in principle be calculated explicitly and this was done first by J. Hoffstein in the case of a rational function field. Here we shall return to his calculations and clarify a number of aspects of them, some of which are important for recent developments.

Keywords

Type
Research Article
Copyright
Copyright © Glasgow Mathematical Journal Trust 2007

References

REFERENCES

1. Anderson, G. W., The evaluation of Selberg sums, Comptes Rendus Acad. Sci. Paris, Ser. I, 311 (1990), 469472.Google Scholar
2. Bass, H., Algebraic K-theory (Benjamin, 1968).Google Scholar
3. Brubaker, B., Bump, D., Chinta, G., Friedberg, S. and Hoffstein, J., Weyl group multiple Dirichlet series I, to appear in The Proceedings of the workshop on multiple Dirichlet series, PSPUM Series (Amer. Math. Soc.)Google Scholar
4. Davenport, H. and Hasse, H., Die, Nullstellen der Kongruenzzetafunktionen in gewissen zyklischen Fällen, J. Reine Angew. Math. 172 (1934), 151182.Google Scholar
5. Eckhardt, C. and Patterson, S. J., On the Fourier coefficients of biquadratic theta series, Proc. London Math. Soc. (3) 64 (1992), 225264.CrossRefGoogle Scholar
6. Denef, J. and Loeser, F., Détermination géométrique des sommes de Selberg-Evans, Bull. Soc. Math. France 122 (1994), 533551.Google Scholar
7. Denef, J. and Loeser, F., Character sums associated to finite Coxeter groups, Trans. Amer. Math. Soc. 350 (1998), 50475066.Google Scholar
8. Evans, R., The evaluation of Selberg character sums, L'Enseign. Math. 37 (1991), 235248.Google Scholar
9. Hoffstein, J., Theta functions on the n-fold metaplectic cover of SL(2) – the function field case, Inv. Math. 107 (1992), 6186.CrossRefGoogle Scholar
10. Kazhdan, D. A. and Patterson, S. J., Metaplectic forms, Publ. Math. IHES 59 (1984), 35142.CrossRefGoogle Scholar
11. Patterson, S. J., A cubic analogue of the theta series, J. Reine Angew. Math. 296 (1977), 125161, 217–220.Google Scholar
12. Roquette, P., Class field theory in characteristic p, its origins and development, Class field theory – Its centenary and prospect, Advanced Studies in Pure Mathematics 30 (2001), 549631.CrossRefGoogle Scholar
13. Schmidt, F. K., Zur, Zahlentheorie in Körpern von Charakteristik p. Vorläufige Mitteilung, Sitz.-Ber. Phys. Med. Soz. Erlangen 58 59 (1926/1927), 159172.Google Scholar
14. Waerden, B. L. van der, Modern algebra, Vol. 1 (Blum, Trans. F.) (Fredrick Unger Publishing Co., New York, 1949).Google Scholar
15. Wamelen, P. B. van, Proof of the Evans-Root conjectures for Selberg character sums, J. London Math. Soc., (2) 48 (1993), 415426.CrossRefGoogle Scholar
16. Wellhausen, G., Fourierkoeffizienten von Thetafunktionen sechster Ordnung, Dissertation (Göttingen, 1996).Google Scholar