Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-19T05:59:51.109Z Has data issue: false hasContentIssue false

Theta Series, Eisenstein Series and Poincaré Series over Function Fields

Published online by Cambridge University Press:  20 November 2018

Ambrus Pál*
Affiliation:
Department of Mathematics and Statistics, McGill University, 805 Sherbrook St. West, Montréal, Quebec, H3A 2K6 e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We construct analogues of theta series, Eisenstein series and Poincaré series for function fields of one variable over finite fields, and prove their basic properties.

Keywords

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2004

References

[1] Casselman, W., On Some Results of Atkin and Lehmer. Math. Ann. 201(1973), 301314.Google Scholar
[2] Curtis, C. W. and Reiner, I., Methods of Representation Theory I. John Wiley & Sons, Inc., New York, 1981.Google Scholar
[3] Deligne, P., Les constantes des équations fonctionnelles des fonctions L. Springer Lecture Notes in Math. 349(1973), 501597.Google Scholar
[4] Gekeler, E.-U. and Reversat, M., Jacobians of Drinfeld Modular Curves. J. Reine Angew. Math. 476(1996), 2793.Google Scholar
[5] Gelbart, S. S., Automorphic Forms on Adele Groups. Princeton University Press, Princeton, 1975.Google Scholar
[6] Godement, R. and Jacquet, H., Zeta Functions of Simple Algebras. Springer, Berlin-Heidelberg-New York, 1972.Google Scholar
[7] Gross, B. H., Heights and the Special Values of L-series. In: Number Theory (Montreal, Que., 1985), Amer. Math. Soc., Providence, 1987, 115187.Google Scholar
[8] Gross, B. H. and Zagier, D., Heegner Points and Derivatives of L-series. Invent. Math. 84(1986), 225320.Google Scholar
[9] Harder, G., Chevalley Groups over Function Fields and Automorphic Forms. Ann. Math. 100(1974), 249306.Google Scholar
[10] Harder, G., Li, W. C. W. and Weisinger, J. R., Dimensions of Spaces of Cusp Forms over Function Fields. J. Reine Angew. Math. 319(1980), 73103.Google Scholar
[11] Jacquet, H. and Langlands, R. P., Automorphic Forms on GL(2). Springer, Berlin-Heidelberg-New York, 1970.Google Scholar
[12] Jacquet, H., Piatevski-Shapiro, I. I. and Shalika, J., Conducteur des représentations du group linéare. Math. Ann. 256(1981), 199214.Google Scholar
[13] Lang, S., Algebraic Number Theory. Addison-Wesley Publishing, Inc., Reading-London, 1970.Google Scholar
[14] Li, W. C. W., Eisenstein Series and Decomposition Theory over Function Fields. Math. Ann. 240(1979), 115139.Google Scholar
[15] Rück, H.-G., Theta Series of Imaginary Quadratic Function Fields. Manuscripta Math. 88(1995), 385407.Google Scholar
[16] Rück, H.-G., Poincaré Series of Drinfeld Type. Arch. Math. 68(1997), 190201.Google Scholar
[17] Rück, H.-G., L-series of Automorphic Cusp Forms of Drinfeld Type. In: Drinfeld Modules, Modular Schemes and Applications (Alden-Biesen, 1996), World Sci. Publishing, River Edge, NJ, 1997, 311329.Google Scholar
[18] Rück, H.-G. and Tipp, U., Heegner Points and L-series of Automorphic Cusp Forms of Drinfeld Type. Doc. Math. 5(2000), 365444.Google Scholar
[19] Serre, J.-P., Corps Locaux. Hermann, Paris, 1971.Google Scholar
[20] Serre, J.-P., Arbres, Amalgames, SL2. Société Mathématique de France, Paris, 1977.Google Scholar
[21] Weil, A., On Some Exponential Sums. Collected Papers 1(1979), 386389.Google Scholar
[22] Weil, A., Dirichlet Series and Automorphic Forms. Springer, Berlin-Heidelberg-New York, 1971.Google Scholar
[23] Weil, A., Basic Number Theory. Springer, Berlin-Heidelberg-New York, 1973.Google Scholar
[24] Williams, K. S., The Kloosterman Sum Revisited. Canad. Math. Bull. 16(1973), 363365.Google Scholar
[25] Zhang, S.-W., Heights of Heegner Points on Shimura Curves. Ann. Math. 153(2001), 27147.Google Scholar