Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

Remote Access
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)


On modular functions in characteristic $ p$

Author: Wen Ch’ing Winnie Li
Journal: Trans. Amer. Math. Soc. 246 (1978), 231-259
MSC: Primary 10D05
MathSciNet review: 515538
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ k\, = \,{{\textbf{F}}_q}\left( T \right)$ be a function field of one variable over a finite field $ {{\textbf{F}}_q}$. For a nonzero polynomial $ A\, \in \,{{\textbf{F}}_q}\left[ T \right]$ one can define the modular group $ \Gamma \left( A \right)$. In this paper, we continue a theme introduced by Weil, and study the $ \lambda $-harmonic modular functions for $ \Gamma \left( A \right)$. The main purpose of this paper is to give a natural definition of $ \lambda $-harmonic Eisenstein series for $ \Gamma \left( A \right)$ so that we obtain a decomposition theory of $ \lambda $-harmonic modular functions, analogous to the classical results of Hecke. That is, we prove

$\displaystyle {\text{Modular}}\,{\text{Function}}\,{\text{ = }}\,{\text{Eisenstein}}\,{\text{series}}\, \oplus \,{\text{Cusp Functions}}{\text{.}}$

Moreover, the dimension of the space generated by $ \lambda $-harmonic Eisenstein series for $ \Gamma \left( A \right)$ is equal to the number of cusps of $ \Gamma \left( A \right)$, and so is independent of $ \lambda $.

For the definition of $ \lambda $-harmonic Eisenstein series and the proof of decomposition theory, we consider two cases: (i) $ \lambda \, \ne \, \pm 2\sqrt q $ and (ii) $ \lambda \, = \, \pm 2\sqrt q $, separately. Case (i) is treated in the usual way. Case (ii), being a ``degenerate'' case, is more interesting and requires more complicated analysis.

References [Enhancements On Off] (What's this?)

  • [1] P. Cartier, Fonctions harmoniques sur un arbre, Symposia Mathematica, Vol. IX (Convegno di Calcolo delle Probabilità, INDAM, Rome, 1971) Academic Press, London, 1972, pp. 203–270 (French). MR 0353467 (50 #5950)
  • [2] P. Deligne, Formes modulaires et représentations l-adiques, Séminaire Bourbaki, 1968/1969, Exp. 355, Lecture Notes in Math., vol. 179, Springer-Verlag, Berlin, 1971, pp. 139-172.
  • [3] P. Deligne, Les constantes des équations fonctionnelles des fonctions 𝐿, Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Springer, Berlin, 1973, pp. 501–597. Lecture Notes in Math., Vol. 349 (French). MR 0349635 (50 #2128)
  • [4] Pierre Deligne and Jean-Pierre Serre, Formes modulaires de poids 1, Ann. Sci. École Norm. Sup. (4) 7 (1974), 507–530 (1975) (French). MR 0379379 (52 #284)
  • [5] G. Harder, Chevalley groups over function fields and automorphic forms, Ann. of Math. (2) 100 (1974), 249–306. MR 0563090 (58 #27799)
  • [6] Erich Hecke, Mathematische Werke, Herausgegeben im Auftrage der Akademie der Wissenschaften zu Göttingen, Vandenhoeck & Ruprecht, Göttingen, 1959 (German). MR 0104550 (21 #3303)
  • [7] Tomio Kubota, Elementary theory of Eisenstein series, Kodansha Ltd., Tokyo; Halsted Press [John Wiley & Sons], New York-London-Sydney, 1973. MR 0429749 (55 #2759)
  • [8] Serge Lang, Algebraic number theory, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London-Don Mills, Ont., 1970. MR 0282947 (44 #181)
  • [9] Wen Ch’ing Winnie Li, Newforms and functional equations, Math. Ann. 212 (1975), 285–315. MR 0369263 (51 #5498)
  • [10] Hans Maass, Lectures on modular functions of one complex variable, Notes by Sunder Lal. Tata Institute of Fundamental Research Lectures on Mathematics, No. 29, Tata Institute of Fundamental Research, Bombay, 1964. MR 0218305 (36 #1392)
  • [11] J.-P. Serre, Abres amalgames et $ {\text{S}}{{\text{L}}_2}$, Collège de France 1968/1969 (note polycopiées, rédigrées avec la collaboration de H. Bass).
  • [12] André Weil, Basic number theory, Die Grundlehren der mathematischen Wissenschaften, Band 144, Springer-Verlag New York, Inc., New York, 1967. MR 0234930 (38 #3244)
  • [13] André Weil, On the analogue of the modular group in characteristic 𝑝, Functional Analysis and Related Fields (Proc. Conf. for M. Stone, Univ. Chicago, Chicago, Ill., 1968) Springer, New York, 1970, pp. 211–223. MR 0271030 (42 #5913)
  • [14] -, Dirichlet series and automorphic forms, Lecture Notes in Math., vol. 189, Springer-Verlag, Berlin and New York, 1971.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 10D05

Retrieve articles in all journals with MSC: 10D05

Additional Information

PII: S 0002-9947(1978)0515538-9
Article copyright: © Copyright 1978 American Mathematical Society

Comments: Email Webmaster

© Copyright , American Mathematical Society
Contact Us · Sitemap · Privacy Statement

Connect with us Facebook Twitter Google+ LinkedIn Instagram RSS feeds Blogs YouTube Podcasts Wikipedia