Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

A simplified trace formula for Hecke operators for $ \Gamma\sb 0(N)$


Author: Shepley L. Ross
Journal: Trans. Amer. Math. Soc. 331 (1992), 425-447
MSC: Primary 11F11; Secondary 11F25, 11F72
MathSciNet review: 1053115
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ N$ and $ n$ be relatively prime positive integers, let $ \chi $ be a Dirichlet character modulo $ N$, and let $ k$ be a positive integer. Denote by $ {S_k}(N,\chi)$ the space of cusp forms on $ {\Gamma _0}(N)$ of weight $ k$ and character $ \chi $, a space denoted simply $ {S_k}(N)$ when $ \chi $ is the trivial character. Beginning with Hijikata's formula for the trace of $ {T_n}$ acting on $ {S_k}(N,\chi)$, we develop a formula which essentially reduces the computation of this trace to looking up values in a table. From this formula we develop very simple formulas for (1) the dimension of $ {S_k}(N,\chi)$ and (2) the trace of $ {T_n}$ acting on $ {S_k}(N)$.


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

  • [A-L] A. O. L. Atkin and J. Lehner, Hecke operators on Γ₀(𝑚), Math. Ann. 185 (1970), 134–160. MR 0268123
  • [Hij] Hiroaki Hijikata, Explicit formula of the traces of Hecke operators for Γ₀(𝑁), J. Math. Soc. Japan 26 (1974), 56–82. MR 0337783
  • [H-P-S$ _{1}$] H. Hijikata, A. Pizer, and T. Shemanske, The basis problem for modular forms on $ {\Gamma _0}(N)$, Mem. Amer. Math. Soc. No. 418, 1989.
  • [H-P-S$ _{2}$] -, Twists of newforms, preprint, 1988.
  • [Li] Wen Ch’ing Winnie Li, Newforms and functional equations, Math. Ann. 212 (1975), 285–315. MR 0369263
  • [Sha] Goro Shimura, Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, No. 11. Iwanami Shoten, Publishers, Tokyo; Princeton University Press, Princeton, N.J., 1971. Kan\cflex o Memorial Lectures, No. 1. MR 0314766

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 11F11, 11F25, 11F72

Retrieve articles in all journals with MSC: 11F11, 11F25, 11F72


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1992-1053115-1
Keywords: Cusp forms, Hecke operators, modular forms, trace formula
Article copyright: © Copyright 1992 American Mathematical Society