Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Congruences between systems of eigenvalues of modular forms
HTML articles powered by AMS MathViewer

by Naomi Jochnowitz PDF
Trans. Amer. Math. Soc. 270 (1982), 269-285 Request permission

Abstract:

We modify and generalize proofs of Tate and Serre in order to show that there are only a finite number of systems of eigenvalues for the Hecke operators with respect to ${\Gamma _0}(N)\bmod l$. We also summarize results for ${\Gamma _1}(N)$. Using these results, we show that an arbitrary prime divides the discriminant of the classical Hecke ring to a power which grows linearly with $k$. In this way, we find a lower bound for the discriminant of the Hecke ring. After limiting ourselves to cusp forms, we also find an upper bound. Lastly we use the constructive nature of Tate and Serre’s result to describe the structure and dimensions of the generalized eigenspaces for the Hecke operators $\bmod l$.
References
    N. Bourbaki, Algèbre, Chap. 8.
  • Pierre Deligne, La conjecture de Weil. I, Inst. Hautes Études Sci. Publ. Math. 43 (1974), 273–307 (French). MR 340258
  • M. Eichler, The basis problem for modular forms and the traces of the Hecke operators, Modular functions of one variable, I (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Lecture Notes in Math., Vol. 320, Springer, Berlin, 1973, pp. 75–151. MR 0485698
  • Naomi Jochnowitz, The index of the Hecke ring, $T_{k}$, in the ring of integers of $T_{k}\otimes \textbf {Q}$, Duke Math. J. 46 (1979), no. 4, 861–869. MR 552529
  • Naomi Jochnowitz, A study of the local components of the Hecke algebra mod $l$, Trans. Amer. Math. Soc. 270 (1982), no. 1, 253–267. MR 642340, DOI 10.1090/S0002-9947-1982-0642340-0
  • —, Congruences between systems of eigenvalues and implications for the Hecke algebra, Harvard Ph.D. Thesis, November, 1976.
  • Nicholas M. Katz, $p$-adic properties of modular schemes and modular forms, Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Lecture Notes in Math., Vol. 350, Springer, Berlin, 1973, pp. 69–190. MR 0447119
  • —, A result on modular forms in characteristic $p$, Lecture Notes in Math., vol. 601, Springer-Verlag, Berlin and New York, 1976, pp. 53-61.
  • Peter George Kluit, Hecke operators on $\Gamma ^{\ast } (N)$ and their traces, Vrije Universiteit te Amsterdam, Amsterdam, 1979. With a computer-aided study of low genus $X^{\ast } (N)$; Dissertation, Vrije Universiteit, Amsterdam, 1979; With a Dutch summary. MR 591854
  • Serge Lang, Introduction to modular forms, Grundlehren der Mathematischen Wissenschaften, No. 222, Springer-Verlag, Berlin-New York, 1976. MR 0429740
  • V. Miller, Diophantine and $p$-adic analysis of elliptic curves and modular forms, Ph.D. Thesis, Harvard, June 1975.
  • Goro Shimura, Introduction to the arithmetic theory of automorphic functions, Kanô Memorial Lectures, No. 1, Iwanami Shoten Publishers, Tokyo; Princeton University Press, Princeton, N.J., 1971. Publications of the Mathematical Society of Japan, No. 11. MR 0314766
  • Jean-Pierre Serre, Congruences et formes modulaires [d’après H. P. F. Swinnerton-Dyer], Séminaire Bourbaki, 24e année (1971/1972), Exp. No. 416, Lecture Notes in Math., Vol. 317, Springer, Berlin, 1973, pp. 319–338 (French). MR 0466020
  • Jean-Pierre Serre, Formes modulaires et fonctions zêta $p$-adiques, Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Lecture Notes in Math., Vol. 350, Springer, Berlin, 1973, pp. 191–268 (French). MR 0404145
  • H. P. F. Swinnerton-Dyer, On $l$-adic representations and congruences for coefficients of modular forms, Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Lecture Notes in Math., Vol. 350, Springer, Berlin, 1973, pp. 1–55. MR 0406931
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC: 10D12
  • Retrieve articles in all journals with MSC: 10D12
Additional Information
  • © Copyright 1982 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 270 (1982), 269-285
  • MSC: Primary 10D12
  • DOI: https://doi.org/10.1090/S0002-9947-1982-0642341-2
  • MathSciNet review: 642341