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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.

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A study of the local components of the Hecke algebra mod $l$
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by Naomi Jochnowitz PDF
Trans. Amer. Math. Soc. 270 (1982), 253-267 Request permission

Abstract:

We use information about modular forms $\bmod l$ to study the local structure of the Hecke ring. In particular, we find nontrivial lower bounds for the dimensions of the Zariski tangent spaces of the local components of the Hecke ring $\bmod l$. These results suggest that the local components of the Hecke ring $\bmod l$ are more complex than originally expected. We also investigate the inverse limits of the Hecke rings of weight $k\bmod l$ as $k$ varies within a fixed congruence class $\bmod l - 1$. As an immediate corollary to some of the above results, we show that when $k$ is sufficiently large, an arbitrary prime $l$ must divide the index of the classical Hecke ring ${{\mathbf {T}}_k}$ in the ring of integers of ${{\mathbf {T}}_k} \otimes {\mathbf {Q}}$.
References
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  • 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
  • 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. 53-56. V. Miller, Diophantine and $p$-adic analysis of elliptic curves and modular forms, Ph. D. Thesis, Harvard, June, 1975.
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  • 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
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Additional Information
  • © Copyright 1982 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 270 (1982), 253-267
  • MSC: Primary 10D12
  • DOI: https://doi.org/10.1090/S0002-9947-1982-0642340-0
  • MathSciNet review: 642340