A study of the local components of the Hecke algebra mod

Author:
Naomi Jochnowitz

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

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Abstract: We use information about modular forms 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 . These results suggest that the local components of the Hecke ring are more complex than originally expected. We also investigate the inverse limits of the Hecke rings of weight as varies within a fixed congruence class .

As an immediate corollary to some of the above results, we show that when is sufficiently large, an arbitrary prime must divide the index of the classical Hecke ring in the ring of integers of .

**[1]**Naomi Jochnowitz,*The index of the Hecke ring, 𝑇_{𝑘}, in the ring of integers of 𝑇_{𝑘}⊗𝑄*, Duke Math. J.**46**(1979), no. 4, 861–869. MR**552529****[2]**Naomi Jochnowitz,*Congruences between systems of eigenvalues of modular forms*, Trans. Amer. Math. Soc.**270**(1982), no. 1, 269–285. MR**642341**, https://doi.org/10.1090/S0002-9947-1982-0642341-2**[3]**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) Springer, Berlin, 1973, pp. 69–190. Lecture Notes in Mathematics, Vol. 350. MR**0447119****[4]**-,*A result on modular forms in characteristic*, Lecture Notes in Math., vol. 601, Springer-Verlag, Berlin and New York, 1976, pp. 53-56.**[5]**V. Miller,*Diophantine and*-*adic analysis of elliptic curves and modular forms*, Ph. D. Thesis, Harvard, June, 1975.**[6]**Jean-Pierre Serre,*Congruences et formes modulaires [d’après H. P. F. Swinnerton-Dyer]*, Séminaire Bourbaki, 24e année (1971/1972), Exp. No. 416, Springer, Berlin, 1973, pp. 319–338. Lecture Notes in Math., Vol. 317 (French). MR**0466020****[7]**Jean-Pierre Serre,*Formes modulaires et fonctions zêta 𝑝-adiques*, Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, 1972) Springer, Berlin, 1973, pp. 191–268. Lecture Notes in Math., Vol. 350 (French). MR**0404145****[8]**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, 1972) Springer, Berlin, 1973, pp. 1–55. Lecture Notes in Math., Vol. 350. MR**0406931**

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DOI:
https://doi.org/10.1090/S0002-9947-1982-0642340-0

Article copyright:
© Copyright 1982
American Mathematical Society