## 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

- 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 - 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**
—, - 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**

*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.

## 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