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Zbl 0686.10022
Diamond, Fred
On congruence modules associated to $\Lambda$-adic forms. (English)
[J] Compos. Math. 71, No.1, 49-83 (1989). ISSN 0010-437X; ISSN 1570-5846/e

Let $\Lambda ={\bbfZ}\sb p[[T]]$. The author studies the existence of congruent ordinary $\Lambda$-adic modular forms. Using ideas of Hida and Ribet the author is able to show that if $f=\sum a\sb nq\sp n$ is a p-stabilized newform in $S\sb k(\Gamma\sb 0(Np\sp r,\chi):{\bar {\bbfQ}}\sb p)$ with $a\sp 2\sb{\ell}-\chi (\ell)\ell\sp{k- 2}(\ell +1)\sp 2\in {\cal M}$ then there is a p-stabilized newform $g\in S\sb k(\Gamma\sb 0(Np\sp r\ell,\chi):{\bar {\bbfQ}}\sb p)$ such that $f\equiv g (mod {\cal M})$. This extends results of {\it K. A. Ribet} [Proc. Int. Congr. Math., Warszawa 1983, 503-514 (1984; Zbl 0575.10024)] for modular forms on $\Gamma\sb 0(N)$.
[S.Kamienny]

MSC 2000:: *11F33 Congruences for (p-adic) modular forms
11F11 Modular forms, one variable

Keywords: ordinary modular form; congruent newforms

Citations: Zbl 0575.10024

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