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Ulmer, Douglas L.
Slopes of modular forms and congruences. Annales de l'institut Fourier, 46 no. 1 (1996), p. 1-32
Full text djvu | pdf | Reviews MR 97i:11046a | Zbl 0834.11024
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Abstract

Our aim in this paper is to prove congruences between on the one hand certain eigenforms of level $pN$ and weight greater than 2 and on the other hand twists of eigenforms of level $pN$ and weight 2. One knows a priori that such congruences exist; the novelty here is that we determine the character of the form of weight 2 and the twist in terms of the slope of the higher weight form, i.e., in terms of the valuation of its eigenvalue for $U_p$. Curiously, we also find a relation between the leading terms of the $p$-adic expansions of the eigenvalues for $U_p$ of the two forms. This allows us to determine the restriction to the decomposition group at $p$ of the Galois representation modulo $p$ attached to the higher weight form.

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