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Slopes of modular forms. (English) Zbl 0853.11037
Childress, Nancy (ed.) et al., Arithmetic geometry. Conference on arithmetic geometry with an emphasis on Iwasawa theory, March 15-18, 1993, Arizona State Univ., Tempe, AZ, USA. Providence, RI: American Mathematical Society. Contemp. Math. 174, 167-183 (1994).
The author studies \(p\)-adic valuations of eigenvalues of Hecke operators on modular forms on \(\Gamma_1 (pN)\) with the aim to develop a program and a strategy to prove that the number of normalized Hecke eigenforms of weight \(w\geq 2\) and level \(pN\) whose eigenvalues for the usual Hecke operator \(U_p\) have a fixed \(p\)-adic valuation \(\lambda\) is bounded independently of \(w\). Such an assertion is known for \(\lambda =0\), cf. chapter 7 of H. Hida [Elementary theory of \(L\)-functions and Eisenstein series, Cambridge University Press, Cambridge (1993)] and in the general case has been conjectured by F. Gouvêa and B. Mazur [Math. Comput. 58, 793-805 (1992; Zbl 0773.11030)]: in fact, their conjecture predicts a simple periodicity property (as a function of the weight) of the number of eigenforms with a given \(\lambda\).
The present paper is a continuation of the author’s previous work [Ann. Sci. Éc. Norm. Supér., IV. Sér. 28, No. 2, 129-160 (1995; Zbl 0827.11024)].
For the entire collection see [Zbl 0802.00017].

11F33 Congruences for modular and \(p\)-adic modular forms
11F85 \(p\)-adic theory, local fields
11F30 Fourier coefficients of automorphic forms
11G18 Arithmetic aspects of modular and Shimura varieties