Families of modular eigenforms. (English) Zbl 0773.11030

Let \(p\) be a prime number and \(N\) an integer prime to \(p\). Suppose that \(f\) is a modular form on \(\Gamma_ 0(pN)\), of some weight \(k\), with coefficients in \(\overline\mathbb{Q}_ p\) which is a cuspidal eigenform for all Hecke operators \(T_ l\) for \(l\nmid pN\) and \(U_ l\) for \(l\mid P^ N\). Then the slope of \(f\) is defined to be valuation of the eigenvalue of \(f\) for \(U_ p\). Precise conjectures are formulated concerning the existenc of eigenforms \(f'\) of certain weights \(k'\) that are congruent to \(f\) modulo a high power of \(p\), if \(f\) has finite slope and \(p\geq 5\). The case of slope zero is part of Hida’s theory of deformations of ordinary \(p\)-adic Galois representations. The analogue for noncuspidal \(f\) is the theory of congruences between Eisenstein series. The conjectures are checked in a number of cases by computer computations. A forthcoming article is announced in which the authors hope to prove a weakened version of the conjectures.


11F33 Congruences for modular and \(p\)-adic modular forms
11Y35 Analytic computations
14G20 Local ground fields in algebraic geometry


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