## 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.

### MSC:

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

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