P-adic Banach spaces and families of modular forms

Abstract

Let p be a prime, Cp the completion of an algebraic closure of the p-adic numbers Qp and K a nite extension of Qp contained in Cp. Let v be the valuation on Cp such that v(p) = 1 and let | | be the absolute value on Cp such that |x| = p−v(x) for x ∈ Cp. Suppose N is a positive integer prime to p. Let X1(Np) denote the modular curve over K which represents elliptic curves with 1(Np)-structure and let Up be the Hecke operator on modular forms on X1(Np) which takes a form with q-expansion ∑ n anq n to the modular form with q-expansion ∑ n anpq n. A modular form F is said to have slope ∈ Q if there is a polynomial R(T ) over Cp such that R(Up)F = 0 and such that the Newton polygon of R(T ) has only one side and its slope is − .