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Published online by Cambridge University Press: 02 January 2014
Given a prime $p\gt 2$, an integer
$h\geq 0$, and a wide open disk
$U$ in the weight space
$ \mathcal{W} $ of
${\mathbf{GL} }_{2} $, we construct a Hecke–Galois-equivariant morphism
${ \Psi }_{U}^{(h)} $ from the space of analytic families of overconvergent modular
symbols over
$U$ with bounded slope
$\leq h$, to the corresponding space of analytic families of overconvergent
modular forms, all with
${ \mathbb{C} }_{p} $-coefficients. We show that there is a finite subset
$Z$ of
$U$ for which this morphism induces a
$p$-adic analytic family of isomorphisms relating overconvergent
modular symbols of weight
$k$ and slope
$\leq h$ to overconvergent modular forms of weight
$k+ 2$ and slope
$\leq h$.