ag.algebraic geometry – Can we attach (formal) abelian varieties to $p$-adic modular forms?

The Jacobian of the modular curve $X_1(N)$ over $mathbb Q$ $J_1(N)$ can be decomposed up to isogeny, as a product of abelian subvarieties $A_f$ corresponding to Galois conjugacy classes of Hecke cuspforms $f$ of level $N$ of weight $2$. $A_f$ is the abelian variety of dimension $(E_f:mathbb Q)$ associated to $f$ by decomposition of the Hecke algebra, where $E_f$ is the coefficient field of $f$.

Can we do similar things for (overconvergent) $p$-adic modular forms of classical weight $2$ by a limiting process? Maybe the dimension will be larger and larger, but can we at least define things over $mathbb Q_p$? At least can we construct a local Galois representation?

This is really a soft question, thank you very much.