# 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.