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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.

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    $\begingroup$ There is certainly a Galois representation associated to an overconvergent modular form. It is even a global Galois representation - there is no need to make it local (why would there be?). If the modular form is not classical then the Galois representation usually doesn't look like the Galois representation coming from an abelian variety - i.e. it is usually not potentially semistable. See Theorem 6.6 of Kisin's paper Overconvergent Modular Forms and the Fontaine-Mazur conjecture for a precise statement. $\endgroup$ – Will Sawin yesterday
  • $\begingroup$ @WillSawin Thank you! The local story is for formal abelian varieties over $\mathbb Q_p$, their $H^1$ (base change to algebraic closure) still give local Galois representations. $\endgroup$ – loos 19 hours ago
  • $\begingroup$ As Will says, you seem to have the wrong idea about what object to expect. Formal ab vars will give you Galois reps which are Barsotti--Tate at p (i.e. HTW are all 0 or 1). You seem to be looking for something which is Barsotti--Tate, but not global; while overconvergent eigenforms actually give you things which are global, but not Barsotti--Tate. $\endgroup$ – David Loeffler 16 hours ago
  • $\begingroup$ @DavidLoeffler I see, thanks. How to construct the global Galois rep without using abelian variety? And do we have some classical result for the eigenform if we know the component at p is from a (formal) abelian variety? $\endgroup$ – loos 9 hours ago
  • $\begingroup$ Most Galois representations do not come from abelian varieties (algebraic or formal). You should read the Coleman--Mazur "Eigencurve" paper, and if you get through that, go on to Kisin's "Overconvergent modular forms and the Fontaine--Mazur conjecture". $\endgroup$ – David Loeffler 2 hours ago

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