Let $\mathbf{f}$ be a Hida family of ordinary $p$-adic modular forms, and let $V(\mathbf{f})$ be the corresponding $\Lambda$-adic Galois representation (a quotient of the inverse limit
$$ \varprojlim\_r H^1\_{et}(X\_1(Np^r), \mathbf{Z}\_p) $$
with respect to the trace maps).


It follows from work of Wiles that if the residual representation of $\mathbf{f}$ is irreducible and $p$-distinguished, then $V(\mathbf{f})$ is free of rank 2 over the Hida--Hecke algebra $\Lambda\_{\mathbf{f}}$ and there is a canonical exact sequence
$$ 0 \to \mathscr{F}^+ V(\mathbf{f}) \to V(\mathbf{f}) \to \mathscr{F}^- V(\mathbf{f}) \to 0 \quad (\star)$$
where the filtration steps are free of rank 1 over $\Lambda\_{\mathbf{f}}$ and $\mathscr{F}^- V(\mathbf{f})$ is unramified.


Ohta has shown (Compos. Math. 115, 1999) that if the weight of $\mathbf{f}$ is not 2 modulo $p-1$, then there is a canonical basis of the module
$$ \mathbf{D}\_{\mathrm{dR}}(\mathscr{F}^- V(\mathbf{f})) := \left( \mathscr{F}^- V(\mathbf{f}) \mathop{\hat\otimes} \widehat{\mathbf{Z}}\_p^{\mathrm{nr}}\right)^{G\_{\mathbf{Q}\_p}}$$
which interpolates the images of the normalized differentials $\omega\_{f^\*}$ under the de Rham comparison isomorphisms for each classical specialization $f$ of $\mathbf{f}$. (Ohta also assumes the weight is not $1$ modulo $p-1$; but from his method is it clear that this can be weakened to assuming p-distinguishedness.)



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> (1) Can the assumption in Ohta's theorem that the weight is not 2 modulo $p-1$ be removed?
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> (2) The exact sequence $(\star)$ exists for arbitrary $\mathbf{f}$ (not necessarily p-distinguished) if we tensor with the field of fractions of the Hecke algebra, by a theorem of Hida and Wiles. Can Ohta's $p$-adic interpolation of periods be generalized to this setting?
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Interpolation of periods for a Hida family of modular forms
Have a look at Ohta's paper "Ordinary p-adic ´etale cohomology groups
attached to towers of elliptic modular curves. II" (Mathematische Annalen); he strengthens there the results from his previous paper to include all congruence classes, using a result of Gross. 


