# Interpolation of periods for a Hida family of modular forms

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.)

(1) Can the assumption in Ohta's theorem that the weight is not 2 modulo $p-1$ be removed?

(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?

(1) The answer is yes. In fact this was done by Ohta himself essentially simultaneously as the other cases. The relevant publication is here (the proof in the case $k\equiv 2$ is around 569/570).
(2) I am not completely sure I understand the generalization you have in mind. Once you have tensored with the fraction field, the relevant condition becomes hard to state because the notion of interpolation becomes harder to define. At any rate, Ohta's result remain true as a statement about modules over the group algebra of the diamond operators (so the usual $\Lambda$, but not the $\Lambda_{\mathbf{f}}$ of your question). This is corollary (2.3.6) of the aforementioned article.