suppose we have a $p$-stabilized newform $f$ of classical level $\Gamma\_0(p)$; then there is a unique ordinary deformation into a Hida family $\mathbf{f}$ defined over a finite flat extension $R/\Lambda$, and a corresponding representation $\rho: G\_{\mathbb{Q}}\to \text{GL}\_2(\text{Frac}(R))$ on a big vector space $\mathbb{V}$. Under some mild conditions, there exists a free rank-two $R$-lattice $\mathbb{T}\subset \mathbb{V}$.


I believe there should be some kind of autoduality for $\mathbb{V}$, and maybe $\mathbb{T}$ as well, in the form of a perfect pairing
$$
\mathbb{T} \times \mathbb{T} \to R(1)
$$
or similar, which is equivariant. Does anyone have a reference where this kind of thing is spelled out?



Reference for auto-duality of nearly ordinary deformations associated to Hida families
[Section 1.6 in Fukaya-Kato](https://www.math.ucla.edu/%7Esharifi/sharificonj.pdf) spells it out very nicely; the same material appeared earlier in section 4.1 of Ohta's *On the $p$-adic Eichler-Shimura isomorphism for $\Lambda$-adic cusp forms*.


