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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?

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    $\begingroup$ My entry point would be Section 7 of Kings, Loeffler, Zerbes, “Rankin-Eisenstein classes and explicit reciprocity laws” (arxiv.org/abs/1503.02888) and references therein, but that may not be optimal. I merely happened to study that paper not too long ago. $\endgroup$
    – Aphelli
    Commented Jan 6, 2023 at 22:27
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    $\begingroup$ theorem 7.2.3(v) is very close to what I want, but it doesn't say anything about the Galois action! $\endgroup$
    – xir
    Commented Jan 7, 2023 at 1:07
  • $\begingroup$ Oops, you’re right (I thought they talked about the Galois action, though…). Maybe in the Ohta paper that they cite, then? $\endgroup$
    – Aphelli
    Commented Jan 7, 2023 at 11:26
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    $\begingroup$ found it in a different ohta paper! $\endgroup$
    – xir
    Commented Jan 7, 2023 at 17:14
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    $\begingroup$ @xir If you put together parts (iii), (iv), and (v) of the theorem from KLZ, then the Galois action drops out; but it isn't literally $R(1)$ that it lands in, it is $R(1 + \mathbf{k})$, where $\mathbf{k}$ is the canonical character $\mathbf{Z}_p^\times \to R^\times$ which specialises to $x \mapsto x^k$ at a weight $k$ point. $\endgroup$
    – David Loeffler
    Commented Jan 9, 2023 at 8:32

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Section 1.6 in Fukaya-Kato 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.

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