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Let $K$ be a finite extension of $\mathbb{Q}_p$, and let $\overline{\rho}:\mathrm{Gal}(\overline{K}/K)\rightarrow \mathrm{GL}_2(\mathbb{F})$ be a characteristic $p$ representation. According to a theorem of Kisin, if $a \neq b$ are integers, there exists a $p$-torsionfree universal (framed) deformation ring $R_\overline{\rho}^{a,b}$, parametrizing lifts whose generic fiber is crystalline of Hodge-Tate weights $a$ and $b$. This can be carried out more generally for potentially semistable representations, but one needs to fix an inertial type for this to work.

Does there exist a $p$-torsionfree universal (framed) deformation ring $R_\overline{\rho}^{a,a}$ parametrizing those lifts whose generic fiber has generalized Hodge-Tate weights equal to the multiset $\{a,a\}$? What if we impose some extra (but not too restrictive) conditions? What about the case $K=\mathbb{Q}_p$?

A rigid analytic space parametrizing this condition certainly exists, but I am asking if the stronger condition that the Hodge-Tate weights are both equal to some number can be cut out inside the universal deformation space by some integral algebraic equation.

Thanks!

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