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Let $\bar{\rho}$ be a residual ordinary and locally split modular Galois Representation (into $GL_{2}(\mathbb{F}_p))$ associated to a weight $k$ and level $1$ form. In the sense of Deformation Theory, let $p$ be an unobstructed prime. What is the dimension of the image of the linear map

$ H^{1}(G_{S}, Ad(\bar{\rho})) \rightarrow H^{1}(I_{p},\mathbb{F}_{p}(\omega^{k-1})) $

where $S=\lbrace p,\infty \rbrace$ and $\omega$ is the mod $p$ cyclotomic character.

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  • $\begingroup$ How is that linear map even defined? $\endgroup$ Apr 14, 2011 at 4:50
  • $\begingroup$ The OP hasn't stated their normalizations, but I will guess that they are homological, so that by assumption $\overline{\rho} = 1 \oplus \omega^{k-1}$. Then $Ad(\overline{\rho}) = \omega^{1-k} \oplus 1 \oplus 1 \oplus \omega^{k-1}$, and the map is defined by projection to the final summand. $\endgroup$
    – Emerton
    Apr 14, 2011 at 6:19
  • $\begingroup$ Yes, $\bar{\rho} \mid I_{p} = 1 \oplus \omega^{k-1}$ and also $\text{dim}H^{1}(G_{S},Ad(\bar{\rho})) = 3$ since it is an unobstructed prime. Is the image inside $H^{1}(I_p, F_{p}(\omega^{k-1}))$ one dimensional $\endgroup$ Apr 14, 2011 at 14:10

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