What is the status of Ihara's lemma for Shimura curves over totally real fields? In particular, why is it not implicit in the exact sequence of Rajaei, "On the levels of mod $l$ Hilbert modular forms" (Crelle 2001), Theorem 3 (3.18)?

To be a little bit more precise (or at least to give the main idea), let $F$ be a totally real field of degree $d$. Let $M(\mathfrak{M}^+, \mathfrak{M}^{-})$ be the Shimura curve of level $\mathfrak{M}^+ \subset \mathcal{O}_F$ associated to the indefinite quaternion algebra of discriminant $\mathfrak{M}^{-} \subset \mathcal{O}_F$, and $M(\mathfrak{m};\mathfrak{M}^+, \mathfrak{M}^{-})$ the Shimura curve with maximal level at primes dividing $\mathfrak{m} \subset \mathcal{O}_F$. Fix two coprime ideals $\mathfrak{N}^+, \mathfrak{N}^{-} \subset \mathcal{O}_F$ such that $\mathfrak{N}^{-}$ is the squarefree product of a number of primes ideals congruent to $d \mod 2$. Suppose that $\mathfrak{N}^{-}$ has at least one prime divisor $\mathfrak{q}$ say. Fix a prime $v \subset \mathcal{O}_F$ that does not divide $\mathfrak{N}^+\mathfrak{N}^{-}$. Let ${\bf{M}}(\mathfrak{q}; \mathfrak{N}^+, \mathfrak{N}/\mathfrak{q})$ and ${\bf{M}}(v\mathfrak{q}; \mathfrak{N}^+, \mathfrak{N}^{-}/\mathfrak{q})$ be the good reduction integral models over $\mathcal{O}_{(\mathfrak{q})}$ of $\bf{M}(\mathfrak{q}; \mathfrak{N}^+, \mathfrak{N}/\mathfrak{q})$ and $M(v\mathfrak{q}; \mathfrak{N}^+, \mathfrak{N}^{-}/\mathfrak{q})$ respectively. Ihara's lemma for Shimura curves over totally real fields, at least as I understand it, is the assertion that for any non-Eisenstein maximal ideal $\mathfrak{m}$ in the algebra $\mathbb{T}(v\mathfrak{q}; \mathfrak{N}^+, \mathfrak{N}^{-})$ generated by Hecke operators acting on $M(v\mathfrak{q}; \mathfrak{N}^+, \mathfrak{N}^{-}/\mathfrak{q})$, the map $$\begin{align*} H^1({\bf{M}}(\mathfrak{q}; \mathfrak{N}^+, \mathfrak{N}/\mathfrak{q}), \mathcal{F})_{\mathfrak{m}}^{2} & \longrightarrow H^1({\bf{M}}(v\mathfrak{q}; \mathfrak{N}^+, \mathfrak{N}^{-}/\mathfrak{q}), \mathcal{F})_{\mathfrak{m}} \\ (f_1, f_2) & \longmapsto 1_* f_1 +\eta_{v, *} f_2 \end{align*}$$ is injective, where $\eta_v = \left( \begin{array}{cccc} 1 & 0 \\ 0 & \pi_v \end{array}\right)$. Here, $\mathcal{F}$ is the usual sheaf defined by Carayol and Jarvis, and $\pi_v$ is a fixed uniformizer at $v$. Rajaei shows that there is an injection of the associated dual character groups, $$\begin{align*} \widehat{\mathcal{X}}_{\mathfrak{q}}(\mathfrak{q}; \mathfrak{N}^+, \mathfrak{N}/\mathfrak{q}))_{\mathfrak{m}}^{2} &\longrightarrow \widehat{\mathcal{X}}_{\mathfrak{q}}(v\mathfrak{q}; \mathfrak{N}^+, \mathfrak{N}^{-}/\mathfrak{q})_{\mathfrak{m}} \\(f_1, f_2) &\longmapsto 1_* f_1 +\eta_{v, *} f_2 .\end{align*}$$ Sorry if the question is naive! It is related to a previous question I asked here (Soft proof of multiplicity one for character groups of Shimura curves?), to the effect of whether or not proofs of these types of results can be streamlined for the case of parallel weight $2$. In particular, consider the Ribet/Rajaei exact sequence \begin{align*} \mathcal{X}_{v_2}(\mathfrak{N}^+, \mathfrak{p}v_1 v_2 \mathfrak{N}^{-}) \longrightarrow \mathcal{X}_{\mathfrak{p}}(\mathfrak{p}v_2; \mathfrak{N}^+; v_1 \mathfrak{N}^{-}) \longrightarrow \mathcal{X}_{\mathfrak{p}}(\mathfrak{p}; \mathfrak{N}^+, v_1 \mathfrak{N}^{-})^2.\end{align*} Here, in the notations of the previous question, $v_1, v_2, \mathfrak{p} \subset \mathcal{O}_F$ are distinct primes that do not divide the product $\mathfrak{N}^+ \mathfrak{N}^{-}$. Rajaei shows that the localization of the natural map $$\begin{align*} \widehat{\mathcal{X}}_{\mathfrak{p}}(v_2; \mathfrak{N}^+, v_1 \mathfrak{N}^{-})^2 &\longrightarrow \widehat{\mathcal{X}}_{\mathfrak{p}}(\mathfrak{p}v_2; \mathfrak{N}^+, v_1 \mathfrak{N}^{-}) \end{align*}$$ given by $1_* \oplus {\eta_{\mathfrak{p}}}_*$ at any non-Eisenstein maximal ideal of the Hecke algebra $\mathbb{T}(\mathfrak{p}v_2; \mathfrak{N}^+, v_1\mathfrak{N}^{-})$ is injective. On the other hand, given the diagram in the previous question (so beautifully compiled by Dror Speiser), we have identifications $$\begin{align*} \mathcal{X}_{\mathfrak{p}}(\mathfrak{p}v_2; \mathfrak{N}^+, v_1 \mathfrak{N}^{-}) &\cong \operatorname{Div}^0 \left({\bf{M}}(\mathfrak{p}v_2; \mathfrak{N}^+, v_1 \mathfrak{N}^{-})^{ss} \otimes \kappa_{\mathfrak{p}} \right) \\ \mathcal{X}_{\mathfrak{p}}(\mathfrak{p}; \mathfrak{N}^+, v_1 \mathfrak{N}^{-}) &\cong \operatorname{Div}^0 \left({\bf{M}}(\mathfrak{p}; \mathfrak{N}^+, v_1 \mathfrak{N}^{-})^{ss} \otimes \kappa_{\mathfrak{p}} \right) \end{align*}$$ Can we not then just take ${\bf{Z}}$-duals to deduce the result? This is the naive question.

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    $\begingroup$ Ihara's lemma usually refers to this statement over the residual field. It is now known in low weight (though probably not published yet), as I know too well, having considered giving it as a PhD. thesis subject. Presumably M.Emerton will confirm this. $\endgroup$ – Olivier Feb 21 '11 at 6:10

Okay, so the answer to the naive question is most likely no for several reasons. For instance, the identifications explained in the second paragraph show that the injection in Rajaei is really an analogue of Ihara's lemma for totally definite quaternion algebras (via Carayol's description of supersingular points). As well, the cocharacter groups only give information about the p-new automorphic forms. In general, it seems very unlikely that Ihara's lemma for Shimura curves over totally real fields can ever be reduced to the totally definite case, at least via naive methods. As for the status of Ihara's lemma itself, there does appear to be at least one work towards a generalization to totally real fields of the method of Diamond and Taylor, for instance in the preprint of Chuangxun Cheng (Ph.D. student of Emerton) ...


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