# Control theory for Kitagawa's $\Lambda$-adic modular symbols

Let $p$ be a prime, $\Gamma=1+p\mathbb Z_p$ and $\Lambda=\mathbb Z_p[[\Gamma]]$ the Iwasawa algebra. Let $\kappa\colon\Gamma\rightarrow\mathbb Z_p^\times$ be the inclusion. For a character $\varepsilon\colon\Gamma\rightarrow\mathbb Z_p^\times$ and $k\in \mathbb N$, let $x_{k,\varepsilon}$ be the morphism $\Lambda\rightarrow\mathbb Z_p$ induced by $\varepsilon\kappa^k$ and let $P_{k,\varepsilon}$ be its kernel.

This question is about Kitagawa's $\Lambda$-adic modular symbols. To recall the most important notation from his article, let $N\in\mathbb{N}$ with $(N,p)=1$, $\mathrm{MS}(\Gamma_1(Np^r),L_{k-2}(\mathbb Z_p))$ the $\mathbb Z_p$-module of classical modular symbols of weight $k$ and level $Np^r$ and $\overline{\mathcal{MS}}({k-2},\mathbb Z_p)$ be the $p$-adic completion of $\varinjlim_r\ \mathrm{MS}(\Gamma_1(Np^r),L_{k-2}(\mathbb Z_p))$. The latter space has an action of the Hida-Hecke-Algebra of level $Np^\infty$, which I denote simply by $\mathcal H$.

Define further $UM(\mathbb Z_p)=\operatorname{Hom}_{\mathbb Z_p}(\overline{\mathcal{MS}}(0,\mathbb Z_p),\mathbb Z_p)$ with the dual Hecke action and $MS^{\mathrm{ord}}(\Lambda)=\operatorname{Hom}_\Lambda(UM^{\mathrm{ord}}(\mathbb Z_p),\Lambda)$ again with the dual Hecke action, following Kitagawa. Fix a character $\varepsilon\colon\Gamma\rightarrow\mathbb Z_p^\times$ with kernel $1+p^r\mathbb Z_p$ and $k\ge2$. Let $F\colon \mathcal H^{\mathrm{ord}}\rightarrow\Lambda$ be a $\Lambda$-linear morphism and $f_{k,\varepsilon}$ its reduction mod $P_{k,\varepsilon}$ which corresponds to a classical cusp form which we denote by the same symbol. I read that the following control statement holds: $$MS^{\mathrm{ord}}(\Lambda)[F]\otimes_\Lambda(\Lambda/P_{k,\varepsilon})\cong \mathrm{MS}^{\mathrm{ord}}(\Gamma_1(Np^r),L_{k-2}(\mathbb Z_p))[f_{k,\varepsilon}].$$ Here $[F]$ resp. $[f_{k,\varepsilon}]$ denotes the subspace where $\mathcal H^{\mathrm{ord}}$ resp. the Hecke algebra of level $Np^r$ and level $k$ act by the eigenvalues given by $F$ resp. $f_{k,\varepsilon}$. I tried to derive this from the statements proved in Kitagawa's article, but I did not succeed. I sketch below my attempt to prove this and my question is: what am I doing wrong?

Start with the exact sequence $0\rightarrow P_{k,\varepsilon}\rightarrow\Lambda\rightarrow\mathbb Z_p\rightarrow0$, where the right map is $x_{k,\varepsilon}$. Consider $\mathbb Z_p$ as a $\Lambda$-algebra via $x_{k,\varepsilon}$ and apply the functor $\operatorname{Hom}_\Lambda(UM^{\mathrm{ord}}(\mathbb Z_p),-)$, which is exact since $UM^{\mathrm{ord}}(\mathbb Z_p)$ is $\Lambda$-free (Prop. 5.7 in Kitagawa). In this way one easily derives isomorphisms $MS^{\mathrm{ord}}(\Lambda)\otimes_\Lambda(\Lambda/P_{k,\varepsilon})\cong\operatorname{Hom}_\Lambda(UM^{\mathrm{ord}}(\mathbb Z_p),\mathbb Z_p)$ and $MS^{\mathrm{ord}}(\Lambda)[F]\otimes_\Lambda(\Lambda/P_{k,\varepsilon})\cong\operatorname{Hom}_\Lambda(UM^{\mathrm{ord}}(\mathbb Z_p),\mathbb Z_p)[F]$.

Next, let $\Gamma$ act on $\overline{\mathcal{MS}}(0,\mathbb Z_p)$ by diamond operators, as in Kitagawa (note that this is not the same as the action through the Hecke algebra $\mathcal H$, which is twisted by a power of $\kappa$). Write $[\varepsilon]$ for the subspace where $\Gamma$ acts by $\varepsilon$. Using the biduality map $\overline{\mathcal{MS}}(0,\mathbb Z_p)\rightarrow\operatorname{Hom}_{\mathbb Z_p}(UM(\mathbb Z_p),\mathbb Z_p)$ one obtains an isomorphism $$\overline{\mathcal{MS}}^{\mathrm{ord}}(0,\mathbb Z_p)[\varepsilon\kappa^{k-2}]\cong\operatorname{Hom}_\Lambda(UM^{\mathrm{ord}}(\mathbb Z_p),\mathbb Z_p)$$ what can be derived from the definitions of the various actions. Further one checks that $\overline{\mathcal{MS}}^{\mathrm{ord}}(0,\mathbb Z_p)[f_{k,\varepsilon}]\subseteq\overline{\mathcal{MS}}^{\mathrm{ord}}(0,\mathbb Z_p)[\varepsilon\kappa^{k-2}]$ (where $[f_{k,\varepsilon}]$ here means the subspace where $\mathcal H$ acts by $x_{k,\varepsilon}\circ F$) and that the restriction of the above isomorphism induces $$\overline{\mathcal{MS}}^{\mathrm{ord}}(0,\mathbb Z_p)[f_{k,\varepsilon}]\cong\operatorname{Hom}_\Lambda(UM^{\mathrm{ord}}(\mathbb Z_p),\mathbb Z_p)[F].$$

On the other hand, starting from finite level, we have $\mathrm{MS}^{\mathrm{ord}}(\Gamma_1(Np^r),L_{k-2}(\mathbb Z_p))[f_{k,\varepsilon}]\subseteq\mathrm{MS}^{\mathrm{ord}}(\Gamma_1(Np^r),L_{k-2}(\mathbb Z_p))[\varepsilon]$ and by Thm. 5.5 in Kitagawa the latter space equals $\overline{\mathcal{MS}}^{\mathrm{ord}}(k-2,\mathbb Z_p)[\varepsilon]$. Now by Thm. 5.3 in Kitagawa, we have an isomorphism $\overline{\mathcal{MS}}^{\mathrm{ord}}(k-2,\mathbb Z_p)\cong\overline{\mathcal{MS}}^{\mathrm{ord}}(0,\mathbb Z_p)(\kappa^{k-2})$, where $(\kappa^{k-2})$ means that the action of $\Gamma$ has been twisted by $\kappa^{k-2}$. This means $\overline{\mathcal{MS}}^{\mathrm{ord}}(k-2,\mathbb Z_p)[\varepsilon]\cong\overline{\mathcal{MS}}^{\mathrm{ord}}(0,\mathbb Z_p)[\varepsilon\kappa^{2-k}]$.

Now $\mathrm{MS}^{\mathrm{ord}}(\Gamma_1(Np^r),L_{k-2}(\mathbb Z_p))[f_{k,\varepsilon}]\subseteq\overline{\mathcal{MS}}^{\mathrm{ord}}(0,\mathbb Z_p)[\varepsilon\kappa^{2-k}]$, but $MS^{\mathrm{ord}}(\Lambda)\otimes_\Lambda(\Lambda/P_{k,\varepsilon})\cong\overline{\mathcal{MS}}^{\mathrm{ord}}(0,\mathbb Z_p)[\varepsilon\kappa^{k-2}]$ and I cannot conclude.

EDIT: From what I wrote above, it follows that $MS^{\mathrm{ord}}(\Lambda)[F]\otimes_\Lambda(\Lambda/P_{k,\varepsilon})\subseteq\overline{\mathcal{MS}}^{\mathrm{ord}}(0,\mathbb Z_p)[\varepsilon\kappa^{k-2}]$, but $\mathrm{MS}^{\mathrm{ord}}(\Gamma_1(Np^r),L_{k-2}(\mathbb Z_p))[f_{k,\varepsilon}]\subseteq\overline{\mathcal{MS}}^{\mathrm{ord}}(0,\mathbb Z_p)[\varepsilon\kappa^{2-k}]$. Note the different exponents of the character $\kappa$ ($k-2$ and $2-k$). These spaces in general intersect trivially (unless $k=2$), so if the statement holds in some situation, I must be mistaken somewhere.